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OPEN ACCESS
Bulletin of Mathematical Sciences
Vol. 10, No. 3 (2020) 2030002 (57 pages)
c
The Author(s)
DOI: 10.1142/S1664360720300029
Smoothness of functions versus smoothness
of approximation processes
Yu. S. Kolomoitsev∗
Universit¨at zu L¨ubeck, Institut f¨ur Mathematik
Ratzeburger Allee 160, 23562 L¨ubeck, Germany
kolomoitsev@math.uni-luebeck.de
S. Yu. Tikhonov
Ce nt re de Recerca M at em`atica, Campus de Bellaterra
Edifici C 08193 Bellaterra, Barcelona, Spain
ICREA, Pg. Llu´ıs Companys 23, 08010 Barcelona
Spain, and Universitat Aut´onoma de Barcelona
stikhonov@crm.cat
Received 15 June 2020
Revised 26 August 2020
Accepted 12 September 2020
Published 28 October 2020
Communicated by Ari Laptev
We provide a comprehensive study of interrelations between different measures of
smoothness of functions on various domains and smoothness properties of approxima-
tion processes. Two general approaches to this problem have been developed: The first
based on geometric properties of Banach spaces and the second on Littlewood–Paley and
H¨ormander-type multiplier theorems. In particular, we obtain new sharp inequalities for
measures of smoothness given by the K-functionals or moduli of smoothness. As exam-
ples of approximation processes we consider best polynomial and spline approximations,
Fourier multiplier operators on Td,Rd,[−1,1], nonlinear wavelet approximation, etc.
Keywords: Measures of smoothness; K-functionals; best approximation; Jackson and
Bernstein inequalities; Littlewood–Paley decomposition; Fourier multipliers.
Mathematics Subject Classification: Primary: 41A65, 41A63, 41A50, 41A17, 42B25;
Secondary: 41A15, 42A45, 41A35, 41A25
∗Corresponding author.
This is an Open Access article published by World Scientific Publishing Company. It is distributed
under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use,
distribution and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
The fundamental problem in approximation theory is to find for a complicated
function fin a quasinormed space Xa close-by, simple approximant Pnfrom a
subset of Xsuch that the error of approximation f−PnXcan be controlled by a
specific majorant. In many cases, this problem is solved completely and necessary
and sufficient conditions are given in terms of smoothness properties of either the
function for approximants Pnof f.
We illustrate this by considering the well-known case of approximation of
periodic functions by trigonometric polynomials on T=[0,2π]. If f∈Lp(T),
1≤p≤∞,and0<α<r, for the best approximant T∗
nand the modulus of
smoothness ωr(f, t)p, the following conditions are equivalent:
(i1)f−T∗
np=O(n−α),
(i2)ωr(f,t)p=O(tα),
(i3)(T∗
n)(r)p=O(nr−α),
see [53, 5; 16, Chap. 7]; for functions on Tdsee [31]. Let us also mention earlier
results by Salem and Zygmund [50], Zamansky [69], and Civin [6]. Similar results
in the case of approximation by algebraic polynomials of functions on [−1,1] can
be found in [20, Chap. 8; 4].
Equivalence (i1)⇔(i2) easily follows from the classical Jackson and Bernstein
approximation theorems, see, e.g. [16, Chap. 7], given by
En(f)pωr(f,1/n)p1
nr
n
k=0
(k+1)
r−1Ek(f)p,1≤p≤∞,
or their sharper versions for 1 <p<∞, see, e.g. [11]
1
nrn
k=0
(k+1)
rτ−1Ek(f)τ
p1
τ
ωr(f,1/n)p1
nrn
k=0
(k+1)
rθ−1Ek(f)θ
p1
θ
,
where En(f)pis the error of the best approximation, τ=max(p, 2) and θ=
min(p, 2).
The equivalence (i2)⇔(i3) follows from the inequalities
n−r(T∗
n)(r)pωr(f,1/n)p
∞
k=n
k−r−1(T∗
k)(r)p,1≤p≤∞.(1.1)
The left-hand side estimate is a corollary of the well-known Nikolskii–Stechkin
inequality T(r)
npnrωr(Tn,1/n)p. The right-hand side estimate was proved
in [70].
Jackson and Bernstein approximation theorems as well as the corresponding
equivalence (i1)⇔(i2) are known to be true in various settings. Surprisingly enough
the results involving the smoothness of approximation processes given in the strong
form, i.e. similar to inequalities (1.1), or, even in the weak form, i.e. similar to
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Smoothness of functions versus smoothness of approximation processes
equivalence (i2)⇔(i3), are much less known in the literature. It is clear that such
results provide additional information on smoothness properties of approximants
and, therefore, they are useful for applications. As an example, we mention that the
smooth function spaces (Lipschitz, Sobolev, Besov) can be characterize in terms of
smoothness of approximation processes.
The main goal of this paper is to present a thoughtful study of interrelations
between smoothness properties of functions on various domains and smoothness
properties of approximation processes. In particular, we extend inequalities (1.1) as
follows: For f∈Lp(T),1<p<∞
∞
k=n+1
2−krτ (T∗
2k)(r)τ
p1
τ
ωrf,2−np∞
k=n+1
2−krθ(T∗
2k)(r)θ
p1
θ
,
where T∗
2kstands for the best approximants, partial sums of the Fourier series, de
la Vall´ee Poussin means, Fej´er means, etc.
In the general form, our main results state that for f∈X
∞
k=n+1
2−kατ P2k(f)τ
Y1
τ
Ω(f,2−nα,X,Y)∞
k=n+1
2−kαθP2k(f)θ
Y1
θ
,
(1.2)
where the parameters τand θare related to geometry of the space X, and, in
particular, for X=Lp,0<p≤∞,aregivenby
τ=max(p, 2),1<p<∞,
∞,otherwise, θ=min(p, 2),p<∞,
1,p=∞.
Here, Yis a smooth function space (Sobolev or Besov spaces), Pn(f) is a suitable
(linear or nonlinear) approximation method, and Ω(f, 2−nα,L
p,Y)issomemeasure
of smoothness related to the spaces Lpand Y. It is worth mentioning that the classi-
cal modulus of smoothness is equivalent to the K-functional for a couple (Lp,Wr
p),
namely, K(f, t;Lp(T),Wr
p(T))pωr(f,t), see, e.g. [16, p. 177]. Therefore, as a
measure of smoothness it is natural to consider the K-functional K(f,2−nα ,L
p,Y)
in the case 1 ≤p≤∞and either an appropriate modulus of smoothness or a
realization of the K-functional for any 0 <p≤∞.
The rest of the paper is organized as follows. In Sec. 2, we consider general
(Banach) spaces and investigate smoothness properties of the best approximants.
Using geometric properties of X, we obtain sharp inequalities (1.2) for appropriate
θand τ. In more detail, if the space Xis θ-uniformly smooth and τ-uniformly
convex, then (1.2) holds.
Section 3 studies the smoothness properties of Fourier means of functions
from Lp,w (D). Our approach is based on Littlewood–Paley-type inequalities and
H¨ormander’s-type multiplier theorems. In particular, inequalities (1.2) are obtained
for a wide class of Fourier multiplier operators, which includes partial sums of
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Yu. S. Kolomoitsev & S. Yu. Tikhonov
Fourier series, de la Val´ee Poussin means, Fej´er means, Riesz means, etc. Sharpness
of the parameters in (1.2) will be discussed in Sec. 9.
In Sec. 4, we deal with general approximation processes {P2n(f)}and abstract
measures of smoothness Ω(f, t)Xin the metric space X. In particular, we treat the
case of X=Lpfor 0 <p<1. We prove that
Ω(P2k(f),2−k)Xk≥n
∞
Ω(f,2−n)X
Ω(P2k(f),2−k)Xk≥n
λ
,
where λis a parameter related to the geometry of X. Let us emphasize that this
result holds under very mild conditions on the approximants P2n(f). Moreover,
these inequalities easily imply the results similar to those given in the equivalence
(i2)⇔(i3).
In Secs. 5–8, we illustrate our main results obtained in Secs. 4–3 by several
important examples. In particular, in Sec. 5, we investigate relationship between
smoothness of periodic functions on Tdand smoothness of the best trigonometric
approximants, various Fourier means, and smoothness of interpolation operators.
Moreover, we consider approximations in Hardy spaces Hp(D), 0 <p≤1, and
smooth (Lipschitz, Sobolev) spaces. Section 6 is devoted to approximation processes
on Rd. In this case, we study smoothness properties of band-limited functions that
approximate functions from Lp(Rd).
In Sec. 7, we deal with functions on Lp,w[−1,1], where wis the Jacobi weight. In
particular, we study smoothness properties of algebraic polynomials and splines of
the best approximation and consider some Fourier means related to Fourier–Jacobi
series.
In Sec. 8, we show that the results of Secs. 4 and 2 can be applied to study
smoothness properties of nonlinear approximation processes. As examples, we treat
nonlinear wavelet approximation and splines with free knots.
Finally, in Sec. 9, we study the optimality of inequalities (1.2), showing that the
parameters τand θcannot be improved in general. Moreover, we define function
classes such that the right-hand side and the left-hand side sums in (1.2) (with
appropriate values of τand θ) are equivalent to the corresponding modulus of
smoothness.
Throughout the paper, we use the notation FG, with F, G ≥0, for the
estimate F≤CG, where Cis a positive constant independent of the essential
var iables in Fand G(usually, f,δ,andn). If FGand GFsimultaneously,
we write FGand say that Fis equivalent to G.
2. K-Functionals and Smoothness of Best Approximants
Let (X, Y ) be a couple of normed function spaces with (semi-)norms ·
Xand
·Y, respectively, and Y⊂X. The Peetre K-functional for this couple is given by
K(f,t;X, Y )=inf{f−gX+tgY:g∈Y}(2.1)
for any f∈Xand t>0.
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Smoothness of functions versus smoothness of approximation processes
Let {Gn}∞
n=1 be a family of subsets of Ysuch that
(i) 0 ∈G1,
(ii) Gn⊂Gn+1,
(iii) Gn=−Gn,
(iv) the closure of {Gn}∞
n=1 in Xis X.
The best approximation of f∈Xby elements from Gnis given by
En(f)X=inf{f−gX:g∈Gn}.
Moreover, we suppose that the family {Gn}is such that Jackson- and Bernstein-
type inequalities are valid. Namely, there are positive constants c1,c2,andαsuch
that for any n∈Nwe have
En(f)X≤c1n−αfY,f∈Y, (2.2)
g1−g2Y≤c2nαg1−g2X,g
1,g
2∈Gn.(2.3)
The latter condition implies that, for every g∈Gn,
gY≤c2nαgX,g∈Gn.(2.4)
Clearly, if Gnis a linear space, then (2.3) and (2.4) are equivalent.
It is also plain to see that the Jackson-type inequality (2.2) implies the direct
approximation theorem given by
En(f)XK(f,n−α;X, Y ),f∈X, n ∈N,(2.5)
where the constant in is independent of fand n.
Our main goal in this section is to obtain inequalities for K(f,t;X, Y )interms
of the best approximation of fby elements from Gn.
In what follows, we denote by Pn(f) an element of the best approximation of
f∈Xby functions from Gn(assuming it exists), i.e.
En(f)X=f−Pn(f)X≤f−gXfor any g∈Gn.
An element of the near best approximation of f∈Xby functions from Gnis
denoted by Qn(f), i.e., there exists a constant c>0 independent of fand nsuch
that
f−Qn(f)X≤cEn(f)X.
One of our main tools is the realization of K-functional given by
R(f,n−α;X, Gn)=inf{f−gX+n−αgY:g∈Gn}.(2.6)
Clearly,
K(f,n−α;X, Y )≤R(f, n−α;X, Gn),f∈X, n ∈N,
but for applications it is important to know when
K(f,n−α;X, Y )R(f, n−α;X, Gn).
The next proposition describes such cases.
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Proposition 2.1. Let inequalities (2.2) and (2.3) hold. Then the following condi-
tions are equivalent:
(i) for every f∈Xand n∈N,
R(f,n−α;X, Gn)K(f,n−α;X, Y ),(2.7)
(ii) for every f∈Xand n∈N,
f−Qn(f)X+n−αQn(f)YK(f,n−α;X, Y ),
where the constant in is independent of fand n.
Even though Proposition 2.1 in this form was not mentioned in [26], its proof
easily follows from [26, Theorem 2.2] taking into account that by (2.5), for the near
best approximation Qn(f), we have
f−Qn(f)XEn(f)XK(f,n−α;X, Y )
for any f∈Xand n∈N.
Remark 2.1. It follows from [26, Theorem 2.2] that under conditions of Proposi-
tion 2.1, assertions (i) and (ii) are equivalent to the following conditions:
(iii) for every f∈Xand n∈N,
Pn(f)YnαK(f,n−α;X, Y ),
(iv) for every g∈Gnand n∈N,
gYnαK(g, n−α;X, Y ).
The next lemma is a crucial result of this section.
Lemma 2.1. Let f∈Xand inequalities (2.2),(2.3),and (2.7) hold.
(A) Suppose that there exist positive constants Aand τsuch that
f−Pn(f)τ
X≤f−gτ
X−Ag−Pn(f)τ
X,(2.8)
for any g∈Gn.Then,for any n∈N,we have
∞
k=n+1
2−kατ P2k(f)τ
Y1
τ
K(f,2−nα;X, Y ),(2.9)
where the constant in is independent of fand n.
(B) Suppose that there exist positive constants Band θsuch that
f−gθ
X≤f−Pn(f)θ
X+Bg−Pn(f)θ
X(2.10)
for all g∈Gn.Then,for any n∈N,we have
K(f,2−nα;X, Y )∞
k=n+1
2−kαθP2k(f)θ
Y1
θ
,(2.11)
where the constant in is independent of fand n.
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Smoothness of functions versus smoothness of approximation processes
Proof. (A) Using the representation
P2k(f)=
k
l=n+1
(P2l(f)−P2l−1(f)) + P2n(f),
we derive
∞
k=n+1
2−kατ P2k(f)τ
Y
∞
k=n+1
2−kατ
k
l=n+1
P2l(f)−P2l−1(f)
τ
Y
+2
−nατ P2n(f)τ
Y
∞
k=n+1
2−kατ k
l=n+1 P2l(f)−P2l−1(f)Yτ
+2
−nατ P2n(f)τ
Y
=: L+2
−nατ P2n(f)τ
Y.(2.12)
Next, by Hardy’s inequality
∞
k=n
2−kα k
s=n
Asq
∞
k=n
2−αkAq
k,A
k≥0,q>0,(2.13)
and Bernstein’s inequality (2.3), we obtain
L
∞
k=n+1
2−kατ P2k(f)−P2k−1(f)τ
Y
∞
k=n+1 P2k(f)−P2k−1(f)τ
X.(2.14)
Using (2.8) with g=P2k−1(f)andn=2
k,wederive
P2k(f)−P2k−1(f)τ
X≤1
A(f−P2k−1(f)τ
X−f−P2k(f)τ
X).(2.15)
Thus, combining (2.14) and (2.15) and taking into account that E2k(f)X=f−
P2k(f)X→0ask→∞,wehave
Lf−P2n(f)τ
X.(2.16)
Finally, combining (2.12) and (2.16) and using Proposition 2.1, we obtain (2.9).
(B) By the definition of the K-functional, we have
K(f,2−nα;X, Y )≤f−P2n+1 (f)X+2
−nαP2n+1 (f)Y.
Thus, to prove (2.11) it is enough to show that
f−P2n+1 (f)θ
X
∞
k=n+1
2−kαθP2k(f)θ
Y.(2.17)
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Since E2k(f)X→0ask→∞,wederive
f−P2n+1 (f)θ
X=
∞
k=n+2 f−P2k−1(f)θ
X−f−P2k(f)θ
X.(2.18)
Next, by the definition of the best approximation
f−P2k−1(f)X≤f−P2k−1(P2k(f))X.
Then, inequality (2.10) with n=2
kand g=P2k−1(P2k(f)) and the Jackson inequal-
ity (2.2) imply
f−P2k−1(f)θ
X−f−P2k(f)θ
X≤f−P2k−1(P2k(f))θ
X−f−P2k(f)θ
X
≤BP2k(f)−P2k−1(P2k(f))θ
X
2−(k−1)αθP2k(f)θ
Y.(2.19)
Thus, (2.18) and (2.19) yield (2.17), completing the proof.
Remark 2.2. (i) It follows from the proof of Lemma 2.1 that conditions (2.8)
and (2.10) can be replaced by the following weaker conditions:
f−P2n(f)τ
X≤f−Pn(f)τ
X−APn(f)−P2n(f)τ
X
and
f−Pn(P2n(f))θ
X≤f−P2n(f)θ
X+BPn(P2n(f)) −P2n(f)θ
X,
respectively.
(ii) Note that by triangle inequality, estimate (2.10) is always valid with θ=B=1.
(iii) Lemma 2.1 remains valid without assumption (2.7) with the realization
R(f,2−nα;X, Y )inplaceoftheK-functional K(f,2−nα ;X, Y )in(2.9)
and (2.11).
In what follows, we need some terminology from the theory of Banach spaces
(see, e.g. [13, Chap. IV]). Let Xbe a Banach space with the norm ·=·
X.
The moduli of convexity and smoothness of Xare defined, respectively, by
δX(ε)=inf1−
x+y
2
:x=y=1andx−y=ε,0≤ε≤2,
and
ρX(t)=sup1
2(x+y+x−y)−1:x=1,y=t,t>0.
Let τ,θ > 1berealnumbers.ThenXis said to be τ-uniformly convex (respec-
tively, θ-uniformly smooth) if there exists a constant c>0 such that δX(ε)≥cετ
(respectively, ρX(t)≤ctθ). Note that by the Day–Nordlander theorem we always
have θ≤2≤τ, see, e.g. [30] or [44].
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Theorem 2.1. Let Gnbe convex,f∈X, and inequalities (2.2),(2.3),and (2.7)
hold.
(A) Suppose Xis τ-uniformly convex for some τ>1.Then,for any n∈N,we
have
∞
k=n+1
2−kατ P2k(f)τ
Y1
τ
K(f,2−nα;X, Y ),(2.20)
where the constant in is independent of fand n.
(B) Suppose Xis θ-uniformly smooth for some θ>1.Then,for any n∈N,we
have
K(f,2−nα;X, Y )∞
k=n+1
2−kαθP2k(f)θ
Y1
θ
,(2.21)
where the constant in is independent of fand n.
Proof. (A) Since Xis τ-uniformly convex, then there exists a constant c>0such
that for all x, y ∈Xand t∈[0,1]
tx +(1−t)yτ≤txτ+(1−t)yτWτ(t)cx−yτ,(2.22)
where ·=·
Xand Wτ(t)=t(1 −t)τ+tτ(1 −t) (see the proof of Theorem 1
in [68], see also [49, 52]). Consider the following Gateaux derivative at yin the
direction x−y:
gτ(y, x −y) = lim
t→+0 y−t(x−y)τ−yτ
t.
Dividing both sides of (2.22) by t∈(0,1) and taking limit as t→+0, we get
gτ(y, x −y)≤xτ−yτ−cx−yτ.
Now, let g∈Gn. Replacing xby f−gand yby f−Pn(f), we have that
gτ(f−Pn(f),P
n(f)−g)≤f−gτ−f−Pn(f)τ−cPn(f)−gτ.
By the Kolmogorov criterion, see, e.g. [51, p. 90], we have gτ(f−Pn(f),P
n(f)−g)=
0, which implies (2.8). Thus, using Lemma 2.1, we get (2.20).
(B) The proof of (2.21) is similar. We only note that by [68, Theorem 1], Xis
θ-uniformly smooth if and only if there exists a constant d>0 such that
tx +(1−t)yθ≥txθ+(1−t)yθWθ(t)dx−yθ.(2.23)
Then, as above, we derive
gτ(f−Pn(f),P
n(f)−g)≥f−gθ−f−Pn(f)θ−cPn(f)−gθ
and apply the Kolmogorov criterion. Lemma 2.1 completes the proof.
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Let us give two important examples of Banach space Xto illustrate Theorem 2.1,
namely, Lebesgue and Orlicz spaces.
Proposition 2.2 (see [39, p. 63]). Let Xbe an abstract Lpspace with 1<p<∞,
i.e. let Xbe a Banach lattice for which
x+yp=xp+yp,
whenever x, y ∈Xand min(x, y )=0. Then there exists a constant c>0such that
δX(ε)≥cεmax(2,p)for all 0≤ε≤2and ρX(t)≤ctmin(2,p)for all t>0.
Making use of Theorem 2.1 and Proposition 2.2, we obtain the following result.
Theorem 2.2. Let inequalities (2.2),(2.3),and (2.7) be valid for X=Lp,1<
p<∞,and let Gnbe con vex. Then,for any f∈Lpand n∈N,we have
∞
k=n+1
2−kατ P2k(f)τ
Y1
τ
K(f,2−nα;Lp,Y),τ=max(2,p),
and
K(f,2−nα;Lp,Y)∞
k=n+1
2−kαθP2k(f)θ
Y1
θ
,θ=min(2,p),
where the constants in are independent of fand n.
In Sec. 9, we will see that the parameters τand θin Theorem 2.2 are optimal.
Corollary 2.1. Let inequalities (2.2),(2.3),and (2.7) be valid f or X=Lp,1<
p<∞,and let Gnbe con vex. Then,for any f∈Lp,the following assertions are
equ iva len t :
(i) for any n∈N
K(f,2−nα;Lp,Y)2−nαθ P2n(f)Y,
where the constants in are independent of fand n,
(ii) for any n∈N
∞
k=n
2−kαθP2k(f)Y2−nαθ P2n(f)Y,
where the constant in is independent of fand n.
Proof. The proof easily follows from Theorem 2.2 and (4.14).
Finally, we consider Orlicz spaces. Recall that the Orlicz function M(t)on[0,∞)
is an increasing convex function satisfying M(0) = 0. We assume that Msatisfies
Δ2-condition, that is, M(2t)≤cM (t) for all t>0. The Orlicz class of functions
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X=XMon some domain Dwith a positive measure dμ(x) is the class of functions
f,forwhich
D
M(|f(x)|)dμ(x)<∞,(2.24)
and the (Luxemburg) norm is
fXM=infσ>0: D
M(|f(x)|/σ)dμ(x)≤1.(2.25)
Proposition 2.3. (A) Suppose that M(u)is an Orlicz function such that M(u1/τ )
is concave for some τ, 2≤τ<∞,and M(lt)≤1
2M(t)for some l<1.Then
thereexistsanOrliczfunctionN(u)such that C−1N(u)≤M(u)≤CN(u)and
δXN(ε)≥cετwith the norm of the space XNgiven by
fXN=infσ>0: D
N(|f(x)|/σ)dμ(x)≤1.(2.26)
(B) Suppose that M(u)is an Orlicz function such that M(u1/θ )is convex for
some θ, 1<θ≤2. Then there exists an Orlicz function N(u)such that
C−1N(u)≤M(u)≤CN(u)and ρXN(t)≤ctθwith the norm of the space XN
given by (2.26).
Proof. The proof of (B) can be found in [19, Lemma 2.2]. Assertion (A) can be
proved similarly employing [43, Theorem 1].
Using Theorem 2.1 and Proposition 2.3, we obtain the following result.
Theorem 2.3. Let inequalities (2.2),(2.3),and (2.7) be valid f or th e Orlicz space
X=XMdefined by (2.24) and (2.25),and let Gnbe convex.
(A) Suppose that the function Mand the parameter τare the same as in Proposi-
tion 2.3(A).Then,for any f∈Xand n∈N,we have
∞
k=n+1
2−kατ P2k(f)τ
Y1
τ
K(f,2−nα;X, Y ),
where the constant in is independent of fand n.
(B) Suppose that the function Mand the parameter θare the same as in Proposi-
tion 2.3(B).Then,for any f∈Xand n∈N,we have
K(f,2−nα;X, Y )∞
k=n+1
2−kαθP2k(f)θ
Y1
θ
,
where the constant in is independent of fand n.
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3. Smoothness of Fourier Multiplier Operators
3.1. Realization and Littlewood–Paley-type inequality
First, we introduce basic notations and collect auxiliary results. We follow the
discussion in the paper [11].
Let Lp,w(D)beaweightedLpspace with the norm
fLp,w(D)=fp,w =D|f|pw1
p
.
We assume that Q(D) is a self-adjoint operator in L2,w(D), that is, Q(D)f,g=
f,Q(D)gwhenever Q(D)f,Q(D)g∈L2,w(D),where, as usual, f, g=Dfgw.
We further assume that the eigenvalues (−1)jλk,jfixed, of Q(D)satisfy0≤λ0<
λk<λ
k+1,G
k={ϕ:Q(D)ϕ=λkϕ}is finite-dimensional, Gk⊂Lp,w(D)for
1≤p≤∞,andspan∪kGkis dense in Lp,w (D)for1≤p<∞.Examplesofsuch
operators and matching spaces are: −d
dx 2for Lp(T); −d
dx (1−x2)d
dx for Lp[−1,1];
−Δ+|x|2, where Δ is the Laplacian for Lp(Rd); and −w−1
α,β d
dx wαβ (1 −x2)d
dx for
Lp,wα,β [−1,1], where wα,β (x)=(1−x)α(1 + x)βwith α, β > −1.
We define
Akf=
dk
=1 f,ψk,ψk,,
where dkis the dimension of Gkand {ψk,}an orthonormal basis of Gkin L2,w(D).
For f∈Lp,w(D),f∼∞
k=0 Akf, we define Q(D)γby
Q(D)γf∼
k
λγ
kAkf
and we say that Q(D)γf∈Lp,w(D)ifthereexistsg∈Lp,w (D) such that λγ
kAkf=
Akg.
In what follows, we suppose that λkkσfor some positive σ>0. Note that
in the example above σ= 2 except for the eigenvalues of −Δ+|x|2where σ=1
(see [17]).
As usual, we define the K-functional Kγf,Q(D),t
σγ p,w by
Kγf,Q(D),t
σγp,w := inf
Q(D)γg∈Lp,w(D){f−gLp,w(D)+tσγ Q(D)γgLp,w (D)}.
(3.1)
In this section, we consider approximation processes, which are defined by means
of the Fourier multiplier operator Tμgiven by
Tμf∼
∞
k=0
μkAkffor f∼
∞
k=0
Akf.
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We will use the following assumption related to a H¨ormander–Mikhlin-type
theorem.
Assumption 3.1. Fo r some 0≥0,the condition
|Δμk|≤A(k+1)
−for 0≤≤0,(3.2)
where
Δ0μk=μk,Δμk=μk+1 −μkand Δμk=Δ(Δ
−1μk),
implies
TμfLp,w(D)≤CA, p, w, {Gk}fLp,w(D),1<p<∞.
It is clear that under Assumption 3.1, the de la Vall´ee Poussin-type operator
ηNf:=
∞
k=0
ηk
NAk,f∼
∞
k=0
Akf,
satisfies
ηNfp,w ≤Afp,w.(3.3)
Here and in what follows, we assume that
η(ξ)∈C∞[0,∞),η(ξ)=1,ξ≤1/2,
0,ξ≥1.
Moreover, the following realization result (see [17, Theorem 7.1]) holds:
Kγf,Q(D),λ
−γ
Np,w f−ηNfp,w +λ−γ
NQ(D)γηNfp,w,(3.4)
where the constants in are independent of fand N.
Denote
θ0(f):=η1fand θj(f):=η2jf−η2j−1ffor j>0.(3.5)
The following Littlewood–Paley-type theorem plays a crucial role in our further
study.
Theorem 3.1 (see [10, Theorem 2.1; 11, Theorem 3.1]). Let f∈Lp,w (D),
1<p<∞,and Assumption 3.1 be satisfied,then
⎧
⎨
⎩
∞
j=0 θj(f)2⎫
⎬
⎭
1/2
Lp,w(D)
fLp,w (D).
If in addition,γ>0,then
⎧
⎨
⎩
∞
j=1 2jγσθj(f)2⎫
⎬
⎭
1/2
Lp,w(D)
Q(D)γfLp,w(D).(3.6)
In the above relations,the constants in are independent of fand n.
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3.2. Smoothness of the de la Vall´ee Poussin means in Lp,w
Theorem 3.2. Let f∈Lp,w(D),1<p<∞,γ>0,τ=max(2,p),θ=min(2,p),
n∈N,and Assumption 3.1 hold. Then
∞
k=n+1
2−σγτ kQ(D)γη2kfτ
p,w1
τ
Kγ(f,Q(D),2−nγσ )p,w (3.7)
and
Kγ(f,Q(D),2−nγσ )p,w ∞
k=n+1
2−σγθkQ(D)γη2kfθ
p,w1
θ
,(3.8)
where the constants in are independent of fand n.
Proof. Denote α=σγ and
Iτ=
∞
k=n+1
2−ατ kQ(D)γη2kfτ
p,w.
Then
Iτ
∞
k=n+1
2−ατ kQ(D)γ(η2kf−η2nf)τ
p,w +2
−nατ Q(D)γη2nfτ
p,w
=J+2
−nατ Q(D)γη2nfτ
p,w.(3.9)
By (3.6), we have
J
∞
k=n+1
2−kατ ⎧
⎪
⎨
⎪
⎩D⎛
⎝
∞
j=1
22αj (θj(η2kf−η2nf))2⎞
⎠
p
2
w⎫
⎪
⎬
⎪
⎭
τ
p
=
∞
k=n+1
2−kατ ⎧
⎪
⎨
⎪
⎩D⎛
⎝
k+1
j=n
22αj (θj(η2kf−η2nf))2⎞
⎠
p
2
w⎫
⎪
⎬
⎪
⎭
τ
p
∞
k=n+1
2−kατ 2αnθn(η2kf−η2nf)p,w +2
α(n+1)θn+1 (η2kf−η2nf)p,wτ
+
∞
k=n+1
2−kατ ⎧
⎪
⎨
⎪
⎩D⎛
⎝
k−1
j=n+2
22jαθj(f)2⎞
⎠
p
2
w⎫
⎪
⎬
⎪
⎭
τ
p
+
∞
k=n+1
2−kατ 2kαθk(η2kf−η2nf)p,w +2
(k+1)αθk+1(η2kf−η2nf)p,w τ
=J1+J2+J3.
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Let us estimate the first sum J1.By(3.3),wehave
θj(η2kf−η2nf)p,w ≤2Aη2kf−η2nfp,w
≤2A(f−η2nfp,w +f−η2kfp,w).
In light of the fact that
η2k(η2nf)=η2nffor k≥n+1,
we derive
f−η2kfp,w =f−η2nf+η2k(η2nf−f)p,w
≤(1 + A)f−η2nfp,w.(3.10)
Therefore,
θj(ηkf−ηnf)p,w f−η2nfp,w
and we get
J1f−η2nfp,w.
Regarding J2,wenotethat
θj(f)=θj(f−η2nf)forj≥n+2,
and, therefore,
J2=
∞
k=n+1
2−kατ ⎧
⎪
⎨
⎪
⎩D⎛
⎝
k−1
j=n+2
22jαθj(f−η2nf)2⎞
⎠
p
2
w⎫
⎪
⎬
⎪
⎭
τ
p
.
Dealing with J3,weobservethatθk(η2nf)=η2n(θk(f)). Then
θk(η2kf−η2nf)p,w ≤θk(η2kf−f)p,w +θk(η2nf−f)p,w
=η2k(θk(f)) −θk(f)p,w +θk(η2nf−f)p,w
θk(η2nf−f)p,w,
where in the last estimate we used (3.10) with θk(f)inplaceoff.
Combining the above inequalities, we obtain that
J
∞
k=n+1
2−kατ ⎧
⎪
⎨
⎪
⎩D⎛
⎝
k+1
j=n
22jα (θj(f−η2nf))2⎞
⎠
p
2
w⎫
⎪
⎬
⎪
⎭
τ
p
+f−η2nfτ
p,w.
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Next, using Minkowski’s inequality with τ
p≥1, Hardy’s inequality (2.13), the
inequality {ak}τ≤{ak}2,andTheorem3.1,weget
∞
k=n+1
2−kατ ⎧
⎪
⎨
⎪
⎩D⎛
⎝
k+1
j=n
22jα (θj(f−η2nf))2⎞
⎠
p
2
w⎫
⎪
⎬
⎪
⎭
τ
p
⎧
⎪
⎪
⎨
⎪
⎪
⎩D⎡
⎢
⎣
∞
k=n+1
2−kατ ⎛
⎝
k+1
j=n
22jα(θj(f−η2nf))2⎞
⎠
τ
2⎤
⎥
⎦
p
τ
w⎫
⎪
⎪
⎬
⎪
⎪
⎭
τ
p
⎧
⎪
⎨
⎪
⎩D⎡
⎣
∞
j=n|θj(f−η2nf)|τ⎤
⎦
p
τ
w⎫
⎪
⎬
⎪
⎭
τ
p
⎧
⎪
⎨
⎪
⎩D⎡
⎣
∞
j=n|θj(f−η2nf)|2⎤
⎦
p
2
w⎫
⎪
⎬
⎪
⎭
τ
p
f−η2nfτ
p,w.
Therefore,
Jf−η2nfτ
p,w.(3.11)
In light of (3.4), estimates (3.9) and (3.11) imply
Iτf−η2nfτ
p,w +2
−nατ Q(D)γη2nfτ
p,w
Kγ(f,Q(D),2−nγσ )τ
p,w,
which proves (3.7).
Let us prove (3.8). By (3.4), we have
Kγ(f,Q(D),2−nγσ )θ
p,w f−η2nfθ
p,w +2
−nαθQ(D)γη2nfθ
p,w.(3.12)
By Theorem 3.1, taking into account that
(θj(f−η2nf))2≤4(θj(f))2+4(θj(η2nf))2,
θj(η2nf)=0 forj≥n+2,
θj(η2nf)p,w ≤Aθj(f)p,w,
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and {ak}2≤{ak}θ,wederive
f−η2nfθ
p,w ⎛
⎜
⎝D⎡
⎣
∞
j=n
θj(f−η2nf)2⎤
⎦
p
2
w⎞
⎟
⎠
θ
p
⎛
⎜
⎝D⎡
⎣
∞
j=n
θj(f)2⎤
⎦
p
2
w⎞
⎟
⎠
θ
p
⎛
⎜
⎝D⎡
⎣
∞
j=n|θj(f)|θ⎤
⎦
p
θ
w⎞
⎟
⎠
θ
p
=⎛
⎜
⎝D⎡
⎣
∞
j=n
2−jαθ θj(f)222jαθ
2⎤
⎦
p
2
w⎞
⎟
⎠
θ
p
⎛
⎝D⎡
⎣
∞
j=n
2−jαθ j
k=n
θk(f)222kα
+θj+1(η2j+1 f)222(j+1)α+θj+2 (η2j+1 f)222(j+2)αθ
2⎤
⎦
p
θ
w⎞
⎟
⎠
θ
p
=⎛
⎜
⎝D⎡
⎣
∞
j=n
2−jαθ j+2
k=n
θk(η2j+1 f)222kαθ
2⎤
⎦
p
θ
w⎞
⎟
⎠
θ
p
.
Next, Minkowski’s inequality with p
θ≥1 and Theorem 3.1 (see (3.6)), yield
f−η2nfθ
p,w
∞
j=n
2−jαθ ⎛
⎝D%j+2
k=n
θk(η2j+1 f)222kα&p
2
w⎞
⎠
θ
p
∞
j=n
2−jαθQ(D)γη2j+1 fθ
p,w
∞
j=n
2−jαθQ(D)γη2jfθ
p,w.
(3.13)
Finally, combining (3.12) and (3.13), we derive (3.8).
Corollary 3.1. Under the conditions of Theorem 3.2, we have
∞
k=n+1
k−σγτ −1Q(D)γηkfτ
p,w1
τ
Kγ(f,Q(D),n
−γσ)p,w
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and
Kγ(f,Q(D),n
−γσ)p,w ∞
k=n+1
k−σγθ−1Q(D)γηkfθ
p,w1
θ
,
where the constants in are independent of fand n.
Proof. The proof easily follows from inequalities (3.7) and (3.8) and the fact that
Q(D)γημfp,w Q(D)γηνfp,w,ν≥2μ.
The latter holds in light of boundedness of the de la Vall´ee Poussin-type operator
in Lp,w given by (3.3) and the fact that ημ(ηνf)=ημffor ν≥2μ.Wealsotake
into account that Kγ(f, Q(D),2t)p,w Kγ(f,Q(D),t)p,w for any t>0.
3.3. General Fourier multiplier operators
In this subsection, we extend Theorem 3.2 considering general Fourier multiplier
operators given by
Ψnf∼
∞
k=0
ψk
nAkf,
where a function ψ:[0,∞)→Ris such that supp ψ⊂[0,1). Together with the
operator Ψn, additionally assuming that ψ(x)=0forallx∈[0,2−m]forsome
m∈Z+, we will also use the operator
'
Ψn∼
∞
k=0 '
ψk
nAkf, '
ψ(ξ)= η(ξ)
ψ(2−mξ),
which plays a role of the inverse operator to Ψn.
Theorem 3.3. Suppose that the conditions of Theorem 3.2 are satisfied.
(A) Let the operators Ψ2nbe such that,for any f∈Lp,w (D)and n∈N,
Ψ2nfp,w ≤C(ψ, p, w)fp,w.(3.14)
Then
∞
k=n+1
2−σγτ kQ(D)γΨ2kfτ
p,w1
τ
Kγ(f,Q(D),2−nγσ )p,w ,(3.15)
where the constant in is independent of fand n.
(B) Suppose that there exists m∈Nsuch that ψ(x)=0for all x∈[0,2−m]and
the operators '
Ψ2nare such that,for any f∈Lp,w(D)and n∈N,
'
Ψ2nfp,w ≤C(ψ, p, w)fp,w.(3.16)
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Then
Kγ(f,Q(D),2−nγσ )p,w ∞
k=n+1
2−σγθkQ(D)γΨ2kfθ
p,w1
θ
,(3.17)
where the constant in is independent of fand n.
Proof. To prove inequality (3.15), it is enough to note that by (3.14) one has
Q(D)γΨ2nfp,w =Q(D)γΨ2n(η2n+1f)p,w ≤CQ(D)γη2n+1 fp,w .
Thus, (3.7) clearly implies (3.15).
To show (3.17), we note that by (3.16), we have
n
k=0
η(2−nk)Ak(f)
p,w
=
n
k=0
η(2−nk)ψ(2−n−mk)(ψ(2−n−mk))−1Ak(f)
p,w
≤C
n+m
k=0
ψ(2−n−mk)Ak(f)
p,w
,
which gives
Q(D)γη2nfp,w Q(D)γΨ2n+mfp,w.(3.18)
This and (3.8) imply
Kγ(f,Q(D),2−nγσ )p,w ∞
k=n+1
2−σγθkQ(D)γΨ2k+mfθ
p,w1
θ
∞
k=n+1
2−σγθkQ(D)γΨ2kfθ
p,w1
θ
,
completing the proof.
Remark 3.1. (i) By Assumption 3.1, condition (3.14) can be replaced by the
condition that the sequence {ψ(k2−n)}k∈Z+satisfies (3.2). Similarly, condi-
tion (3.16) can be replaced by the condition that the sequence {'
ψ(k2−n)}k∈Z+
satisfies (3.2).
(ii) If ψ∈Cr([0,∞), then both sequences {ψ(k2−n)}k∈Z+and {'
ψ(k2−n)}k∈Z+
satisfy (3.2) with 0=r.
(iii) Inequalities (3.15) and (3.17) can be written similarly to those in Corollary 3.1.
Example. Many classical Fourier means are covered by Theorem 3.3. In particular,
these cases include the following operators Ψnf∼n
k=0 ψk
nAkf:
(1) Partial sums of Fourier series, the case ψ(x)=χ[0,1](x);
(2) Fej´er means that are generated by the function ψ(x)=(1−x)+
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(3) More generally, Riesz means for which ψ(x)=(1−xα)δ
+,α, δ > 0;
(4) Rogosinskii means that are generated by
ψ(x)=⎧
⎪
⎨
⎪
⎩
cos (πx
2),0≤x≤1,
0,x>1.
(5) Jackson means, the case ψ(x)= 3
2(1 −|x|)+∗(1 −|x|)+.
The precise formulation of the corresponding results in the periodic case will be
given in Corollary 5.1.
4. General Approximation Processes and Measures of Smoothness
For a fi x e d po s it i v e λ, we consider a metric space (X, ρ) with the metric ρ:X×X→
R+defined by
ρ(f,g)=f−gλ
X,
where the functional ·
X:X→ R+is such that for all f, g ∈Xthe following
properties hold:
(i) fX=0ifandonlyiff=0,
(ii) −fX=fX,
(iii) f+gλ
X≤fλ
X+gλ
X.
Note that the metric ·
X=ρ(f,0) is not a norm in general since the homo-
geneity property is not assumed. A typical example of ·
Xwith λ=1isgiven
by fX=Aϕ(|f(t)|)dμ, where ϕis a positive continuous function such that
ϕ(0) = 0 and ϕ(x+y)≤ϕ(x)+ϕ(y) for all x, y ∈R+. Other examples of ·
X
concerns the standard (quasi-)norm defined in the Lorentz space Lp,q, the Orlicz
spaces XMgiven in Sec. 2, the Wiener-type spaces Ap, and related spaces.
Let us consider the following functional, which to some extend, plays a role of
a measure of smoothness (abstract modulus of smoothness)
Ω(f,δ)X:X×(0,∞)→ R+,
which satisfies the following conditions: For any f,g ∈Xand δ>0,
Ω(f,δ)X→0asδ→+0,(4.1)
Ω(f,δ)X≤C1fX,(4.2)
Ω(f+g, δ)X≤C2(Ω(f,δ)X+Ω(g, δ)X),(4.3)
Ω(f,2δ)X≤C3Ω(f, δ)X,(4.4)
where Cj=Cj(X, λ), j=1,2,3. A typical example is the modulus of smoothness
defined by
Ω(f,δ)X=sup
|h|≤δΔr
hfX,
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where Δ1
hf(x)=f(x+h)−f(x), Δr
h=Δ
1
hΔr−1
h(here, we suppose that Δr
hf∈X).
Other examples of Ω(f,δ)Xare given by the K-functionals or realizations mentioned
in the previous sections as well as by any modulus of smoothness, which will be
introduced in Secs. 5–7.
As an approximation tool, we consider the family of operators Pn:X→ X,
n∈N, such that the following two properties hold: For any f∈Xand n∈N,
f−Pn(f)X≤f−Pn(P2n(f))X,(4.5)
f−Pn(f)X≤C4Ωf,n−1X,(4.6)
where C4=C4(X, λ).
Inequality (4.5) trivially holds when Pn(f) is a best approximant to fin Xor
Pn(f) is such that Pn(P2n(f)) = Pn(f), for example, take a de la Vall´ee Poussin-
type operator or a projection operator. The second inequality is the Jackson-type
theorem.
Theorem 4.1. Let f∈Xand n∈N.Then
Ω(P2n(f),2−n)XΩ(f,2−n)X∞
k=n+1
Ω(P2k(f),2−k)λ
X1
λ
,(4.7)
where the left-hand side inequality holds if we assume only (4.2),(4.3),and (4.6).
Here,the constants in are independent of fand n.
Note that in the case of the Banach space X, a similar result for K-functionals
and holomorphic semi-groups was obtained in [3, Lemmas 3.5.4 and 3.5.5].
Proof of Theorem 4.1. By (4.3),
Ω(P2n(f),2−n)XΩ(P2n(f)−f,2−n)X+Ω(f, 2−n)X,
and the left-hand side estimate in (4.11) follows from (4.2) and (4.6).
Let us prove the right-hand side inequality. Denote
I2n:= P2n+1 (f)−P2n(P2n+1(f))X.
Then by (4.6) and (4.4), we have
I2nΩ(P2n+1(f),2−n)XΩ(P2n+1 (f),2−n−1)X.(4.8)
At the same time, by (4.5) we get
Iλ
2n=P2n+1 (f)−f+f−P2n(P2n+1(f))λ
X
≥f−P2n(P2n+1 (f))λ
X−f−P2n+1 (f)λ
X
≥f−P2n(f)λ
X−f−P2n+1 (f)λ
X
=: Eλ
2n−Eλ
2n+1 .(4.9)
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By (4.6) and (4.1), E2k→0ask→∞. Thus, (4.8) and (4.9) imply
Eλ
2n=
∞
k=nEλ
2k−Eλ
2k+1 ≤
∞
k=n
Iλ
2k
∞
k=n
Ω(P2k+1 (f),2−k−1)λ
X.(4.10)
Then, using properties of the modulus of smoothness, namely (4.4), (4.3), and (4.2),
we obtain
Ω(f,2−n)λ
XΩ(f,2−n−1)λ
X
Ω(f−P2n+1 (f),2−n−1)λ
X+Ω(P2n+1 (f),2−n−1)λ
X
f−P2n+1 (f)λ
X+Ω(P2n+1 (f),2−n−1)λ
X
=Eλ
2n+1 +Ω(P2n+1 (f),2−n−1)λ
X.
Finally, taking into account (4.10),
Ω(f,2−n)λ
X
∞
k=n
Ω(P2k+1 (f),2−k−1)λ
X+Ω(P2n+1 (f),2−n−1)λ
X
∞
k=n
Ω(P2k+1 (f),2−k−1)λ
X,
which is the right-hand side inequality of (4.11).
Remark 4.1. Under the conditions of Theorem 4.1, we have
Ω(f,n−1)X∞
k=1
Ω(P2kn(f),2−kn−1)λ
X1
λ
,(4.11)
where the constant in is independent of fand n.
This inequality can be obtained by using a slight modification of the proof of
Theorem 4.1. See also the proof in [28, Lemma 8]. Similar assertions are also valid
for Theorems 2.1–2.3, 3.2, 3.3 as well as for the corresponding examples in Secs. 5–8.
As a simple corollary of Theorem 4.1 and Remark 4.1, we have the following ver-
sion of Jackson’s inequality written in terms of measure of smoothness of P2kn(f).
Corollary 4.1. Let f∈Xand n∈N.Then
f−Pn(f)X∞
k=1
Ω(P2kn(f),2−kn−1)λ
X1
λ
,
where the constant in is independent of fand n.
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Remark 4.2. If we assume the more general condition than (4.6), namely,
f−Pn(f)X≤C4ξ(n)Ω f,n−1X,
where C4=C4(X, λ)andξis a positive non-decreasing function on [1,∞),then
repeating the proof of Theorem 4.1 gives the following estimates:
ξ−1(2n)Ω(P2n(f),2−n)XΩ(f,2−n)X∞
k=n+1
ξλ(2k)Ω(P2k(f),2−k)λ
X1
λ
(4.12)
and
f−P2n(f)X∞
k=n+1
ξλ(2k)Ω(P2k(f),2−k)λ
X1
λ
.
A typical example when Remark 4.2 can be applied is considering the partial sums of
Fourier series Pn(f)=Sn(f)inthecaseX=Lp(T), p=1,∞,andξ(t) = log(t+1);
for details see Corollary 5.2.
In what follows, we say that ω:R+→R+is the modulus a continuity if ωis
a positive non-decreasing function, ω(0) = 0, and ω(x+y)≤ω(x)+ω(y) for any
x, y ∈R+.
Corollary 4.2. For any modulus of continuity ωsuch that
∞
k=n
ω(2−k)ω(2−n),(4.13)
the following assertions are equivalent:
(1) Ω(P2n(f),2−n)Xω(2−n),
(2) Ω(f, 2−n)Xω(2−n).
Proof. The proof follows from (4.11) and the simple fact that (4.13) is equivalent to
∞
k=n
ω(2−k)λ1
λ
ω(2−n) for any λ>0,(4.14)
see, e.g. [59].
For a given modulus of continuity ω, we define the function class
Ξω={f∈X:Ω(f, δ)Xω(δ),δ→0}.
The next corollary provides sharpness of Theorem 4.1.
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Corollary 4.3. Let f∈Ξωand ωsatisfy (4.13).Then,for large enough n∈N,
Ω(f,2−n)XΩ(P2n(f),2−n)X∞
k=n+1
Ω(P2k(f),2−k)λ
X1
λ
,(4.15)
where the constants in are independent of fand n.
Proof. First, we prove that
Ω(f,2−n)XΩ(P2n(f),2−n)X.(4.16)
The part in (4.16) is given by (4.11). To show the part , we note that by (4.13)
and monotonicity of ω, for any m<n,wehave
ω(2−n+m)
∞
k=n−m
ω(2−k)
n
k=n−m
ω(2−k)(m+1)ω(2−n).
Then, taking onto account (4.2)–(4.4), and (4.6) and choosing large enough m∈N,
we derive
Ω(P2n(f),2−n)λ
X≥C−mλ
3Ω(P2n(f),2−n+m)λ
X
≥C−mλ
3C−λ
2Ω(f,2−n+m)λ
X−Ω(f−P2n(f),2−n−m)λ
X
≥C−mλ
3C−λ
2Ω(f,2−n+m)λ
X−Cλ
1f−P2n(f)λ
X
≥C−mλ
3C−λ
2Ω(f,2−n+m)λ
X−(C1C4)λΩ(f,2−n)λ
X
≥C−mλ
3cω(2−n+m)λ−cω(2−n)λ
≥C−mλ
3c(m+1)
λ−cω(2−n)λ
ω(2−n)λΩ(f,2−n)λ
X.
To prove the second equivalence in (4.15), we note the part follows from the
right-hand side inequality of (4.11) and (4.16) while the part follows from (4.14),
the left-hand side inequality in (4.11), and (4.16),
∞
k=n+1
Ω(P2k(f),2−k)λ
X1
λ
∞
k=n
ω(2−k)λ1
λ
ω(2−n)Ω(P2n(f),2−n)X.
Remark 4.3. Corollaries 4.2 and 4.3 imply that if ω(δ)=δα,α>0, then, for any
f∈Xand n∈N,wehave
Ω(f,2−n)Xω(2−n) if and only if Ω(P2n(f),2−n)Xω(2−n).
If, in addition, f∈Ξω,then
Ω(f,2−n)XΩ(P2n(f),2−n)Xω(2−n).
The results of Remark 4.3 can be extended to Besov-type spaces.
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For a given modulus of smoothness Ω, s>0, and 0 <q≤∞, we define the
Besov-type space as follows:
Bs
X,q =f∈X:|f|Bs
X,q =1
0t−sΩ(f,t)X)qdt
t1
q
<∞*(4.17)
with the usual modification in the case q=∞.
We have the following characterization of Bs
X,q.
Corollary 4.4. Let s>0and 0<q≤∞,we have
|f|Bs
X,q ∞
k=1
2sqk ΩP2k(f),2−kq
X1
q
,
where the constants in are independent of f.
Proof. The proof easily follows from Theorem 4.1 and the Hardy-type inequality
∞
ν=n
2νs ∞
k=ν
Akq
∞
ν=n
2νsAq
ν,
where Aν≥0ands, q > 0.
5. Smoothness of Approximation Processes on Td
5.1. Smoothness of best approximants
In this subsection, we give analogues of Theorems 4.1 and 2.2 for best trigonometric
approximants in Lp(Td) spaces. We recall some basic notations. Denote the set of
all trigonometric polynomials of degree at most nby
Tn=span{ei(k,x):|k|≤n},
where |k|=(k2
1+···+k2
d)1/2.The best approximation by trigonometric polynomials
is given by
En(f)Lp(Td)=inf{f−ϕLp(Td):ϕ∈T
n}.
As above, by Pn(f) we denote the best approximant of a function fin Lp(Td),
that is,
f−Pn(f)Lp(Td)=En(f)Lp(Td),
where Pn(f)∈T
n.
In what follows, we will use the well-known Jackson-type inequality, see, e.g.
[62, 56]:
En(f)Lp(Td)≤Cωrf, 1
nLp(Td)
,f∈Lp(Td),0<p≤∞,r∈N,(5.1)
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where ωr(f,h)pis the classical modulus of smoothness,
ωr(f,δ)p=sup
|h|<δ Δr
hfLp(Td),
Δhf(x)=f(x+h)−f(x),Δr
h=Δ
hΔr−1
h,h∈Rd,d≥1,
and C=C(r, p, d).
We will also need the following Stechkin–Nikolskii-type inequality (see [36, The-
orem 3.2]), which states that, for any n∈Nand 0 <δ≤π/n,
Tn˙
Wr
p(Td)δ−rωr(Tn,δ)Lp(Td),T
n∈T
n,0<p≤∞,r∈N,(5.2)
where the constants in this equivalence are independent of Tnand δ. Here, the
homogeneous Sobolev norm is given by
f˙
Wr
p(Td)=
|ν|1=rDνfLp(Td).
Using Theorem 4.1 with X=Lp(Td), 0 <p≤∞,andΩ(f, δ)X=ωr(f, δ)Lp(Td)
for some r∈N, one can easily verify that properties (4.1)–(4.6) are valid. Therefore,
applying Stechkin–Nikolskii-type inequality (5.2), we obtain the following result.
Theorem 5.1. Let f∈Lp(Td),0<p≤∞,and r∈N.Then
2−nrP2n(f)˙
Wr
p(Td)ωr(f,2−n)Lp(Td)∞
k=n+1
2−krλP2k(f)λ
˙
Wr
p(Td)1
λ
,
(5.3)
where λ=min(p, 1) and the constants in are independent of fand n.
The above theorem can be also formulated in terms of the fractional smoothness.
For this, we recall the following assertion from [36, Corollary 3.1]: Let 0<p≤∞,
α>0,n∈N,and 0<δ≤π/n.Then,for any Tn∈T
n,we have
sup
ξ∈Rd,|ξ|=1
∂
∂ξα
Tn
Lp(Td)δ−αωα(Tn,δ)Lp(Td),(5.4)
where the constants in are independent of Tnand the fractional modulus of
smoothness ωα(f, δ)Lp(Td)is given by
ωα(f,δ)Lp(Td)=sup
|h|≤δ
∞
ν=0
(−1)να
νf·+(α−ν)h
Lp(Td)
and α
ν=α(α−1)...(α−ν+1)
ν!,α
0=1,see[48].
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Our next goal is to obtain a sharp version of (5.3) in the case 1 <p<∞.For
this, we use Theorem 2.2 with Gn=Tn,X=Lp(Td), and Y=Hα
p(Td), where
Hα
p(Td)={g∈Lp(Td):g˙
Hα
p(Td)=(−Δ)α/2gLp(Td)<∞}
is the fractional Sobolev space. Recall that
Kf,tα,L
p(Td); Hα
p(Td)=inf
f−gLp(Td)+tαg˙
Hα
p(Td):g∈Hα
p(Td)(5.5)
and
Rf,tα;Lp(Td),T[1/t]=inf
f−TLp(Td)+tαT˙
Hα
p(Td):T∈T
[1/t](5.6)
(cf. (2.1) and (2.6)). For any f∈Lp(Td), 1 <p<∞,andα>0 we have (see,
e.g. [36])
K(f,tα;Lp(Td),Hα
p(Td)) R(f,tα;Lp(Td),T[1/t])ωα(f,t)Lp(Td),
which, in particular, implies (2.7). Here, the constants in are independent of f
and t.
Jackson and Bernstein inequalities (2.2) and (2.3) are given by (5.1) and the
following inequality, see, e.g. [67]:
(−Δ)α/2TnLp(Td)nαTnLp(Td),T
n∈T
n,1<p<∞,α>0.
Thus, Theorem 2.2 implies the following result.
Theorem 5.2. Let f∈Lp(Td),1<p<∞,and α>0.Then
∞
X
k=n+1
2−kατ (−Δ)α/2P2k(f)τ
Lp(Td)!1
τ
ωα(f, 2−n)Lp(Td)
∞
X
k=n+1
2−kαθ(−Δ)α/2P2k(f)θ
Lp(Td)!1
θ
,
where τ=max(2,p),θ=min(2,p),and the constants in are independent of f
and n.
5.2. The case of Fourier multiplier operators
In this subsection, we give an analogue of Theorem 3.3 in the case D=Td.We
start by recalling the multiplier theorem (Assumption 3.1) and the Littlewood–
Paley-type theorem in Lp(Td)for1<p<∞.
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Concerning Assumption 3.1, the well-known Mikhlin–H¨ormander multiplier the-
orem (see [24, p. 224]) states that the condition
|Δβ1
e1...Δβd
edm(k1,...,k
d)|≤A|k|−|β|,|β|≡β1+···+βd<[d/2] + 1,(5.7)
where Δeim(k1,...,k
i,k
d)=m(k1,...,k
i+1,...,k
d)−m(k1,...,k
i,...,k
d),
implies
TmfLp(Td)≤C(A, p, d)fLp(Td),
where
(Tmf)∧(k)=m(k)+
f(k)
and +
f(k)= 1
(2π)dTdf(y)e−i(k,y)dy.
We define the de la Vall´ee Poussin-type multiplier operator by
(ηnf)∧(k)=η|k|
n+
f(k)
and similarly to (3.5), we set
θ0(f)=η1fand θj(f)=η2jf−η2j−1ffor j≥1.
An analogue of the Littlewood–Paley theorem in the case D=Tdis given
by the following two inequalities, see, e.g. [11, Theorem 4.1] or [25, Chap. 6]: For
f∈Lp(Td), 1 <p<∞,and α>0, we have
⎧
⎨
⎩
∞
j=0
(θj(f))2⎫
⎬
⎭
1/2
Lp(Td)
fLp(Td)
and
⎧
⎨
⎩
∞
j=1
22jαθj(f)2⎫
⎬
⎭
1/2
Lp(Td)
(−Δ)α/2fLp(Td),
where the constants in are independent of f.
Let us consider the Fourier means given by
Ψnf(x)=
k∈Zd
ψk
n+
f(k)ei(k,x),
'
Ψnf(x)=
k∈Zd'
ψk
n+
f(k)ei(k,x),'
ψ(ξ)= η(|ξ|)
ψ(2−mξ),
where the function ψ:Rd→Cis such that supp ψ⊂[−1,1]dand for some m∈Z+,
ψ(x)=0forallx∈[−2−m,2−m]d.
We derive the following analogue of Theorem 3.3 in the case D=Td.
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Theorem 5.3. Let f∈Lp(Td),1<p<∞,n∈N,α>0,τ=max(2,p),and
θ=min(2,p).
(A) If {Ψ2k}are uniformly bounded operators in Lp(Td),then
∞
k=n+1
2−kατ (−Δ)α/2Ψ2kfτ
Lp(Td)1
τ
ωα(f,2−n)Lp(Td),
where the constant in is independent of fand n.
(B) If {'
Ψ2k}are uniformly bounded operators in Lp(Td),then
ωα(f,2−n)Lp(Td)∞
k=n+1
2−kαθ(−Δ)α/2Ψ2kfθ
Lp(Td)1
θ
,
where the constant in is independent of fand n.
Remark 5.1. (i)Notethatifψ∈A(Rd)={f:f=+g, g ∈L1(Rd)}(the Wiener
class of absolutely convergent Fourier integrals), then the operators {Ψn}are
uniformly bounded in Lp(Td) for all 1 ≤p≤∞, see, e.g. [54, Chap. VII]. Vari-
ous useful conditions to insure that ψ∈A(Rd) can be found in the survey [41],
see also [64, Chaps. 4 and 6].
(ii) Concerning the uniform boundedness of {'
Ψn}, one can use following version
of 1
f-Wiener theorem (see [42, p. 102]): Let f∈A(Rd). If f(x)=0onaclosed
bounded set V⊂Rd,then 1
f(x)is extendable to a function in A(Rd), i.e. there
exists a function g∈A(Rd) such that f(x)≡g(x)onV.
(iii) To verify the uniform boundedness of {Ψn}and {'
Ψn}in Lp(Td)for1<p<∞,
one can use the Mikhlin–H¨ormander multiplier condition (5.7), which is less
restrictive than the conditions given in parts (i) and (ii) of this remark.
(iv) Under conditions of Theorem 5.3, we have that for any f∈Hβ
p(Td), β>0,
∞
k=n+1
2−kατ (−Δ)(α+β)/2Ψ2kfτ
Lp(Td)1
τ
ωα((−Δ)β/2f,2−n)Lp(Td)
and
ωα((−Δ)β/2f,2−n)Lp(Td)∞
k=n+1
2−kαθ(−Δ)(α+β)/2Ψ2kfθ
Lp(Td)1
θ
.
As examples, let us consider the following approximation processes:
(1) the q-partial Fourier sums
Sn;qf(x)=
kq≤n+
f(k)ei(k,x),1≤q≤∞;
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(2) the de la Vall´ee Poussin-type means
ηnf(x)=
k∈Zd
η|k|
n+
f(k)ei(k,x);
(3) the Riesz spherical means
Rβ,δ
nf(x)=
|k|≤n1−|k|
nβδ
++
f(k)ei(k,x),β,δ>0.
Corollary 5.1. Let f∈Lp(Td),1<p<∞,α>0,τ =max(2,p),and θ=
min(2,p).Then
∞
k=n+1
2−kατ (−Δ)α/2T2kfτ
p1
τ
ωαf, 1
2np
∞
k=n+1
2−kαθ(−Δ)α/2T2kfθ
p1
θ
,(5.8)
where T2kf=S2k;qfwith q=1,∞,η
2kf, or Rβ,δ
2kfwith δ>(d−1)/2,and the
constants in are independent of fand n.
Proof. It is enough to note that these means are uniformly bounded in Lp(Td),
1<p<∞, see, e.g. [54, Chap. VII; 66], and to apply the Mikhlin–H¨ormander
multiplier condition to show that the corresponding inverse operators {'
Ψn}are
also uniformly bounded in Lp(Td).
Remark 5.2. In the univariate case of the Fej´er means T2kf=R1,1
2kf,theright-
hand side of inequality (5.8) was obtained earlier by Zhuk and Natanson in [70].
Note that for α∈Nand 1 <p<∞inequality (5.8) can be equivalently written
as follows:
∞
k=n+1
2−kατ T2kfτ˙
Wα
p(Td)1
τ
ωαf, 1
2np
∞
k=n+1
2−kαθT2kfθ˙
Wα
p(Td)1
θ
.
We give its analogue for the cases p=1,∞.
Corollary 5.2. Let f∈Lp(Td),p=1,∞,and α∈N.Then
2−nαξ−1
q(2n)S2n;qf˙
Wα
p(Td)ωαf, 1
2np
∞
k=n+1
2−kαξq(2k)S2k;qf˙
Wα
p(Td),
(5.9)
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where the constants in are independent of fand n,
ξq(t)=logd(t+1),q=1,∞,
td−1
2,1<q<∞,q=1,
and
2−nαT2nf˙
Wα
p(Td)ωαf, 1
2np
∞
k=n+1
2−kαT2kf˙
Wα
p(Td),(5.10)
where T2kf=η2kfor Rβ,δ
2kfwith δ>(d−1)/2.
Proof. Estimates (5.9) follow from Remark 4.2 with ξ(t)=ξq(t)since
f−Sn;qfLp(Td)Sn;qL1→L1Ecn(f)Lp(Td)ξq(n)ωα(f, n−1)Lp(Td).
For calculation of ξq(t) see, e.g. [40, 21] for the case 1 <q<∞and [64, Sec. 9.2;
34] for the case q=1,∞.
The proof of (5.10) for T2kf=η2kffollows from Theorem 4.1 and the uniform
boundedness of the de la Val´ee Poussin means in L1(Td), see also Remark 5.1. The
case T2kf=Rβ,δ
2kfcan be proved similarly using the uniform boundedness of Rβ,δ
2k,
see, e.g. [54, Chap. VII], the inequality f−Rβ,δ
2kfLp(Td)ωα(f,2−n)p,see[67],
and applying the same arguments as in the proof of (3.17).
5.3. Inequalities in the Hardy spaces Hp(D),0<p≤1
For simplicity, we only consider the analytic Hardy spaces on the unit disc D=
{z∈C:|z|<1}. By definition, an analytic function fon Dbelongs to the space
Hp=Hp(D)if
fHp=sup
0<ρ<12π
0|f(ρeit)|pdt1
p
<∞.
Set
ηnf(x)=
n
k=0
ηk
nckeikx,
where ck=ck(f) are the Taylor coefficients of f. Then, the realization result is
given as follows (see [36, Sec. 11]):
f−η2nfHp+2
−αn(η2nf)(α)Hpωα(f, 2−n)Hp,
where the constants in are independent of fand n.
Using the scheme of the proof of Theorem 3.2 and the Littlewood–Paley theorem
in the Hardy spaces Hp(D), 0 <p≤1, see, e.g. [25, Chap. 6], we obtain the following
result.
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Theorem 5.4. Let f∈Hp(D),0<p≤1,α∈N∪(1/p −1,∞),n∈N.Then
∞
k=n+1
2−2αk(η2kf)(α)2
Hp1
2
ωα(f,2−n)Hp(5.11)
and
ωα(f,2−n)Hp∞
k=n+1
2−αpk(η2kf)(α)p
Hp1
p
,(5.12)
where the constants in are independent of fand n.
Remark 5.3. (i) Note that the restriction α>1/p−1 is needed to correctly define
the modulus of smoothness ωα(f, δ)Hp.
(ii) Inequalities (5.11) and (5.12) are also valid if we replace the de la Vall´ee Poussin
means η2kfby the corresponding means Ψ2kfwith the properties similar to
those indicated in Theorem 3.3.
(iii) Inequality (5.12) also follows from Theorem 4.1 and the Stechkin–Nikolskii
inequality (5.4).
5.4. Approximation in smooth function spaces
We will say that f∈Lip(α, p)(T), 0 <p≤∞,α>0, if f∈Lp(T)and
fLip(α,p)=fLp(T)+|f|Lip(α,p)<∞,
where
|f|Lip(α,p)=sup
h>0
Δr
hfLp(T)
hα=sup
h>0
ωr(f,h)p
hα,r=[α]+1.
Let 0 <p≤∞,0<α<,and, n ∈N. The best approximation in Lip(α, p)(T)
and the modulus of smoothness are given by
En(f)Lip(α,p)=inf
T∈Tnf−TLip(α,p)
and
ϑ,α(f, δ)p=sup
0<h≤δ
ω(f,h)p
hα.
In light of the Jackson inequality (see [35])
En(f)Lip(α,p)ϑ,α f, 1
np
,n∈N,
by (5.2), the realization result can be written as follows:
ϑ,α(f, δ)pf−TnLip(α,p)+δ−αT()
nLp(T),n=[1/δ],(5.13)
where Tn∈T
nis such that En(f)Lip(α,p)=f−TnLip(α,p).
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Therefore, making use of Theorem 4.1 with X=Lip(α, p)andΩ(f, δ)X=
ϑ,α(f, δ)p,α<,∈N, and (5.13), we obtain the following result.
Theorem 5.5. Let f∈Lip(α, p),0<p≤∞,∈N,0<α<,and λ=min(p, 1).
Then
2−n(−α)T()
2nLp(T)ϑ,α(f, 2−n)p∞
k=n+1
2−k(−α)λT()
2kλ
Lp(T)1
λ
,
(5.14)
where T2k∈T
2kis the best approximant of fin Lip(α, p)and the constants in
are independent of fand n.
In view of Theorem 2.2, we sharpen (5.14) for 1 <p<∞as follows.
Theorem 5.6. Let f∈Lip(α, p),1<p<∞,∈N,0<α<,and τ=max(2,p),
θ=min(2,p).Then
∞
k=n+1
2−k(−α)τT()
2kτ
Lp(T)1
τ
ϑ,α(f, 2−n)p
∞
k=n+1
2−k(−α)θT()
2kθ
Lp(T)1
θ
,
where T2k∈T
2kis the best approximant of fin Lip(α, p)and the constants in
are independent of fand n.
Remark 5.4. Using the well-known facts about simultaneous approximation of
functions and their derivatives in Lp(T), see, e.g. [9; 16, Chap. 7, Theorem 2.7], it
is not difficult to obtain analogues of Theorems 5.5 and 5.6 in the Sobolev spaces
Wr
p(T), 1 ≤p≤∞,andr∈N,cf.Remark5.1(iv).
5.5. Interpolation operators
In the above sections, we deal with polynomials of the best approximation and
Fourier means. It turns out that Theorem 4.1 can be also applied for interpolation
operators. As an example, let us consider an interpolation analogue of the de la
Vall´ee Poussin means:
Vnf(t)= 1
3n
6n−1
k=0
f(tk)Kn(t−tk),t
k=πk
3n,t∈T,
where
Kn(t)= 1
2+
2n
k=1
cos kt +
4n−1
k=2n+1
4n−k
2ncos kt.
Recall some basic properties of Vnf(see [57]).
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Proposition 5.1. The following assertions hold :
(1) deg Vnf≤4n−1;
(2) Vnf(tk)=f(tk),k=0,...,6n−1;
(3) VnT(t)=T(t)for any T∈T
2n;
(4) for all f∈C(T)and r, n ∈N, we have
f−VnfL∞(T)ωr(f,1/n)∞.
Thus, noting that Vn(V2nf)=Vnfand using Theorem 4.1, Proposition 5.1, and
the Nikolskii–Stechkin-type inequality (5.2), we derive the following result.
Theorem 5.7. Let f∈C(T)and r, n ∈N.Then
2−nr(V2nf)(r)L∞(T)ωr(f, 2−n)∞
∞
k=n+1
2−kr(V2kf)(r)L∞(T),
where the constants in are independent of fand n.
6. Smoothness of Approximation Processes on Rd
6.1. Smoothness of best approximants
In what follows, the class of band-limited functions Bσ
p,1≤p≤∞,σ>0, is
given by
Bσ
p=ϕ∈Lp(Rd) : supp +ϕ(x)⊂{x:|x|<σ},
where
+g(x)=Rd
g(y)e−i(x,y)dy.
Let
Eσ(f)Lp(Rd)=inf
f−ϕLp(Rd):ϕ∈B
σ
p
be the best approximation of fand Pσ(f)∈B
σ
pbe a best approximant of fin
Lp(Rd), that is,
f−Pσ(f)Lp(Rd)=Eσ(f)Lp(Rd).
We will use the following Jackson and Nikolskii–Stechkin inequalities, see,
e.g. [62, 5.3.2; 67, Theorem 3] for the case 1 ≤p≤∞and [37] for the case
0<p<1:
Eσ(f)pωrf, 1
σp
,f∈Lp(Rd),σ>0,r∈N,(6.1)
Pσ˙
Wr
p(Rd)δ−rωr(Pn,δ)Lp(Rd),P
σ∈B
σ
p,σ>0,0<δ≤π/σ. (6.2)
In the above relations, the constants in and are independent of f,σ,andδ.
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Then, Theorem 4.1 together with inequalities (6.1) and (6.2) imply the following
result.
Theorem 6.1. Let f∈Lp(Rd),0<p≤∞,and r∈N.Then
2−nrP2n(f)˙
Wr
p(Rd)ωr(f,2−n)Lp(Rd)
∞
k=n+1
2−krP2k(f)˙
Wr
p(Rd),
where the constants in are independent of fand n.
To sharpen this result in the case 1 <p<∞, we will use Theorem 2.2 with
Gn=Bn
p,X=Lp(Rd), and Y=Hα
p(Rd), α>0, where
Hα
p(Rd)={g∈Lp(Rd):g˙
Hα
p(Rd)=(−Δ)α/2gLp(Rd)<∞}
is the fractional Sobolev spaces. The corresponding K-functional and its realization
are defined similarly to (5.5) and (5.6) and, moreover, for any f∈Lp(Rd), 1 <p<
∞,andα>0,
K(f,tα;Lp(Rd),Hα
p(Rd)) R(f,tα;Lp(Rd),B1/t
p)ωα(f,t)Lp(Rd),
see [67]. This, in particular, implies
(−Δ)α/2PσLp(Rd)nαPσLp(Rd),P
σ∈B
n
p,1<p<∞.
Thus, by Theorem 2.2, we obtain the following theorem.
Theorem 6.2. Let f∈Lp(Rd),1<p<∞,α>0,τ =max(2,p),and θ=
min(2,p).Then
∞
k=n+1
2−kατ (−Δ)α/2P2k(f)τ
Lp(Rd)1
τ
ωα(f,2−n)Lp(Rd)∞
k=n+1
2−kαθ(−Δ)α/2P2k(f)θ
Lp(Rd)1
θ
,
where the constants in are independent of fand n.
6.2. The case of Fourier multipliers operators.
The Mikhlin–H¨ormander multiplier theorem (cf. Assumption 3.1) states that the
condition
,,,,
∂β
∂β1x1...∂βdxd
μ(x),,,,≤A|x|−|β|,|β|≡β1+···+βd<-d
2.+1
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(see [24, p. 366]) implies
TμfLp(Rd)≤C(A, p, d)fLp(Rd),
where (Tμf)∧(x)=μ(x)+
f(x).Setting
(ησf)∧(x)=η|x|
σ+
f(x)
and
θ0(f)=η1fand θj(f)=η2jf−η2j−1ffor j≥1,
we have the following analogue of the Littlewood–Paley theorem in the case D=Rd
(see [25, p. 20; 11, Theorem 4.1]): for f∈Lp(Td), 1 <p<∞,and γ>0,
⎧
⎨
⎩
∞
j=0
(θj(f))2⎫
⎬
⎭
1/2
Lp(Rd)
fLp(Rd)
and
⎧
⎨
⎩
∞
j=1
22jαθj(f)2⎫
⎬
⎭
1/2
Lp(Rd)
(−Δ)α/2fLp(Rd),
where the constants in are independent of f.
We introduce the operators Ψσand '
Ψσas follows:
(Ψσf)∧(x)=ψ|x|
σ+
f(x),
('
Ψσf)∧(x)= '
ψ|x|
σ+
f(x),'
ψ(ξ)= η(|ξ|)
ψ(2−mξ),
where a function ψ:Rd→Cis such that supp ψ⊂[−1,1]dand for some m∈Z+,
ψ(x)=0forallx∈[−2−m,2−m]d.
We are now in a position to give a version of Theorem 3.3 in the case D=Rd.
Theorem 6.3. Let f∈Lp(Rd),1<p<∞,α>0,τ =max(2,p),and θ=
min(2,p).
(A) If {Ψ2k}are uniformly bounded in Lp(Rd),then
∞
k=n+1
2−kατ (−Δ)α/2Ψ2kfτ
Lp(Rd)1
τ
ωα(f,2−n)Lp(Rd).
(B) If {'
Ψ2k}are uniformly bounded in Lp(Rd),then
ωα(f,2−n)Lp(Rd)∞
k=n+1
2−kαθ(−Δ)α/2Ψ2kfθ
Lp(Rd)1
θ
.
In the above relations,the constants in are independent of fand n.
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An analogue of Corollary 5.1 on Rd, namely, inequality (5.8) holds for the fol-
lowing Fourier means:
(1) the q-Fourier means given by
/
Sn,qf(ξ)=χ{ξ∈Rd:ξq≤n}(ξ)+
f(ξ),q=1,∞;
(2) the de la Vall´ee Poussin-type means ηnf(x);
(3) the Riesz spherical means Rβ,δ
ngiven by
/
Rβ,δ
nf(ξ)=1−|ξ|
nβδ
++
f(ξ)
for β>0andδ>(d−1)/2.
At the same time, an analogue of Corollary 5.2 on Rdis valid only for the
de la Vall´ee Poussin-type means and the Riesz spherical means. Namely, for any
f∈Lp(Rd), p=1,∞,andα∈N,wehave
2−nαT2nf˙
Wα
p(Rd)ωαf, 1
2np
∞
k=n+1
2−kαT2kf˙
Wα
p(Rd),
where T2kf=η2kfor Rβ,δ
2kfwith δ>(d−1)/2.
Finally, in this section, we give a characterization of the classical Besov spaces
Bs
p,q(Rd) in terms of best approximants and Fourier means. Using Theorems 6.1–6.3
and the same arguments as in Corollary 4.4, we derive the following corollary.
Corollary 6.1. Let 1<p<∞,0<q≤∞,and 0<s<α. We have
|f|Bs
p,q(Rd)∞
k=1
2(s−α)qk(−Δ)α/2P2k(f)q
Lp(Rd)1
q
,(6.3)
where P2k(f)stands for the best approximants or the Fourier means Ψ2kfwith the
properties given in Theorem 6.3.
In the case,p=1 or ∞and α∈N,s<α,we have
|f|Bs
p,q(Rd)∞
k=1
2(s−α)qkP2k(f)q
˙
Wα
p(Rd)1
q
,
where P2k(f)stands for the best approximants,thedelaVall´ee Poussin-type means
ηnf(x),or the Riesz spherical means Rβ,δ
nwith δ>(d−1)/2.
In the above relations,the constants in are independent of f.
Note that a similar assertion for the Gauss–Weierstrass semi-group Wtf(x)=
(4πt)d/2Rde−|x−y|2
4tf(y)dy =(e−t|ξ|2+
f(ξ))(x), t>0, was obtained in [3, Theo-
rem 3.4.6, p. 198; 63, Sec. 1.13.2, pp. 76–81].
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7. Smoothness of Approximation Processes on [−1,1]
7.1. Sharp inequalities for algebraic polynomials
Let Lw,p =Lp([−1,1]; w), 0 <p≤∞, be the space of all functions fwith the finite
(quasi-)norm
fw,p =fLp([−1,1];w)=1
−1|f(x)|pw(x)dx1
p
,
where
w(x)=wa,b(x)=(1−x)a(1 + x)b,a,b>−1,
is the Jacobi weight on [−1,1]. In the unweighted case, w(x)≡1, we write Lp=
Lp[−1,1], fp=fLp[−1,1].
Further, let Pnbe the set of all algebraic polynomials of degree at most n.As
usual, the error of the best approximation of a function f∈Lw,p by algebraic
polynomials is defined as follows:
En(f)w,p =inf
P∈Pnf−Pw,p.
Let f∈Lp[−1,1], 0 <p<∞,r∈N,ϕ(x)=√1−x2,andσ≥0. Recall that
the Ditzian–Totik modulus of smoothness ωϕ
r(f,δ)pis given by
ωϕ
r(f,δ)p=sup
|h|≤δ¯
Δr
hϕfLp[−1,1] ,
where
¯
Δr
hϕ(x)f(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
r
k=0
(−1)kr
kf(x+(r
2−k)hϕ(x)),x±r
2hϕ(x)∈[−1,1],
0,otherwise.
The Jackson-type theorem for the Ditzian–Totik moduli of smoothness is
given by
En(f)p≤C(r, p)ωϕ
rf,n−1p,f∈Lp[−1,1],0<p<∞,n>r,
(see [15, Theorem 1.1] for the case 0 <p<1 and [20, p. 79, Theorem 7.2.1] for the
case p≥1). It is also well known, see, e.g. [18], that ωϕ
r(f,δ)p≤C(r, p)fpand
ωϕ
r(f, 2t)p≤C(r, p)ωϕ
r(f,t)p.Thus, taking into account the following Nikolskii–
Stechkin-type inequality (see [18, 28])
ωϕ
r(Pn,δ)pδrϕrP(r)
np,0<p<∞,P
n∈P
n,0<δ≤n−1,
we see that Theorem 4.1 implies the following result (see also [28]).
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Theorem 7.1. For any f∈Lp[−1,1],0<p≤∞,and n>r,we have
2−rnϕrP(r)
2nLp[−1,1] ωϕ
r(f,2−n)Lp[−1,1] ∞
k=n+1
2−rλkϕrP(r)
2kλ
Lp[−1,1]1
λ
,
where λ=min(1,p),P
nis a polynomial of the best approximation of fin Lp[−1,1],
and the constants in are independent of fand n.
Now, we are going to apply Theorems 2.2 and 3.2 in the case of the weighted
Lpspaces for 1 <p<∞.First,weintroducesomenotations.
For a, b > −1, denote by P(a,b)
k(x), k∈Z+, the system of Jacobi polynomials,
orthogonal on [−1,1], such that P(a,b)
k(1) = k+a
k,k∈Z+.LetalsoR(a,b)
kbe the
normalized Jacobi polynomials, R(a,b)
k(x)=P(a,b)
k(x)/P (a,b)
k(1), k∈Z+.
The Fourier–Jacobi series of f∈Lw,p,1≤p≤∞,a, b > −1, is given by
f(x)∼
∞
k=0
c(a,b)
k(f)μ(a,b)
kR(a,b)
k(x),
with the Fourier coefficients
c(a,b)
k(f)=1
−1
f(x)R(a,b)
k(x)w(x)dx, k ∈Z+,
and μ(a,b)
k=R(a,b)
k−2
Lw,2k2a+1.
Note that the Jacobi polynomials are the eigenfunctions of the differential
operator
Q(D)=Qα,β (D)= −1
w(x)
d
dxw(x)(1 −x2)d
dx,
Q(D)P(a,b)
k=λ(a,b)
kP(a,b)
k,λ
(a,b)
k=k(k+a+b+1).
Then the corresponding K-functional is given by (3.1) with σ=2andD=[−1,1].
Recall that by (3.4) and [10, Sec. 6], we have
Kγf,Q(D),n
−2γLp,w[−1,1] f−ηnfLp,w[−1,1] +n−2γQ(D)γηnfLp,w [−1,1],
where the constants in are independent of fand nand the de la Vall´ee Poussin
means ηnfare given by
ηnf(x)=
∞
k=0
ηk
nc(a,b)
k(f)μ(a,b)
kR(a,b)
k(x).
Thus, employing Theorems 2.2 and 3.2, and the needed facts from [10, Sec. 6],
we obtain the following result.
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Theorem 7.2. Let f∈Lp,w [−1,1],1<p<∞,γ >0,τ =max(2,p),and
θ=min(2,p).Then
∞
k=n+1
2−2γτkQ(D)γη2kfτ
Lp,w[−1,1] 1
τ
Kγ(f,Q(D),2−2nγ)Lp,w [−1,1],(7.1)
Kγ(f,Q(D),2−2nγ)Lp,w [−1,1] ∞
k=n+1
2−2γθkQ(D)γη2kfθ
Lp,w[−1,1]1
θ
,(7.2)
where the constants in are independent of fand n.
Inequalities (7.1) and (7.2) are also valid if we replace the de la Vall´ee Poussin
means η2kfby the best approximants P2k(f),or by the Fourier–Jacobi means Ψ2kf
with the properties similar to those indicated in Theorem 3.3.
Remark 7.1. Note that the results given in Theorems 7.1 and 7.2 essentially
improve the corresponding results for the best approximants in Lp,w[−1,1], 1 ≤
p<∞, obtained early in [20, Theorem 8.3.1; 4, 28, 38, 65].
7.2. Sharp inequalities for splines
In this subsection, we consider approximation of functions by splines in the space
Lp[0,1] with the (quasi-)norm ·
p=·
Lp[0,1].
Denote by Sm,n the set of all spline functions of degree m−1withtheknots
tj=tj,n =j/n,j=0,...,n, i.e. S∈S
m,n if S∈Cm−2[0,1] and Sis some algebraic
polynomial of degree m−1ineachinterval(tj−1,t
j), j=1,...,n.
Let
Em,n(f)p=inf
S∈Sm,n f−SLp[0,1]
be the best approximation of a function fby splines S∈S
m,n in Lp[0,1].
The Jackson-type inequality is given by ([46, Theorem 1], see also [16, Chap.
12, p. 379])
Er,n (f)p≤C(r, p)ωr(f, n−1)p,(7.3)
where f∈Lp[0,1], 0 <p≤∞,n∈N,and
ωr(f,δ)p=sup
0<h≤δΔr
hfLp[0,1−rh]
is the modulus of smoothness of order r∈N.
Note that any spline Sn∈S
r,n can be represented (see [46]) as follows:
Sn(x)=P(x)+
n−1
j=1
aj(x−tj)r−1
+,
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where P∈P
r−1,x+=xif x≥0andx+=0ifx<0. Moreover, one has
C(r, p)−1n−(1+(r−1)p)
n−1
j=1 |aj|p≤ωr(Sn,n
−1)p
p≤C(r, p)n−(1+(r−1)p)
n−1
j=1 |aj|p,
(7.4)
Inequalities (7.4) were proved in [29, Lemma 2.1] (see also [27]) in the case 1 ≤p<
∞. It is easy to see that the same also holds in the case 0 <p<1.
It is important to mention that (7.4) implies that for any Sn∈S
r,n ,n, r ∈N,
one has
ωr(Sn,n
−1)pn−(r−1)−1
pV(S(r−1)
n)p,0<p<∞,(7.5)
where V(f)pdenotes the p-variation of the function f,thatis,
V(f)p=sup
0=x0<x1<...<xn=1 n−1
k=0 |f(xk+1)−f(xk)|p1
p
.
In its turn, (7.5) implies the following analogue of the Bernstein inequality:
n−(r−1)−1
pV(S(r−1)
n)p≤C(r, p)Snp,(7.6)
Moreover, by (7.3) and (7.5), for any Sn∈S
r,n ,n, r ∈N, such that f−SnLp[0,1] =
Er,n (f)p,wehave
f−Snp+n−(r−1+ 1
p)V(S(r−1)
n)pωr(f,n−1)p,(7.7)
where the constants in do not depend on f,Sn,andn.
The above results allow us to apply Theorem 4.1 to obtain the following
result.
Theorem 7.3. Let f∈Lp[0,1],0<p<∞,r,n∈N,and λ=min(1,p).Then
2−n(r−1+ 1
p)V(S(r−1)
2k)pωr(f,2−n)p∞
k=n+1 (2−k(r−1+ 1
p)V(S(r−1)
2k)p)λ1
λ
,
where S2k∈S
r,2kis such that f−S2kLp[0,1] =Er, 2k(f)pand the constants in
are independent of fand n.
Inthecase1<p<∞, using (7.5)–(7.7) and Theorem 2.2, we arrive at the next
statement.
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Theorem 7.4. Let f∈Lp[0,1],1<p<∞,r,n∈N,and τ=max(2,p),θ=
min(2,p).Then
∞
k=n+1
2−k(r−1+ 1
p)τV(S(r−1)
2k)τ
p1
τ
ωr(f,2−n)p
∞
k=n+1
2−k(r−1+ 1
p)θV(S(r−1)
2k)θ
p1
θ
,
where S2k∈S
r,2kis such that f−S2kLp[0,1] =Er, 2k(f)pand the constants in
are independent of fand n.
8. Nonlinear Methods of Approximation
8.1. Nonlinear wavelet approximation
We restrict ourselves to the case of compactly supported biorthogonal wavelets
and follow the discussion in [14, Sec. 7]. Let ϕand 'ϕbe two refinable compactly
supported functions in L2(R). Suppose that ϕand 'ϕgenerate two multiresolution
analysis (see, e.g. [45]) and are in duality as follows:
R
ϕ(x−j)'ϕ(x−k)dx =δjk ,
where δjk is the Kronecker delta. For such functions ϕand 'ϕ,wehave
ϕ(x)=
k∈Z
ckϕ(2x−k),'ϕ(x)=
k∈Z'ck'ϕ(2x−k).
Then the corresponding wavelet functions ψand '
ψare given by
ψ(x)=
k∈Z
(−1)k'c1−kϕ(2x−k),'
ψ(x)=
k∈Z
(−1)kc1−k'ϕ(2x−k).
The classical example of wavelet functions is the Haar system. Set ϕ='ϕ=
χ[0,1], then (see, e.g. [45, p. 23])
ψ(x)= '
ψ(x)=ϕ(2x+1)−ϕ(2x+2)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
−1,−1≤x<−1/2;
1,−1/2<x≤0;
0,x=−1/2,x>0,x<−1.
It is well known that each function f∈Lp(R) has the following wavelet decom-
position:
f=
I∈D
cI,p(f)ψI,p,c
I,p(f)=f, '
ψI,p/(p−1),
see, e.g. [7, 12]. In the above formula, Dis the set of all dyadic intervals in R,I
denotes the dyadic cube I=2
−k(j+[0,1]) associated with j, k ∈Zand
ψI,p(x)=|I|−1/pψ(2kx−j).
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Let Σw
ndenote the set of all functions
S=
I∈Λ
aIψI,
where Λ ⊂Dis a set of dyadic intervals of cardinality #Λ ≤n.ThusΣ
w
nis the set
of all functions which are a linear combination of nwavelet functions. We define
σw
n(f)p=inf
S∈Σw
nf−SLp(R).
Let Br
p,q(R), r>0, 0 <p,q≤∞, be the classical Besov spaces. The Jackson-
and Bernstein-type inequalities are given in the following two propositions (see [8,
Corollary 4.1 and Theorem 4.3]).
Proposition 8.1. Let 1<p<∞,r>0,and f∈Lp(R),1/γ =r+1/p.Ifψhas
mvanishing moments with m>rand ψis in Bρ
γ,q(R)for some q>0and some
ρ>r,then
σw
n(f)pKf,n−r;Lp(R),Br
γ,γ(R),n∈N,
where the constant in is independent of fand n.
Proposition 8.2. Let 1<p<∞,r>0,1/γ =r+1/p.IfS=I∈ΛcI,p(f)ψI,p,
with #Λ ≤n,then
|S|Br
γ,γ(R)nrSLp(R),
where the constant in is independent of Sand n.
We will also use the fact that there exists Qnf∈Σw
nsuch that f−QnfLp(R)
σw
n(f)pand
Kf,n−r;Lp(R),Br
γ,γ(R)f−QnfLp(R)+n−r|Qnf|Br
γ,γ(R),
where the constants in are independent of fand n(see for details [8]). This
realization result in particular implies the Nikolskii–Stechkin-type inequality
KS, n−r;Lp(R),Br
γ,γ(R)n−r|S|Br
γ,γ(R),S∈Σw
n.
Thus, in light of Theorem 2.2, Propositions 8.1 and 8.2, we obtain the following
result.
Theorem 8.1. Under conditions of Proposition 8.1, we have
∞
k=n+1
2−rτk|P2kf|τ
Br
γ,γ(R)1
τ
Kf,2−rn;Lp(R),Br
γ,γ(R)
∞
k=n+1
2−rθk|P2kf|θ
Br
γ,γ(R)1
θ
,
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where P2kf∈Σw
2kis such that f−P2kfLp(R)=σw
2k(f)p,τ=max(2,p),θ=
min(2,p),and the constants in are independent of fand n.
As a corollary, we obtain the characterization of the Besov space Br
X,q
(interpolation space) given in (4.17) with X=Lp(R)andΩ(f,2−k)X=
K(f,2−rk,L
p(R),Br
γ,γ(R)).
Corollary 8.1. Under conditions of Proposition 8.1, if 0<σ<rand 0<q≤∞,
then
|f|Bσ
X,q(R)∞
k=1
2(σ−r)qk|P2kf|q
Br
γ,γ(R)1
q
,
where P2kf∈Σw
2kis such that f−P2kfLp(R)=σw
2k(f)pand the constants in
are independent of f.
8.2. Free knot piecewise polynomial approximation
Let r∈Nbe fixed and for each n=1,2,...,letΣ
r,n be the set of piecewise
polynomials of degree rwith npieces on [0,1]. That is, for each element S∈
Σr,n there is a partition Λ of [0,1] consisting of ndisjoint intervals I⊂[0,1] and
polynomials PI∈P
rsuch that
S=
I∈Λ
PIχI.
For e a ch 0 <p<∞, we define the error of the best approximation by
σr,n (f)p=inf
S∈Σr,n f−SLp[0,1].
Recall the well-known Jackson-type inequality (see [47, Theorem 2.3]).
Proposition 8.3. Let f∈Lp[0,1],0<p<∞,r>0,k∈N,and1/γ =r+1/p.
Then
σr,n (f)pKf, n−r;Lp[0,1],Br
γ,γ;k[0,1],n∈N,(8.1)
where Br
γ,γ;k[0,1] is the non-periodic Besov space, which consists of f∈Lγ[0,1]
such that
|f|Br
p,q;k[0,1] =1/k
0t−rωk(f,t)Lγ[0,1]γdt
t1/γ
<∞.
The constant in is independent of fand n.
Now, using (8.1) and Theorem 4.1, we derive the following result.
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Theorem 8.2. Under conditions of Proposition 8.3, we have
K“S2n,2−rn;Lp[0,1],B
r
γ,γ;k[0,1]”K“f,2−rn;Lp[0,1],B
r
γ,γ;k[0,1]”
0
@
∞
X
k=n+1
K“S2k,2−rk;Lp[0,1],B
r
γ,γ;k[0,1]”λ1
A
1
λ
,
where S2k∈Σr,2kis such that f−S2kLp[0,1] =σr,2k(f)p,λ=min(p, 1),and the
constants in are independent of fand n.
Finally, we characterize the Besov space Br
X,q given in (4.17) with X=Lp[0,1]
and Ω(f,2−k)X=K(f,2−rk,L
p[0,1],Br
γ,γ;k[0,1]).
Corollary 8.2. Let 0<σ<r and 0<q≤∞,we have
|f|Bσ
X,q[0,1] ∞
k=1
2σqkKS2k,2−rk;Lp[0,1],Br
γ,γ;k[0,1]q1
q
,
where S2k∈Σ2k,r is such that f−S2kLp[0,1] =σr,2k(f)pand the constants in
are independent of f.
9. Optimality
In the previous sections, we derived the following inequalities:
∞
k=n+1
2−kατ P2k(f)τ
Y1
τ
K(f,2−nα;Lp,Y)∞
k=n+1
2−kαθP2k(f)θ
Y1
θ
,
(9.1)
where f∈Lp,1≤p≤∞,
τ=max(p, 2),1<p<∞,
∞,otherwise,θ=min(p, 2),p<∞,
1,p=∞,
Yis an appropriate smooth function space, and Pn(f) is a suitable approximation
method. In this section, we show that the parameters θand τare optimal.
For this, we restrict ourselves to the case of D=Tand approximation of periodic
Lp-functions by Sn(f), the nth partial sums of the Fourier series of f,andthede
la Vall´ee Poussin means ηnf.
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Recall that if f∈Lp(T),1<p<∞,then inequality (9.1) in particular implies
∞
k=n+1
2−kατ S(α)
2k(f)τ
p1
τ
ωαf, 1
2np
∞
k=n+1
2−kαθS(α)
2k(f)θ
p1
θ
.
(9.2)
If f∈Lp(T),p=1,∞,and Pn(f)=ηnf, estimate (9.1) can be written by
2−αn(η2nf)(α)Lp(T)ωα(f, 2−n)Lp(T)
∞
k=n
2−2αk(η2kf)(α)Lp(T).
9.1. Optimality of (9.1) in the case 1<p<∞
In this subsection, we deal with not only sharpness of the parameters τ=max(2,p)
and θ=min(2,p) but we also show that for the classes of functions with lacunary
and general monotone Fourier coefficients, inequality (9.1) becomes an equivalence
with τ=θ=2andτ=θ=p, respectively.
We start with lacunary series and first give a simple proof of Zygmund’s theorem
in Lp,1<p<∞, based on the Littlewood–Paley technique given in Sec. 3.1. We
deal with the general case of functions represented by
f∼
∞
k=0
Akf, Akf=
dk
=1 f,ψk,ψk,.
For convenience, we suppose that the dimension dk=1forallk∈Z+.
We will say that the Fourier expansion of f∈Lp,w(D) is lacunary, written
f∈Λ, if f∼∞
j=0 A2jf, i.e. Akf=0fork=2
j,j∈Z+.
Let us first derive an analogue of Zygmund’s theorem.
Lemma 9.1. Let 1<p<∞,f ∈Λ,and Assumption 3.1 hold. Suppose that
w∈L1(D)and the functions ψk=ψk,1are such that
0<ξ
2≤ψkp,w ≤ξ1<∞for any k∈Z+.(9.3)
Then
fp,w ∞
k=0
c2k(f)21
2
,c
k(f)=D
fψkw.
In particular,fp,w f2,w.Here,the constants in are independent of f.
Proof. First, let us prove the estimate from above. Let 1 <p≤2. Then
by H¨older’s inequality and Parseval’s inequality, we obtain fp,w f2,w
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∞
k=0 c2k(f)21
2.If p≥2, noting that
θj(A2kf)=(η2j−η2j−1)(A2kf)=A2j−1f, j =k+1,
0,j=k+1,
and using the Littlewood–Paley decomposition (Theorem 3.1), Minkowski’s inequal-
ity, and (9.3), we derive
fp,w
∞
k=0
θk(f)21
2
p,w
=
∞
k=0
A2k(f)21
2
p,w
=⎛
⎝D∞
k=0
(c2k(f)ψ2k)2p
2
w⎞
⎠
1
p
≤∞
k=0 D|c2k(f)ψ2k|pw2
p1
2
≤∞
k=0 |c2k(f)|21
2
max
kD|ψ2k|pw1
p
∞
k=0 |c2k(f)|21
2
.
To show the inverse inequality for p≤2, we similarly obtain
fp,w ⎛
⎝D∞
k=0
(c2k(f)ψ2k)2p
2
w⎞
⎠
1
p
≥∞
k=0 D|c2k(f)ψ2k|pw2
p1
2
≥∞
k=0
c2k(f)21
2
min
kψ2kp,w ∞
k=0
c2k(f)21
2
.
If p≥2, H¨older’s inequality implies f2,w fp,w ,which proves the lemma.
Remark 9.1. As an example of the system {ψk}in Lemma 9.1, one can take the
trigonometric system, the Walsh system, systems of the Chebyshev polynomials
and, more generally, the system of normalized Jacobi polynomials for specific range
of parameters α, β > −1indicatedin[2].
Theorem 9.1. Under all assumptions of Lemma 9.1, we have for f∈Lp,w(D)∩Λ
∞
k=n+1
2−2γσkQ(D)γη2kf2
p,w1
2
Kγ(f,Q(D),2−nγσ )p,w ,γ>0,
where the constants in are independent of fand n.
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Proof. Using the realization result (3.4) and Lemma 9.1, we get
Kγ(f,Q(D),2−nγσ )p,w f−η2nfp,w +2
−γσnQ(D)γη2nfp,w
∞
k=n−1
c2k(f)21
2
+2
−γσn n−1
k=1
22γσkc2k(f)21
2
(9.4)
and
2−2γσkQ(D)γη2kf2
p,w 2−2γσk
k−1
l=1
22γσlc2l(f)2.
Then
∞
k=n+1
2−2γσkQ(D)γη2kf2
p,w
∞
k=n+1
2−2γσk
k−1
l=1
22γσlc2l(f)2
=
∞
k=n+1
2−2γσk n
l=1
+
k−1
l=n+1
22γσlc2l(f)2
2−2γσn
n
l=1
22γσlc2l(f)2+
∞
l=n+1
22γσlc2l(f)2
Kγ(f,Q(D),2−nγσ )2
p,w.
In particular, for the classical Fourier series on D=T,weobtain
ωα(f,2−n)Lp(T)∞
k=n
2−2αk(S2kf)(α)2
Lp(T)1
2
,f∈Lp(T)∩Λ,(9.5)
where 1 <p<∞and α>0; cf. (9.2).
Remark 9.2. It is clear that (9.5) gives the sharpness of the parameter θfor p≥2
and τfor p≤2 in inequality (9.2).
Proof. Assume that p≥2 and there holds
ωα(f,2−n)Lp(T)∞
k=n(2−αk(S2kf)(α)Lp(T))2+ε1
2+ε
(9.6)
with some ε>0. Consider f(x)=∞
n=1 a2ncos 2nx,wherea2n=1/n.Then
f∈Lp(T)∩Λ and, by (9.4), one has
ωα(f,2−n)Lp(T)∞
k=n
a2
2k1
2
+2
−αn n
k=1
22αka2
2k1
2
1
n1/2,
2−αk(S2kf)(α)Lp(T)1
k,∞
k=n
2−(2+ε)αk(S2kf)(α)2+ε
Lp(T)1
2+ε
n−1+ε
2+ε,
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Smoothness of functions versus smoothness of approximation processes
which contradicts (9.6). Similarly, if p≤2, then the inequality
ωα(f,2−n)Lp(T)∞
k=n(2−αk(S2kf)(α)Lp(T))2−ε1
2−ε
with some ε∈(0,2) does not hold for f(x)=∞
n=1 a2ncos 2nx∈Lp,where
a2n=n−1/(2−ε).
Now, let us consider the case of the classical Fourier series with general monotone
coefficients. In what follows, we say (see [60]) that a (complex) sequence {dn}is
general monotone, written {dn}∈GM,if
2n
k=n|dk−dk+1|≤C|dn|,
where Cdoes not depend on n. Note that any monotone (quasi-monotone)
sequences are general monotone. We denote by /
GM the class of integrable functions
such that f(x)∼∞
n=1(ancos nx +bnsin nx)with{an},{bn}∈GM .
Theorem 9.2. Let f∈Lp(T)∩/
GM, 1<p<∞,and α>0.Then
ωα(f,2−n)Lp(T)∞
k=n
2−pαk(S2kf)(α)p
Lp(T)1
p
,(9.7)
where the constants in are independent of fand n.
Proof. First, we recall the following Hardy–Littlewood theorem:
fLp(T)∞
n=1
(|an|+|bn|)pnp−21
p
,f∈Lp(T)∩/
GM, 1<p<∞.
This is a well-known fact for functions with monotone coefficients, see [71,
Chap. XII]. For the class /
GM (in fact for a more general class and for Lorentz
spaces) this has been recently proved in [22]. Moreover, it is also shown in [22] that
ωα(f,n−1)Lp(T)n−αn
k=0
(|ak|+|bk|)pkpα+p−21
p
+∞
k=n
(|ak|+|bk|)pkp−21
p
.
Now, we note that the sequences {d1,...,d
n,0,0,...}and {nαdn}belong to
GM whenever {dn}∈GM, which implies that the Hardy–Littlewood theorem can
be applied for the partial Fourier sums of f. Moreover, since any general monotone
sequence {dn}satisfies the following property, see [60]: |dk|≤C|dn|for n≤k≤2n,
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we have
(S2nf)(α)Lp(T)n
k=0
(|a2k|+|b2k|)p2k(pα+p−1)1
p
.
Thus, we derive
ωα(f,2−n)Lp(T)2−αn n
k=0
(|a2k|+|b2k|)p2k(pα+p−1)1
p
+∞
k=n
(|a2k|+|b2k|)p2k(p−1)1
p
∞
k=n
2−pαk
k
l=0
(|a2l|+|b2l|)p2l(pα+p−1)1
p
∞
k=n
2−pαk(S2kf)(α)p
Lp(T)1
p
,
completing the proof.
Remark 9.3. Similarly to Remark 9.2, equivalence (9.7) provides the sharpness of
the parameter θfor p≤2andτfor p≥2in(9.2).
9.2. Optimality of the right-hand inequality in (9.1) for p=1and
p=∞
We start by obtaining two simple results for lacunary Fourier series.
Theorem 9.3. Let f∈L1(T)∩Λand α>0.Then
ωα(f,2−n)L1(T)∞
k=n
2−2αk(η2kf)(α)2
L1(T)1
2
,
where the constants in are independent of fand n.
Proof. The proof repeats the one of Theorem 9.1 since by Zygmund’s theorem
(see [24, Theorem 3.7.4]), we have
ωα(f,2−n)L1(T)∞
k=n|c2k|21
2
+2
−αn n
k=1
22αk|c2k|21
2
,
where {ck}are the Fourier coefficients of f.
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Smoothness of functions versus smoothness of approximation processes
Theorem 9.4. Let f∈L∞(T)∩Λand α>0.Then
ωα(f,2−n)L∞(T)
∞
k=n
2−αkη(α)
2kfL∞(T),
where the constants in are independent of fand n.
Proof. By Stechkin’s theorem (see [24, Theorem 3.7.6]), we have
∞
k=n
2−αk(η2kf)(α)L∞(T)
∞
k=n
2−αk
n−1
s=1
2αs|c2s|+
∞
k=n
2−αk
k
s=n
2αs|c2s|
2−αn(η2nf)(α)L∞(T)+
∞
k=n|c2k|
2−αn(η2nf)(α)L∞(T)+E2n(f)∞ωα(f, 2−n)L∞(T).
Note that Theorem 9.4 shows that in the case p=∞, the right-hand inequal-
ity (9.1) is sharp for θ= 1, in other words this inequality cannot be improved for
some θ>1 in the general case. At the same time, we remark that Theorem 9.3
only shows that in the case p= 1, the right-hand inequality (9.1) is sharp for θ=2,
that is, (9.1) cannot be sharpen with any θ>2.
Now, we show that (9.1) is in fact sharp for θ=1.
Theorem 9.5. Let α∈N. Then for any q>1there exists a function f∈L1(T)
such that
ωα(f,2−n)L1(T)≤C∞
k=n+1
2−qαk(η2kf)(α)q
L1(T)1
q
(9.8)
is not valid with a constant Cindependent of fand n.
Proof. We will use the following well-known Kolmogorov’s estimates for the L1-
norms of trigonometric series:
π
0,,,,,
∞
k=1
akcos kx,,,,,dx
∞
k=1
k|Δ2ak|,(9.9)
π
0,,,,,
∞
k=1
aksin kx,,,,,dx
∞
k=1
k|Δ2ak|+
∞
k=1
|ak|
k,(9.10)
where Δ2ak=ak+2 −2ak+1 +ak. Inequality (9.9) was obtained in [32], see also [58];
for inequality (9.10) see [58].
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Yu. S. Kolomoitsev & S. Yu. Tikhonov
We will also need the following estimate for the error of the best approximation
given by (see [23, Lemma 2]):
En(g)L1(T),,,,,
∞
k=n+1
ak
k,,,,,,g(x)∼
∞
k=1
aksin kx ∈L1(T).(9.11)
Now, consider the function
fN(x)=
N
k=1
sin kx
logγ(k+1),
where N>2nand 0 <γ<1/q. By the Jackson inequality and (9.11), we obtain
ωα(fN,2−n)L1(T)E2n(fN)L1(T)
N
k=2n+1
1
klogγ(k+1) log1−γN−log1−γ2n.(9.12)
Next, if αis odd, by (9.9), we derive
(η2mfN)(α)L1(T)=
η2mN
k=1
kα
logγ(k+1)cos kx
L1(T)
2m
k=1
kα−1
logγ(k+1) 2αm
mγ.
Similarly, if αis even, (9.10) implies that
(η2mfN)(α)L1(T)2αm
mγ.
Thus, for all α∈N,wehave
∞
m=n(2−αm(η2mfN)(α)L1(T))q
[log N]
m=n
1
mγq +
∞
m=[log N]+1
2−αqm Nαq
(log N)γq
7log N
(log N)γq .(9.13)
Combining (9.12) and (9.13), it is easy to see that inequality (9.8) is not valid
for f=fNwith sufficiently large N.
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Smoothness of functions versus smoothness of approximation processes
9.3. Optimality of the left-hand inequality in (9.1) for p=1and
p=∞
In this subsection, we show that the left-hand inequality in (9.1) cannot be improved
in general. In particular, for p=1orp=∞, the following inequality is not valid
for any q>0:
∞
k=n+1
2−qαk(η2kf)(α)q
Lp(T)1
q
≤Cωα(f,2−n)Lp(T).(9.14)
Theorem 9.6. Let p=1or ∞and α∈N. Then for any q>0there exists
afunctionf∈Lp(T)such that inequality (9.14) is not valid with a constant C
independent of fand n.
Proof. Let p=∞.Wetake
f(x)=
∞
m=1
amsin mx, am=1
mlogγ(m+1),γ>1.
Since am0andmam→0, we have f∈C(T), see, e.g. [71, Chap. V].
By [61], we get
En(f)L∞(T)max
ν≥1νaν+nmax
ν≥1
ν
(ν+n)log
γ(ν+n+1) 1
logγn.(9.15)
Next,
(η2kf)(α)L∞(T)=
η2k∞
m=1
mα−1cos(mx +απ)
logγ(m+1)
L∞(T)
.
If αis even, we obviously have
(η2kf)(α)L∞(T)
2k
m=1
η(m
2k)mα−1
logγ(m+1) 2αk
kγ.(9.16)
For o d d α, using Bernstein’s inequality, we derive
(η2kf)(α)L∞(T)≥1
2k(η2kf)(α+1)L∞(T)
=1
2k
η2k∞
m=1
mαcos mx
logγ(m+1)
L∞(T)2αk
kγ.(9.17)
Due to (9.15)–(9.17), and using the realization result, we have
ωα(f,2−n)L∞(T)1
nγ.
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At the same time, by (9.16) and (9.17), we derive
∞
k=n+1
2−qαk(η2kf)(α)q
L∞(T)1
q
n1
q
nγ.
The last two formula imply that inequality (9.14) is not valid in the case p=∞.
Now, let us consider the case p= 1. We put
f(x)=
∞
m=1
amcos mx, am=1
logγ(m+1),γ>1.
Since am0andΔ
2am≥0, we have f∈L1(T), see, e.g. [71, Chap. V].
Recall that if a convex sequence {am}is the sequence of cosine Fourier coeffi-
cients of an even function f∈L1(T), then applying [1, Theorem 1], we have
ωα(f,2−n)L1(T)1
2αn
2n
m=1
mα−1am1
nγ.(9.18)
Next, since for any g∈L1(T)andk∈N, one has gL1(T)≥2π|+g(2k)|, it follows
that
(η2kf)(α)L1(T)=
η2k∞
m=1
mαcos(mx +απ)
logγ(m+1)
L1(T)
2αk
kγ
and, therefore,
∞
k=n+1
2−qαk(η2kf)(α)q
L1(T)1
q
n1
q
nγ.(9.19)
Finally, combining (9.18) and (9.19), we obtain contradiction to (9.14).
Acknowledgments
The first author was supported by the DFG Project KO 5804/1-1. The second
author was partially supported by the MTM 2017-87409-P, 2017 SGR 358, and
by the CERCA Programme of the Generalitat de Catalunya. The authors would
like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge,
for support and hospitality during the programme “Approximation, sampling and
compression in data science” where part of the work on this paper was undertaken.
This work was supported by the EPSRC Grant No. EP/K032208/1.
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