Basis properties of Fucik eigenfunctions

Abstract

We establish sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$.

Full text

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS
FALKO BAUSTIAN AND VLADIMIR BOBKOV
Abstract. We establish sufficient assumptions on sequences of Fučík eigenvalues of the one-
dimensional Laplacian which guarantee that the corresponding Fučík eigenfunctions form a
Riesz basis in 𝐿2(0, 𝜋).
1. Introduction
The classical spectral theorem for compact self-adjoint (linear) operators on a Hilbert space
𝑋asserts the existence of a sequence of eigenfunctions of such an operator that forms an
orthogonal basis of 𝑋, see, e.g., [30]. One of the simplest examples is given by the sine
functions sin(𝑛𝑥),𝑛∈N, which are eigenfunctions of the one-dimensional Dirichlet Laplacian,
i.e.,
−𝑢′′ =𝜆𝑢 in (0, 𝜋), 𝑢(0) = 𝑢(𝜋)=0,
and hence they form an orthogonal basis in 𝐿2(0, 𝜋). Although the assumptions of the spectral
theorem can not be weakened in general, eigenfunctions of certain operators which do not
satisfy the imposed requirements might still form a basis of a space 𝑋. Classes of such
operators have been an active topic of research, and a great amount of significant results have
been obtained. We refer to [1,13,22,24,26,32] and the extensive bibliographies therein for
a deeper discussion.
The references indicated above are concerned mainly with linear operators. At the same
time, the conclusion of the spectral theorem can be valid even for some nonlinear operators.
As an example, there exists 𝑝0>1such that the generalized trigonometric functions sin𝑝(𝑛𝑥)
defined as eigenfunctions of the one-dimensional Dirichlet 𝑝-Laplacian with 𝑝>𝑝0, i.e.,
−(|𝑢′|𝑝−2𝑢′)′=𝜆|𝑢|𝑝−2𝑢in (0, 𝜋𝑝), 𝑢(0) = 𝑢(𝜋𝑝)=0,
form a Riesz basis in 𝐿2(0, 𝜋𝑝), see [4], and [6,7,16] for further developments. Here,
𝜋𝑝=2𝜋
𝑝sin(𝜋/𝑝). While this basis is non-orthogonal except for 𝑝= 2, it has certain advan-
tages, in particular, in a numerical study of nonlinear equations, see [5]. Let us remark that
it remains unknown whether the same basisness result holds true for all 𝑝 > 1. Notice that
the generalized trigonometric functions sin𝑝(𝑛𝑥)share with sin(𝑛𝑥)the anti-periodic struc-
ture which is significantly used in the proof of [4]. In general, there exist several results on
assumptions for a function 𝑓to guarantee that the corresponding system of dilated functions,
i.e., the sequence of the form {𝑓(𝑛𝑥)}, is a basis, see, e.g., [3,6,18,19,35], to mention a few.
2010 Mathematics Subject Classification. 34L10, 34B25, 34B08, 47A70.
Key words and phrases. Fucik specturm, Fucik eigenfunctions, Riesz basis, Paley-Wiener stability.
V. Bobkov was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports,
and by the grant 18-03253S of the Grant Agency of the Czech Republic.
1
arXiv:2012.10368v1 [math.CA] 18 Dec 2020

2 F. BAUSTIAN AND V. BOBKOV
In the present work, we investigate the basisness of eigenfunctions for another type of
eigenvalue problems for the classical Dirichlet Laplacian in one dimension which does not fit
into the framework of the spectral theorem, namely,
−𝑢′′ =𝛼𝑢+(𝑥)−𝛽𝑢−(𝑥)in (0, 𝜋), 𝑢(0) = 𝑢(𝜋)=0,(1.1)
where both 𝛼, 𝛽 are spectral parameters. This so-called Fučík (or Dancer-Fučík)eigenvalue
problem was originated in the works [11] and [17] in the context of studies of elliptic equations
with “jumping” nonlinearities. Various aspects of the Fučík eigenvalue problem for second-
order elliptic operators where intensively studied afterwards. We refer to [8,33] for the local
existence of curves in the Fučík spectrum emanated from classical eigenvalues 𝜆𝑘, to [9,14]
for variational characterizations of such curves, to [29,31] for their asymptotic behaviour, to
[10,23] for nodal properties of the corresponding Fučík eigenfunctions, and to [12,28] for
algebro-topological properties of the associated energy functional. Although this list is far
from being complete, many other relevant references can be found in the indicated papers.
Nevertheless, the structure and properties of the Fučík spectrum and Fučík eigenfunctions are
understood only to a rather limited extent, and their investigation can be a hard task even
in the case of operators acting in finite-dimensional spaces, see the discussion in [20,25]. In
particular, we are not aware of previous results on the basis properties of Fučík eigenfunctions
of the problem (1.1) for the one-dimensional Laplacian.
Fučík eigenfunctions associated with Fučík eigenvalues (𝛼, 𝛽)of the problem (1.1) can be
interpreted as specific alterations of the sine functions that, in general, do not lead to a system
of dilated or orthogonal functions when 𝛼̸=𝛽, although they preserve a certain periodic
structure. By establishing explicit formulas for the distances between Fučík eigenfunctions and
corresponding sine functions, and making use of the periodic structure, we obtain sequences
of Fučík eigenvalues (𝛼(𝑛), 𝛽(𝑛)) whose corresponding Fučík eigenfunctions form a Riesz basis
in 𝐿2(0, 𝜋). To this end, we employ several classical results from the stability theory of Paley
and Wiener [27], which provides basic information on the basisness of Fučík eigenfunctions,
and we leave more developed approaches for further investigation.
Let us outline the structure of this article. In the remainder of the introduction, we rig-
orously introduce normalized Fučík eigenfunctions and Fučík systems as sequences of such
functions in Section 1.1, and we state our main results in Section 1.2. We derive explicit
formulas for the norms of the normalized Fučík eigenfunctions, their distances to the cor-
responding sine functions, and some scalar products between Fučík eigenfunctions and sine
functions in Section 2. In Section 3, we establish upper bounds on the distances which are
essential for the proof of Theorem 1.4. The proof of Theorem 1.7 is given in Section 4, and
the proof of Theorem 1.9 is given in Section 5. We conclude the article with final remarks in
Section 6. Appendices Aand Bcontain several auxiliary technical details.
1.1 Fučík eigenvalues and eigenfunctions. The Fučík spectrum Σ(0, 𝜋)⊂R2of the
linear Dirichlet Laplacian in one dimension is defined as the set of all pairs of parameters
(𝛼, 𝛽)∈R2for which the problem
−𝑢′′(𝑥) = 𝛼𝑢+(𝑥)−𝛽𝑢−(𝑥)in (0, 𝜋),
𝑢(0) = 𝑢(𝜋) = 0,(1.1)
has a non-trivial solution. Here, 𝑢+= max(𝑢, 0) and 𝑢−= max(−𝑢, 0) denote the positive and
negative parts of 𝑢, respectively. Clearly, for 𝛼=𝛽=: 𝜆we obtain the standard eigenvalue

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 3
problem −𝑢′′ =𝜆𝑢 with zero Dirichlet boundary conditions, which possesses the complete
sequence of eigenvalues 𝜆𝑛=𝑛2,𝑛∈N, and the sine functions 𝜑𝑛= sin(𝑛𝑥)as corresponding
eigenfunctions. Since 𝜑1has definite sign, the lines {1} × Rand R× {1}are the trivial part
of Σ(0, 𝜋)and we have
Σ(0, 𝜋)∖(({1} × R)∪(R× {1})) ⊂ {(𝛼, 𝛽)∈R2:𝛼 > 1, 𝛽 > 1}
by the variational characterization of the first eigenvalue 𝜆1= 1. Any (𝛼, 𝛽)∈Σ(0, 𝜋)is called
Fučík eigenvalue and any corresponding non-zero solution of (1.1) is called Fučík eigenfunction.
Hereinafter, under a solution of (1.1) we mean the classical (i.e., 𝐶2-regular) solution of that
problem. Any Fučík eigenfunction consists of positive bumps of length 𝑙1=𝜋
√𝛼alternating
with negative bumps of length 𝑙2=𝜋
√𝛽, and has the form of translations and multiples of
sin(√𝛼𝑥)at positive bumps and sin(√𝛽𝑥)at negative bumps. For each even number 𝑛of
bumps there exists one curve
Γ𝑛=(𝛼, 𝛽)∈R2:𝑛
2
𝜋
√𝛼+𝑛
2
𝜋
√𝛽=𝜋
in the Fučík spectrum Σ(0, 𝜋)that contains (𝜆𝑛, 𝜆𝑛), while for each odd number 𝑛≥3of
bumps there are two curves
Γ𝑛=(𝛼, 𝛽)∈R2:𝑛+ 1
2
𝜋
√𝛼+𝑛−1
2
𝜋
√𝛽=𝜋,

Γ𝑛=(𝛼, 𝛽)∈R2:𝑛−1
2
𝜋
√𝛼+𝑛+ 1
2
𝜋
√𝛽=𝜋
in Σ(0, 𝜋)intersecting at (𝜆𝑛, 𝜆𝑛), see Figure 1. These curves completely describe the non-
trivial part of Σ(0, 𝜋), see [17, Lemma 2.8]. Observe that if 𝑢is a Fučík eigenfunction for the
pair of parameters (𝛼, 𝛽), then so is 𝑡𝑢 for any 𝑡 > 0, while −𝑡𝑢 is a Fučík eigenfunction for
(𝛽, 𝛼). As a consequence, since (𝛼, 𝛽 )∈Γ𝑛for odd 𝑛implies (𝛽, 𝛼)∈
Γ𝑛, we will neglect the
curve 
Γ𝑛from our further consideration. Moreover, notice that if 𝑢is a Fučík eigenfunction
for the pair of parameters (𝛼, 𝛽)∈Γ𝑛for even 𝑛, then 𝑣(𝑥) = 𝑢(𝜋−𝑥)is also a Fučík
eigenfunction for (𝛼, 𝛽)and 𝑣̸=𝑢.
Figure 1. Several curves of the Fučík spectrum
In order to uniquely specify a Fučík eigenfunction for each point of Σ(0, 𝜋), let us introduce
the following special choice of Fučík eigenfunctions, see Figure 2.

4 F. BAUSTIAN AND V. BOBKOV
Definition 1.1. Let 𝑛≥2and (𝛼, 𝛽 )∈Γ𝑛. The normalized Fučík eigenfunction 𝑓𝑛
𝛼,𝛽 is the
𝐶2-solution of the boundary value problem (1.1) with (𝑓𝑛
𝛼,𝛽)′(0) >0which is normalized by
‖𝑓𝑛
𝛼,𝛽‖∞= sup
𝑥∈[0,𝜋]|𝑓𝑛
𝛼,𝛽(𝑥)|= 1.
For 𝑛= 1, we set 𝑓1
𝛼,𝛽 =𝜑1for every (𝛼, 𝛽)∈({1} × R)∪(R× {1}).
The normalized Fučík eigenfunctions can be described more explicitly by the following
piecewise definition. Let 𝑛≥2. For 𝛼≥𝑛2≥𝛽we have
𝑓𝑛
𝛼,𝛽(𝑥) = √𝛽
√𝛼sin(√𝛼(𝑥−𝑘𝑙)) for 𝑘𝑙 ≤𝑥 < 𝑘𝑙 +𝑙1,
−sin(√𝛽(𝑥−𝑘𝑙 −𝑙1)) for 𝑘𝑙 +𝑙1≤𝑥 < (𝑘+ 1)𝑙, (1.2)
and for 𝛽 > 𝑛2> 𝛼 we have
𝑓𝑛
𝛼,𝛽(𝑥) = sin(√𝛼(𝑥−𝑘𝑙)) for 𝑘𝑙 ≤𝑥 < 𝑘𝑙 +𝑙1,
−√𝛼
√𝛽sin(√𝛽(𝑥−𝑘𝑙 −𝑙1)) for 𝑘𝑙 +𝑙1≤𝑥 < (𝑘+ 1)𝑙, (1.3)
where 𝑙=𝑙1+𝑙2and 𝑘∈N0=N∪ {0}. Notice that (1.2) and (1.3) define 𝑓𝑛
𝛼,𝛽 on the whole
R+. We also remark that 𝑓𝑛
𝛼,𝛽 ̸∈ 𝐶3[0, 𝜋]provided 𝛼̸=𝛽.
(a) 𝑛= 2 (b) 𝑛= 3
Figure 2. A normalized Fučík eigenfunction 𝑓𝑛
𝛼,𝛽 (solid) and sin(𝑛𝑥)(dotted)
In our purpose to form a basis in 𝐿2(0, 𝜋)that consists solely of normalized Fučík eigen-
functions, we pick one normalized Fučík eigenfunction from each curve Γ𝑛,𝑛≥2, and the sine
function as the normalized Fučík eigenfunction of the trivial part of Σ(0, 𝜋). In this regard,
the following definition will be useful.
Definition 1.2. We define a Fučík system 𝐹𝛼,𝛽 ={𝑓𝑛
𝛼(𝑛),𝛽(𝑛)}as a sequence of normal-
ized Fučík eigenfunctions with the mappings 𝛼, 𝛽 :N→Rsatisfying 𝛼(1) = 𝛽(1) = 1 and
(𝛼(𝑛), 𝛽(𝑛)) ∈Γ𝑛for every 𝑛≥2.
1.2 Main results. In this section, we summarize our main results for the basisness of
Fučík systems given by Definition 1.2. We present growth conditions on the mappings 𝛼, 𝛽
which guarantee that the Fučík system 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋). We call a complete
system {𝜓𝑛}in 𝐿2(0, 𝜋)aRiesz basis if there exist positive constants 𝑐, 𝐶 > 0such that the
inequalities
𝑐
𝑁

𝑛=1 |𝛼𝑛|2≤⃦
⃦
𝑁

𝑛=1
𝛼𝑛𝜓𝑛⃦
⃦≤𝐶
𝑁

𝑛=1 |𝛼𝑛|2

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 5
are satisfied for arbitrary 𝑁∈Nand any constants 𝛼1, . . . , 𝛼𝑁. In fact, Riesz bases are images
of an orthonormal basis under a linear homeomorphism. Several equivalent characterizations
of a Riesz basis can be found in [36, Theorem 9]. Hereinafter, ‖·‖ denotes the standard norm
in 𝐿2(0, 𝜋).
We make use of several methods from the stability theory of Paley and Wiener to show
that a Fučík system 𝐹𝛼,𝛽 inherits the basis properties from the system of sine functions {𝜑𝑛}
provided that the two sequences are sufficiently close to each other. There exist various
concepts of nearness between systems of functions, among which we will be interested in the
following two most classical notions, see, e.g., [34].
Definition 1.3. Let {𝜙𝑛}and {𝜓𝑛}be two sequences of functions. The sequence {𝜓𝑛}is
quadratically near to {𝜙𝑛}if
∞

𝑛=1 ‖𝜙𝑛−𝜓𝑛‖2=𝑟 < ∞(1.4)
for a constant 𝑟≥0, and strongly quadratically near if (1.4) holds for 𝑟 < 1. The sequence
{𝜓𝑛}is Paley-Wiener near to {𝜙𝑛}if there exits a constant 𝜆∈(0,1) such that
⃦
⃦
𝑁

𝑛=1
𝛼𝑛(𝜙𝑛−𝜓𝑛)⃦
⃦< 𝜆⃦
⃦
𝑁

𝑛=1
𝛼𝑛𝜙𝑛⃦
⃦
holds for arbitrary 𝑁∈Nand any constants 𝛼1, . . . , 𝛼𝑁.
We introduce our results, each one connected to one of these nearness concepts. The first
general result, Theorem 1.4, allows a rather free choice of the Fučík system 𝐹𝛼,𝛽 and utilizes
the strong quadratic nearness of that system to the sine functions {𝜑𝑛}. In Theorem 1.7,
we significantly improve the constants of Theorem 1.4 by means of the quadratic nearness of
special Fučík systems 𝐹𝛼,𝛽 with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for odd 𝑛to {𝜑𝑛}. For the final result the
normalized Fučík eigenfunctions 𝑓𝑛
𝛼,𝛽 for even 𝑛are chosen in such a way that they form a
sequence of dilated functions, while for odd 𝑛we just pick the sine functions as in the previous
case. This specific choice of the Fučík system allows us to apply the separation of variables
approach of Duffin and Eachus [15] in order to establish the Paley-Wiener nearness to the
sine functions. In view of the nature of Riesz bases, both approaches are intrinsically based
on the construction of a bounded invertible operator 𝑇:𝐿2(0, 𝜋)→𝐿2(0, 𝜋)which maps the
trigonometric system {𝜑𝑛}to the Fučík system 𝐹𝛼,𝛽 .
The basisness of systems that are quadratically near to a complete orthonormal system
was first studied by Bary in [2]. A system {𝜓𝑛}which is quadratically near to a complete
orthonormal system {𝜙𝑛}is a Riesz basis provided that it is 𝜔-linearly independent, i.e., if
the strong convergence
∞

𝑛=1
𝜂𝑛𝜓𝑛= lim
𝑚→∞ ‖
𝑚

𝑛=1
𝜂𝑛𝜓𝑛‖= 0
for a sequence of scalars {𝜂𝑛}implies 𝜂𝑛= 0 for every 𝑛∈N. A proof of this stability result
by means of compact operators is given, e.g., in [21, Theorem V-2.20]. If the system {𝜓𝑛}
satisfies the more restrictive assumption of being strongly quadratically near to a complete
orthonormal system {𝜙𝑛}, then {𝜓𝑛}is also a Riesz basis, see, e.g., [21, Corollary V-2.22].
We establish a summation criterium for bounds on the mappings 𝛼and 𝛽of the Fučík system

6 F. BAUSTIAN AND V. BOBKOV
𝐹𝛼,𝛽 that yields the basisness of that system by means of the strong quadratic nearness to the
system of sine functions {𝜑𝑛}.
Theorem 1.4. Let 𝐹𝛼,𝛽 be a Fučík system. For any natural 𝑛≥2, we set
𝐶𝑛(𝑥, 𝑦) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
4(3 + 𝜋2)𝜋
9max(√𝑥, √𝑦)
𝑛−12
for even 𝑛,
4𝜋𝑛2(𝑛2+ 1)
(𝑛−1)4√𝑥
𝑛−12
for odd 𝑛with 𝑥≥𝑦,
5𝜋𝑛2(𝑛2+ 1)
(𝑛+ 1)4√𝑦
𝑛−12
for odd 𝑛with 𝑦 > 𝑥.
(1.5)
If the summation formula ∞

𝑛=2
𝐶𝑛(𝛼(𝑛), 𝛽(𝑛)) <𝜋
2(1.6)
is satisfied, then 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
The definition (1.5) of 𝐶𝑛is given by the bounds (3.2), (3.4), (3.5), (3.6) on the distances
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2that we will derive in Section 3below. The summation formula (1.6) guarantees
that the Fučík system 𝐹𝛼,𝛽 is quadratically near to the system of sine functions {𝜑𝑛}in the
sense that ∞

𝑛=1 ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2<𝜋
2.(1.7)
This inequality implies that the rescaled system 2/𝜋 𝐹𝛼,𝛽 is strongly quadratically near to
the complete orthonormal system {2/𝜋 𝜑𝑛}in 𝐿2(0, 𝜋), and thus it is a Riesz basis by [21,
Corollary V-2.22]. Hence, the initial Fučík system 𝐹𝛼,𝛽 is also a Riesz basis in 𝐿2(0, 𝜋).
We can make use of Theorem 1.4 to give asymptotic bounds on the mappings 𝛼and 𝛽.
Corollary 1.5. Let 𝐹𝛼,𝛽 be a Fučík system and let 𝜀 > 0be fixed. Let the mappings 𝛼and 𝛽
satisfy
max 𝛼(𝑛),𝛽(𝑛)≤𝑛+√𝑐𝑛𝑛(1−𝜀)/2(1.8)
for every 𝑛≥2with non-negative constants
𝑐𝑛<9
8(3 + 𝜋2)·1
𝜁(1 + 𝜀)−1for even 𝑛,
𝑐𝑛<(𝑛−1)4
8𝑛2(𝑛2+ 1) ·1
𝜁(1 + 𝜀)−1for odd 𝑛with 𝛼(𝑛)≥𝛽(𝑛),(1.9)
𝑐𝑛<(𝑛+ 1)4
10𝑛2(𝑛2+ 1) ·1
𝜁(1 + 𝜀)−1for odd 𝑛with 𝛽(𝑛)> 𝛼(𝑛),(1.10)
where 𝜁is the Riemann zeta function defined by 𝜁(𝑠) = ∞
𝑛=1 1
𝑛𝑠. Then 𝐹𝛼,𝛽 is a Riesz basis
in 𝐿2(0, 𝜋).
Remark 1.6. The upper bounds (1.9) and (1.10) can be replaced by the following weaker
ones which are independent from 𝑛:
𝑐𝑛<1
46 ·1
𝜁(1 + 𝜀)−1for odd 𝑛with 𝛼(𝑛)≥𝛽(𝑛),(1.9’)

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 7
𝑐𝑛<1
10 ·1
𝜁(1 + 𝜀)−1for odd 𝑛with 𝛽(𝑛)> 𝛼(𝑛).(1.10’)
On Figure 3we depict two regions of Fučík eigenvalues described by the inequality (1.8) with
the uniform constants 𝑐𝑛given by (1.9’).
(a) 𝜖= 0.1(b) 𝜖= 0.5
Figure 3. Shaded regions depict the result of Corollary 1.5 with 𝑐𝑛=1
46 ·
1
𝜁(1+𝜀)−1
Clearly, the general result given by Theorem 1.4 also covers such Fučík systems 𝐹𝛼,𝛽 in which
only some Fučík eigenfunctions differ from the sine functions. However, in the particular case
when 𝑓𝑛
𝛼,𝛽 =𝜑𝑛for all odd 𝑛, we can obtain the following stronger result.
Theorem 1.7. Let 𝐹𝛼,𝛽 be a Fučík system with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for every odd 𝑛. If the
mappings 𝛼and 𝛽satisfy
∞

𝑛=2 max(𝛼(𝑛),𝛽(𝑛))
𝑛−12
<∞,(1.11)
then 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
Theorem 1.7 will be proven in Section 4by showing that the corresponding Fučík system
𝐹𝛼,𝛽 is 𝜔-linearly independent and it is quadratically near to the system of sine functions
{𝜑𝑛}.
Corollary 1.8. Let 𝐹𝛼,𝛽 be a Fučík system with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for every odd 𝑛. Let
𝜀, 𝑐 > 0be fixed. If the mappings 𝛼and 𝛽satisfy
max 𝛼(𝑛),𝛽(𝑛)≤𝑛+√𝑐 𝑛(1−𝜖)/2(1.12)
for any even 𝑛, then 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
Let us now discuss the basisness of Fučík systems by means of the Paley-Wiener nearness to
the system of sine functions. We consider a Fučík system 𝐹𝛼,𝛽 for which the points (𝛼(𝑛), 𝛽(𝑛))

8 F. BAUSTIAN AND V. BOBKOV
for even 𝑛are on a line through the origin and, as in Theorem 1.7, the normalized Fučík
eigenfunctions 𝑓𝑛
𝛼,𝛽 are just 𝜑𝑛for odd 𝑛. We apply the method of separation of variables
from [15] to this specific Fučík system to obtain mappings 𝛼and 𝛽with better asymptotics
as 𝑛→ ∞ than in Theorem 1.7, see Figure 4.
Theorem 1.9. Let 𝐹𝛼,𝛽 be a Fučík system with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for every odd 𝑛and
𝛼(𝑛) = 𝑛2𝛾
4, 𝛽(𝑛) = 𝑛2𝛾
(2√𝛾−2)2(1.13)
for every even 𝑛, where 𝛾∈[4,5.682] is an arbitrary fixed constant. Then 𝐹𝛼,𝛽 is a Riesz basis
in 𝐿2(0, 𝜋).
The choice of the Fučík eigenfunctions for even 𝑛in Theorem 1.9 guarantees that these
functions form a dilated system in the sense that
𝑓𝑛
𝛼(𝑛),𝛽(𝑛)(𝑥) = 𝑓2
𝛼(2),𝛽(2) 𝑛𝑥
2for any even 𝑛.
This property is important in the proof of Theorem 1.9 which we give in Section 5.
The formulas (1.13) in Theorem 1.9 obviously guarantee that 𝛼(𝑛)> 𝛽(𝑛)for even 𝑛.
Moreover, the points (𝛼(𝑛), 𝛽(𝑛)) defined by (1.13) are on the line
𝛽=4𝛼
(2√𝛾−2)2.(1.14)
We observe that (1.13) for the mapping 𝛼can be written as
𝛼(𝑛) = 𝑛+√𝛾
2−1𝑛for any even 𝑛,
which provides a better asymptotic than (1.12) of Corollary 1.8. Notice also that Theorem
1.9 remains valid if we exchange 𝛼and 𝛽due to the symmetry properties of the Fučík curves
Γ𝑛with even 𝑛.
(a) 𝜖= 0.1,
𝑐= 0.25;𝛾= 5.6
(b) 𝜖= 0.5,
𝑐= 0.4;𝛾= 5.6
Figure 4. Comparison of the results of Corollary 1.8 and Theorem 1.9

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 9
Let us emphasize that constants which appear in the function 𝐶𝑛in Theorem 1.4 and
the admissible range for the constant 𝛾in Theorem 1.9 are non-optimal since they follow
from a series of estimates convenient for simplification of the derived expressions. Thus, we
anticipate that these constants might be substantially improved. On the other hand, the
asymptotic growth rate of 𝐶𝑛(𝛼(𝑛), 𝛽(𝑛)) is expected to be sharp for the quadratic nearness
considerations.
2. Norms – distances – scalar products
In this section, we derive explicit expressions for the 𝐿2-norms of the normalized Fučík
eigenfunctions 𝑓𝑛
𝛼,𝛽, the distances ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2which are important for the proof of Theorem
1.4, and some scalar products ⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑚⟩which will be used in the proof of Theorem 1.7. We
write these formulas for the case 𝛼≥𝑛2≥𝛽in dependence of 𝛼and 𝑛, and for the case
𝛽 > 𝑛2> 𝛼 in dependence of 𝛽and 𝑛. We will thoroughly treat only the first case, and omit
details for the second case to shorten the exposition.
Recall the following notations from Section 1.1:
𝑙1=𝜋
√𝛼, 𝑙2=𝜋
√𝛽, 𝑙 =𝑙1+𝑙2.
Moreover, we have
𝛼=𝑛2𝛽
(2√𝛽−𝑛)2and 𝛽=𝑛2𝛼
(2√𝛼−𝑛)2
for all points (𝛼, 𝛽)∈Γ𝑛with even 𝑛, and
𝛼=(𝑛+ 1)2𝛽
(2√𝛽−(𝑛−1))2and 𝛽=(𝑛−1)2𝛼
(2√𝛼−(𝑛+ 1))2
for all points (𝛼, 𝛽)∈Γ𝑛with odd 𝑛.
2.1 The case 𝛼≥𝑛2≥𝛽.We begin with the derivation of the norms of the normalized
Fučík eigenfunctions 𝑓𝑛
𝛼,𝛽. We obtain
‖𝑓𝑛
𝛼,𝛽‖2=𝑛
2𝑙1
0
𝛽
𝛼sin2(√𝛼𝑥) d𝑥+𝑛
2𝑙2
0
sin2(𝛽𝑥) d𝑥=𝑛
4
𝛽
𝛼
𝜋
√𝛼+𝑛
4
𝜋
√𝛽
=𝜋𝑛3
4√𝛼(2√𝛼−𝑛)2+1
2𝜋−𝑛
2
𝜋
√𝛼=𝜋
2−𝜋𝑛(√𝛼−𝑛)
(2√𝛼−𝑛)2
for even 𝑛. In a similar way, we have
‖𝑓𝑛
𝛼,𝛽‖2=𝑛+ 1
4
𝛽
𝛼
𝜋
√𝛼+𝑛−1
4
𝜋
√𝛽=𝜋
2−𝜋(𝑛+ 1)(√𝛼−1)(√𝛼−𝑛)
√𝛼(2√𝛼−(𝑛+ 1))2
for odd 𝑛. Notice that all the sine functions have the same norm ‖𝜑𝑛‖=𝜋
2,𝑛∈N.
Let us now derive expressions for the distances ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2. We start with the case 𝑛= 2.
Using the formulas from Appendix A, we get
‖𝑓2
𝛼,𝛽 −𝜑2‖2=𝑙1
0√𝛽
√𝛼sin(√𝛼 𝑥)−sin(2𝑥)2
d𝑥+𝜋
𝑙1sin(𝛽(𝑥−𝑙1)) + sin(2𝑥)2d𝑥

10 F. BAUSTIAN AND V. BOBKOV
=𝜋
2+1
2
𝛽
𝛼𝑙1+1
2𝑙2+ 2 √𝛽
4−𝛼sin(2𝑙1)−2√𝛽
4−𝛽sin(2𝑙1)
=𝜋
2+1
2
𝛽
𝛼
𝜋
√𝛼+1
2
𝜋
√𝛽+ 2𝛽𝛼−𝛽
(4 −𝛼)(4 −𝛽)sin 2𝜋
√𝛼
=𝜋−𝜋2(√𝛼−2)
(2√𝛼−2)2−4𝛼2
(2√𝛼−2)(3√𝛼−2)(√𝛼+ 2)
sin 2𝜋
√𝛼
√𝛼−2.(2.1)
Now, for a general even 𝑛we have
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2=
𝑛
2−1

𝑖=0 (𝑖+1)𝑙
𝑖𝑙
(𝑓𝑛
𝛼,𝛽(𝑥)−sin(𝑛𝑥))2d𝑥=𝑛
2𝑙
0
(𝑓𝑛
𝛼,𝛽(𝑥)−sin(𝑛𝑥))2d𝑥
=𝜋
0
(𝑓2
4𝛼/𝑛2,4𝛽/𝑛2(𝑦)−sin(2𝑦))2d𝑦=‖𝑓2
4𝛼/𝑛2,4𝛽/𝑛2−𝜑2‖2,
where we used that 𝑓𝛼,𝛽(𝑥)−sin(𝑛𝑥)realizes the same values on each interval (𝑖𝑙, (𝑖+ 1)𝑙)for
every 𝑖∈N. Therefore, we deduce from (2.1) that
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2=𝜋−𝜋𝑛(√𝛼−𝑛)
(2√𝛼−𝑛)2−4𝛼2
(2√𝛼−𝑛)(3√𝛼−𝑛)(√𝛼+𝑛)
sin 𝑛𝜋
√𝛼
√𝛼−𝑛(2.2)
holds for any even 𝑛.
Now we consider the case of odd 𝑛≥3. This case requires more extensive calculations, and
we put the derivation of the following formula to Appendix A. We have
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2=𝜋−𝜋(𝑛+ 1)(√𝛼−1)
√𝛼(2√𝛼−(𝑛+ 1))2(√𝛼−𝑛)
−16(𝑛−1)√𝛼3
(2√𝛼−(𝑛+ 1))
(√𝛼−1)
(𝑛+√𝛼)(𝑛+ 1)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))×
×
cos 𝜋
2
𝑛
√𝛼cos 𝜋
2
𝑛2+𝑛−2√𝛼
(𝑛−1)√𝛼
(√𝛼−𝑛) sin 𝜋√𝛼−𝑛
(𝑛−1)√𝛼.
(2.3)
for any odd 𝑛≥3. Notice that the assumption 𝛼≥𝑛2guarantees that each multiplier on the
right-hand side of (2.3) is nonnegative.
Finally, we derive some scalar products ⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑚⟩needed for the proof of Theorem 1.7. For
the special case 𝑛=𝑚≥2, we can express the scalar product in terms of the formulas above
as
⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑛⟩=1
2‖𝑓𝑛
𝛼,𝛽‖2+‖𝜑𝑛‖2− ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2,(2.4)
and then easily obtain
⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑛⟩=2𝛼2
(2√𝛼−𝑛)(3√𝛼−𝑛)(√𝛼+𝑛)
sin 𝑛𝜋
√𝛼
√𝛼−𝑛(2.5)

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 11
for even 𝑛, and
⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑛⟩=8(𝑛−1)√𝛼3
(2√𝛼−(𝑛+ 1))
(√𝛼−1)
(𝑛+√𝛼)(𝑛+ 1)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))×
×
cos 𝜋
2
𝑛
√𝛼cos 𝜋
2
𝑛2+𝑛−2√𝛼
(𝑛−1)√𝛼
(√𝛼−𝑛) sin 𝜋√𝛼−𝑛
(𝑛−1)√𝛼
for odd 𝑛≥3.
If 𝑛̸=𝑚, then the scalar product ⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑚⟩vanishes for some combinations of 𝑛and 𝑚.
In particular, we have ⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑚⟩= 0 for odd 𝑛and even 𝑚by a simple symmetry argument.
When both 𝑛and 𝑚are even with 𝑛>𝑚, the scalar product also vanishes. Indeed, we get
⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑚⟩=(𝛽−𝛼)√𝛽
(𝛼−𝑚2)(𝛽−𝑚2)sin 𝑛
2−1𝑚
2𝑙+ sin 𝑛
2−1𝑚
2𝑙+𝑚𝑙1sin 𝑚𝑛
4𝑙
sin 𝑚
2𝑙= 0
(2.6)
due to 𝑛
2𝑙=𝜋. Details on the derivation of this formula are given in Appendix A.
2.2 The case 𝛽 > 𝑛2> 𝛼.For the norm of the normalized Fučík eigenfunctions 𝑓𝑛
𝛼,𝛽, we
have
‖𝑓𝑛
𝛼,𝛽‖2=𝑛
2𝑙1
0
sin2(√𝛼𝑥) d𝑥+𝑛
2𝑙2
0
𝛼
𝛽sin2(𝛽𝑥) d𝑥
=𝑛
4
𝜋
√𝛼+𝑛
4
𝛼
𝛽
𝜋
√𝛽=𝜋
2−𝜋𝑛(√𝛽−𝑛)
(2√𝛽−𝑛)2
for even 𝑛, and
‖𝑓𝑛
𝛼,𝛽‖2=𝑛+ 1
4
𝜋
√𝛼+𝑛−1
4
𝛼
𝛽
𝜋
√𝛽=𝜋
2−𝜋(𝑛−1)(√𝛽+ 1)(√𝛽−𝑛)
√𝛽(2√𝛽−(𝑛−1))2
for odd 𝑛.
Now we derive expressions for the distances ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2. As in Section 2.1, we start with
the case 𝑛= 2:
‖𝑓2
𝛼,𝛽 −𝜑2‖2=𝑙1
0
(sin(√𝛼 𝑥)−sin(2𝑥))2d𝑥+𝜋
𝑙1√𝛼
√𝛽sin(𝛽(𝑥−𝑙1)) + sin(2𝑥)2
d𝑥
=𝜋
2+1
2
𝜋
√𝛼+1
2
𝛼
𝛽
𝜋
√𝛽+ 2√𝛼𝛼−𝛽
(4 −𝛼)(4 −𝛽)sin 2𝜋
√𝛼
=𝜋−𝜋2(√𝛽−2)
(2√𝛽−2)2−4𝛽2
(2√𝛽−2)(3√𝛽−2)(√𝛽+ 2)
sin 2𝜋
√𝛽
√𝛽−2.
Therefore, we obtain
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2=𝜋−𝜋𝑛(√𝛽−𝑛)
(2√𝛽−𝑛)2−4𝛽2
(2√𝛽−𝑛)(3√𝛽−𝑛)(√𝛽+𝑛)
sin 𝑛𝜋
√𝛽
√𝛽−𝑛(2.7)

12 F. BAUSTIAN AND V. BOBKOV
for even 𝑛. Furthermore, we have
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2=𝜋−𝜋(𝑛−1)(√𝛽+ 1)
√𝛽(2√𝛽−(𝑛−1))2(𝛽−𝑛)
−16(𝑛+ 1)√𝛽3
(2√𝛽−(𝑛−1))
(√𝛽+ 1)
(𝑛+√𝛽)(𝑛−1)((3𝑛+ 1)√𝛽−𝑛(𝑛−1))×
×
cos 𝜋
2
𝑛
√𝛽cos 𝜋
2
(2√𝛽+𝑛2−𝑛)
(𝑛+1)√𝛽
(√𝛽−𝑛) sin 𝜋𝑛(√𝛽+1)
(𝑛+1)√𝛽
(2.8)
for odd 𝑛≥3. As in (2.3), each multiplier on the right-hand side of (2.8) is nonnegative in
view of the assumption 𝛽 > 𝑛2.
Finally, for the derivation of the scalar product ⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑛⟩, we use (2.4) and get
⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑛⟩=2𝛽2
(2√𝛽−𝑛)(3√𝛽−𝑛)(√𝛽+𝑛)
sin 𝑛𝜋
√𝛽
√𝛽−𝑛(2.9)
for even 𝑛, and
⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑛⟩=8(𝑛+ 1)√𝛽3
(2√𝛽−(𝑛−1))
(√𝛽+ 1)
(𝑛+√𝛽)(𝑛−1)((3𝑛+ 1)√𝛽−𝑛(𝑛−1))×
×
cos 𝜋
2
𝑛
√𝛽cos 𝜋
2
(2√𝛽+𝑛2−𝑛)
(𝑛+1)√𝛽
(√𝛽−𝑛) sin 𝜋𝑛(√𝛽+1)
(𝑛+1)√𝛽
for odd 𝑛≥3. The scalar product ⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑚⟩vanishes for even 𝑚provided that either 𝑛is
odd, or 𝑛is even with 𝑛>𝑚, as in the case 𝛼≥𝑛2≥𝛽in Section 2.1.
3. Asymptotics of distances
In this section, we establish upper bounds on the distances ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2which allows to
describe their asymptotic behaviour for 𝑛→ ∞, as well as for √𝛼→𝑛or √𝛽→𝑛.
3.1 The case 𝛼≥𝑛2≥𝛽.We begin with estimating the distance ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2given by
(2.2) for even 𝑛. Using the lower bound
sin 𝑥≥𝑥−𝑥3
6=𝑥
6(√6−𝑥)(√6 + 𝑥), 𝑥 ≥0,(3.1)
and the assumption 𝛼≥𝑛2, we see that
sin 𝑛𝜋
√𝛼= sin 𝜋−𝜋𝑛
√𝛼≥𝜋
6√𝛼3(√𝛼−𝑛)((√6−𝜋)√𝛼+𝑛𝜋)((√6 + 𝜋)√𝛼−𝑛𝜋).
Therefore, we get
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤𝜋−𝜋𝑛(√𝛼−𝑛)
(2√𝛼−𝑛)2−2𝜋√𝛼((√6−𝜋)√𝛼+𝑛𝜋)((√6 + 𝜋)√𝛼−𝑛𝜋)
3(2√𝛼−𝑛)(3√𝛼−𝑛)(√𝛼+𝑛)
=𝜋
3
4(3 + 𝜋2)𝛼+√𝛼𝑛(15 −2𝜋2)−6𝑛2
(2√𝛼−𝑛)2(3√𝛼−𝑛)(√𝛼+𝑛)(√𝛼−𝑛)2

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 13
≤𝜋
3
4(3 + 𝜋2)𝛼−(2𝜋2−9)𝑛2
𝑛2(3√𝛼−𝑛)(√𝛼+𝑛)(√𝛼−𝑛)2
=4(3 + 𝜋2)𝜋
93√𝛼−3√2𝜋2−9
2√3+𝜋2𝑛√𝛼+√2𝜋2−9
2√3+𝜋2𝑛
𝑛2(3√𝛼−𝑛)(√𝛼+𝑛)(√𝛼−𝑛)2
≤4(3 + 𝜋2)𝜋
9
(√𝛼−𝑛)2
𝑛2=4(3 + 𝜋2)𝜋
9√𝛼
𝑛−12
(3.2)
for even 𝑛.
Let us now estimate ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2given by (2.3) for odd 𝑛. Recalling that all multipliers in
(2.3) are nonnegative and applying the rough upper bound
sin 𝜋√𝛼−𝑛
(𝑛−1)√𝛼≤𝜋√𝛼−𝑛
(𝑛−1)√𝛼,
we get
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤𝜋−𝜋(𝑛+ 1)(√𝛼−1)
√𝛼(2√𝛼−(𝑛+ 1))2(√𝛼−𝑛)
−16(𝑛−1)2𝛼2
𝜋(2√𝛼−(𝑛+ 1))
(√𝛼−1)
(𝑛+√𝛼)(𝑛+ 1)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))×
×
cos 𝜋
2
𝑛
√𝛼cos 𝜋
2
𝑛2+𝑛−2√𝛼
(𝑛−1)√𝛼
(√𝛼−𝑛)2.
(3.3)
Then, using (3.1), we have
cos(𝑥) = sin 𝜋
2−𝑥≥1
6𝜋
2−𝑥√6−𝜋
2+𝑥√6 + 𝜋
2−𝑥, 𝑥 ≤𝜋
2,
and hence, by 𝛼 > 𝑛2, we obtain
cos 𝜋
2
𝑛
√𝛼≥𝜋(√𝛼−𝑛)
48√𝛼3((2√6−𝜋)√𝛼+𝜋𝑛)((2√6 + 𝜋)√𝛼−𝜋𝑛)
and
cos 𝜋
2
𝑛2+𝑛−2√𝛼
(𝑛−1)√𝛼≥𝜋(𝑛+ 1)(√𝛼−𝑛)
48(𝑛−1)3√𝛼3((2√6(𝑛−1) −𝜋(𝑛+ 1))√𝛼+𝜋𝑛(𝑛+ 1))
×((2√6(𝑛−1) + 𝜋(𝑛+ 1))√𝛼−𝜋𝑛(𝑛+ 1)).
Substituting these estimates into (3.3), we get
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤𝜋−𝜋(𝑛+ 1)(√𝛼−1)
√𝛼(2√𝛼−(𝑛+ 1))2(√𝛼−𝑛)
−𝜋(√𝛼−1)((2√6−𝜋)√𝛼+𝜋𝑛)((2√6 + 𝜋)√𝛼−𝜋𝑛)
144(𝑛−1)𝛼(2√𝛼−(𝑛+ 1))(𝑛+√𝛼)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))×
×((2√6(𝑛−1) −𝜋(𝑛+ 1))√𝛼+𝜋𝑛(𝑛+ 1))
×((2√6(𝑛−1) + 𝜋(𝑛+ 1))√𝛼−𝜋𝑛(𝑛+ 1))
≤4𝜋𝑛2(𝑛2+ 1)
(𝑛−1)4√𝛼
𝑛−12
≤23𝜋√𝛼
𝑛−12
(3.4)

14 F. BAUSTIAN AND V. BOBKOV
for odd 𝑛. Several intermediate estimates used to derive these upper bounds are given in
Appendix A.
3.2 The case 𝛽 > 𝑛2> 𝛼.Let us estimate ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2given by (2.7) for even 𝑛. Noting
that (2.7) is the same formula as (2.2) up to a replacement of 𝛼by 𝛽, we get
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤4(3 + 𝜋2)𝜋
9√𝛽
𝑛−12
(3.5)
for even 𝑛, as in (3.2).
Now we provide an upper bound on ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2given by (2.8) for odd 𝑛. Using (3.1) as in
Section 3.1 and recalling that 𝛽 > 𝑛2, we estimate the trigonometric terms in (2.8) as follows:
cos 𝜋
2
𝑛
√𝛽≥𝜋(√𝛽−𝑛)
48𝛽3((2√6−𝜋)𝛽+𝜋𝑛)((2√6 + 𝜋)𝛽−𝜋𝑛),
cos 𝜋
2
(2√𝛽+𝑛2−𝑛)
(𝑛+ 1)√𝛽≥𝜋(𝑛−1)(√𝛽−𝑛)
48(𝑛+ 1)3𝛽3((2√6(𝑛+ 1) −𝜋(𝑛−1))𝛽+𝜋𝑛(𝑛−1))
×((2√6(𝑛+ 1) + 𝜋(𝑛−1))𝛽−𝜋𝑛(𝑛−1)),
sin 𝜋𝑛(√𝛽+ 1)
(𝑛+ 1)√𝛽= sin 𝜋−𝜋𝑛(√𝛽+ 1)
(𝑛+ 1)√𝛽= sin 𝜋(√𝛽−𝑛)
(𝑛+ 1)√𝛽≤𝜋(√𝛽−𝑛)
(𝑛+ 1)√𝛽.
Substituting these estimates into (2.8), we deduce that
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤𝜋−𝜋(𝑛−1)(√𝛽+ 1)
√𝛽(2√𝛽−(𝑛−1))2(𝛽−𝑛)
−𝜋(√𝛽+ 1)((2√6−𝜋)√𝛽+𝜋𝑛)((2√6 + 𝜋)√𝛽−𝜋𝑛)
144(𝑛+ 1)𝛽(2√𝛽−(𝑛−1))(𝑛+√𝛽)((3𝑛+ 1)√𝛽−𝑛(𝑛−1))
×((2√6(𝑛+ 1) −𝜋(𝑛−1))𝛽+𝜋𝑛(𝑛−1))
×((2√6(𝑛+ 1) + 𝜋(𝑛−1))𝛽−𝜋𝑛(𝑛−1))
≤5𝜋𝑛2(𝑛2+ 1)
(𝑛+ 1)4√𝛽
𝑛−12
≤5𝜋√𝛽
𝑛−12
(3.6)
for odd 𝑛. Several intermediate estimates in the derivation of these upper bounds can be
found in Appendix A.
4. Proof of Theorem 1.7
To prove Theorem 1.7, we first establish the 𝜔-linear independence of Fučík systems 𝐹𝛼,𝛽
with 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for every odd 𝑛. As the second step, we use the bounds (3.2) and
(3.5) to deduce that the assumption (1.11) guarantees the quadratic nearness of 𝐹𝛼,𝛽 to the
system of sine functions {𝜑𝑛}.
Let {𝜂𝑛}be a sequence of scalars such that
lim
𝑚→∞ ‖
𝑚

𝑛=1
𝜂𝑛𝑓𝑛
𝛼,𝛽‖= 0.

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 15
In order to show that 𝐹𝛼,𝛽 is 𝜔-linear independent, we have to prove 𝜂𝑛= 0 for each 𝑛. Fix
any 𝑘∈Nand consider the functions
𝑔𝑚,𝑘(𝑥) = sin(𝑘𝑥)
𝑚

𝑛=1
𝜂𝑛𝑓𝑛
𝛼,𝛽(𝑥)for 𝑚∈N.
We see that
⃒⃒⃒𝜋
0
𝑔𝑚,𝑘(𝑥) d𝑥⃒⃒⃒≤𝜋
0|sin(𝑘𝑥)|⃒⃒⃒
𝑚

𝑛=1
𝜂𝑛𝑓𝑛
𝛼,𝛽(𝑥)⃒⃒⃒d𝑥≤𝜋
2‖
𝑚

𝑛=1
𝜂𝑛𝑓𝑛
𝛼,𝛽‖ → 0as 𝑚→ ∞,
which yields
∞

𝑛=1
𝜂𝑛⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑘⟩= lim
𝑚→∞ 𝜋
0
𝑔𝑚,𝑘(𝑥) d𝑥= 0.(4.1)
Taking 𝑘= 2 and recalling that ⟨𝑓𝑛
𝛼,𝛽, 𝜑2⟩= 0 for any 𝑛̸= 2 (see Section 2), we get
𝜂2⟨𝑓2
𝛼,𝛽, 𝜑2⟩= 0. Since ⟨𝑓2
𝛼,𝛽, 𝜑2⟩ ̸= 0 by (2.5) and (2.9), we obtain 𝜂2= 0. By an induc-
tive argument, we derive in the same way 𝜂2𝑚= 0 for every 𝑚≥1. Let us remark that until
now we did not use the special form of the Fučík system 𝐹𝛼,𝛽 , namely, 𝛼(𝑛) = 𝛽(𝑛) = 𝑛2for
all odd 𝑛≥3. Since this assumption reads as 𝑓𝑛
𝛼,𝛽 =𝜑𝑛for any odd 𝑛, we further obtain
from (4.1) that 𝜂2𝑚−1= 0 for every 𝑚≥1by the orthogonality of {𝜑𝑛}. This concludes the
𝜔-linear independence of the Fučík system 𝐹𝛼,𝛽.
Using the uppers bounds (3.2) and (3.5) on the distances ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2, we get
∞

𝑛=1 ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤4(3 + 𝜋2)𝜋
9
∞

𝑛=2 max(𝛼(𝑛),𝛽(𝑛))
𝑛−12
.
Since the right-hand side is bounded in view of the assumption (1.11), the Fučík system 𝐹𝛼,𝛽
is quadratically near to the system of sine functions {𝜑𝑛}. Therefore, the rescaled system
2/𝜋 𝐹𝛼,𝛽 is quadratically near to the complete orthonormal system {2/𝜋 𝜑𝑛}, and hence
it is a Riesz basis in 𝐿2(0, 𝜋)by [21, Theorem V-2.20]. Clearly, 𝐹𝛼,𝛽 is also a Riesz basis.
5. Proof of Theorem 1.9
In this section, we provide the proof of our third main result, Theorem 1.9. For this purpose,
we use the method of separation of variables from [15] to show that a specific Fučík system 𝐹𝛼,𝛽
is Paley-Wiener near to the system of sine functions {𝜑𝑛}, see Definition 1.2. The classical
result of Paley and Wiener [27] then yields the basisness of 𝐹𝛼,𝛽 in 𝐿2(0, 𝜋).
Recall that we choose 𝐹𝛼,𝛽 to satisfy 𝑓𝑛
𝛼(𝑛),𝛽(𝑛)=𝜑𝑛for any odd 𝑛, and
𝛼(𝑛) = 𝑛2𝛾
4, 𝛽(𝑛) = 𝑛2𝛾
(2√𝛾−2)2(1.13)
for every even 𝑛, where 𝛾is a fixed constant in the interval [4,5.682]. Denoting 𝑙1(𝑛) = 𝜋
√𝛼(𝑛)
and 𝑙2(𝑛) = 𝜋
√𝛽(𝑛), we see that (1.13) implies
𝑙1(2)
𝑙1(𝑛)=𝑛
2and 𝑙2(2)
𝑙2(𝑛)=𝑛
2

16 F. BAUSTIAN AND V. BOBKOV
for every even 𝑛. It can be easily deduced from the piecewise definition (1.2) that 𝑓𝑛
𝛼(𝑛),𝛽(𝑛)
for even 𝑛form a sequence of dilated functions in the sense that
𝑓𝑛
𝛼(𝑛),𝛽(𝑛)(𝑥) = 𝑓2
𝛼(2),𝛽(2) 𝑛𝑥
2≡𝑓2
𝛾,𝛾/(√𝛾−1)2𝑛𝑥
2.(5.1)
We claim that the assumptions of [15, Theorem D] (with 𝑓𝑛=𝜑𝑛and 𝑔𝑛=𝑓𝑛
𝛼(𝑛),𝛽(𝑛)) are
satisfied for any 𝛾∈[4,5.682], namely, there exist a matrix of constants {𝐶𝑛,𝑘}and a sequence
of bounded linear operators {𝑇𝑘}such that each 𝑓𝑛
𝛼(𝑛),𝛽(𝑛)has the representation
𝑓𝑛
𝛼(𝑛),𝛽(𝑛)(𝑥) = sin(𝑛𝑥) + ∞

𝑘=1
𝐶𝑛,𝑘𝑇𝑘sin(𝑛𝑥),(5.2)
where we have |𝐶𝑛,𝑘 | ≤ 𝑐𝑘and ‖𝑇𝑘‖*≤𝑡𝑘with constants 𝑐𝑘and 𝑡𝑘satisfying ∞
𝑘=1 𝑐𝑘𝑡𝑘<1.
Here, ‖ · ‖*is the operator norm. If our claim is true, then the Fučík system 𝐹𝛼,𝛽 is a Riesz
basis in 𝐿2(0, 𝜋). We remark that although the system {𝜑𝑛}is not orthonormal, [15, Theorem
D] is applicable by simple rescaling arguments since all 𝜑𝑛are of the same norm.
Let 𝑛be even. Then, in view of (5.1), we have
𝑓𝑛
𝛼(𝑛),𝛽(𝑛)(𝑥) = 𝑓2
𝛼(2),𝛽(2) 𝑛𝑥
2=∞

𝑘=1
𝐴𝑘sin 𝑘𝑛𝑥
2,(5.3)
where 𝐴𝑘are the coefficients of the odd Fourier expansion of 𝑓2
𝛼(2),𝛽(2). The comparison of
(5.2) and (5.3) suggests to define bounded linear operators 𝑇𝑘,𝑘∈N, that satisfy the following
property:
𝑇𝑘sin(𝑛𝑥) = sin 𝑘𝑛𝑥
2for every even 𝑛. (5.4)
To this end, for any 𝑔∈𝐿2(0, 𝜋)we define its antiperiodic extension 𝑔*as
𝑔*(𝑥)=(−1)𝜅𝑔(𝑥−𝜋𝜅)for 𝜋𝜅 ≤𝑥≤𝜋(𝜅+ 1), 𝜅 ∈N0.
In particular, we see that if 𝑔(𝑥) = sin(𝑥)for 𝑥∈(0, 𝜋), then 𝑔*(𝑥) = sin(𝑥)for any 𝑥∈R+.
Now we choose 𝑇𝑘:𝐿2(0, 𝜋)→𝐿2(0, 𝜋)as
𝑇𝑘𝑔(𝑥) = 𝑔*𝑘𝑥
2.(5.5)
Clearly, 𝑇2is just the identity operator and each 𝑇𝑘satisfies (5.4). Moreover, each 𝑇𝑘is linear
with the norm ‖𝑇𝑘‖*= 1 for even 𝑘and ‖𝑇𝑘‖*=1+1/𝑘 for odd 𝑘, see Appendix B.
Thus, in accordance with (5.2) and (5.3), for even 𝑛we set 𝐶𝑛,1=𝐴1,𝐶𝑛,2=𝐴2−1, and
𝐶𝑛,𝑘 =𝐴𝑘for 𝑘≥3, while for odd 𝑛we simply choose 𝐶𝑛,𝑘 = 0 for 𝑘∈N. For this choice of
constants, we can set 𝑐1=|𝐴1|,𝑐2=|𝐴2−1|, and 𝑐𝑘=|𝐴𝑘|for 𝑘≥3.
Let us now estimate the constants 𝑐𝑘. Notice that
𝐴𝑘=2
𝜋𝜋
0
𝑓2
𝛼(2),𝛽(2)(𝑥) sin(𝑘𝑥) d𝑥=2
𝜋
𝛾2
√𝛾−1
(2 −√𝛾) sin 𝑘𝜋
√𝛾
(𝑘2−𝛾)(𝑘2(√𝛾−1)2−𝛾).
Thus, for 𝑐1we easily obtain
𝑐1=|𝐴1| ≤ 2
𝜋
𝛾2(√𝛾−2)
(√𝛾−1)2(√𝛾+ 1)(2√𝛾−1).

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 17
For 𝑐2we recall that 𝛾≥4and use the upper bound
sin 2𝜋
√𝛾= sin 𝜋−2𝜋
√𝛾= sin 𝜋(√𝛾−2)
√𝛾≤𝜋(√𝛾−2)
√𝛾(5.6)
to deduce
𝐴2−1 = 2
𝜋
𝛾2
√𝛾−1
sin 2𝜋
√𝛾
(𝛾−4)(3√𝛾−2) −1≤2√𝛾3
(√𝛾−1)(√𝛾+ 2)(3√𝛾−2) −1≤0.
Hence, using (3.1) instead of (5.6), we derive that
𝑐2= 1 −𝐴2≤1−√𝛾((√6−𝜋)√𝛾+ 2𝜋)((√6 + 𝜋)√𝛾−2𝜋)
3(√𝛾−1)(√𝛾+ 2)(3√𝛾−2)
=((3 + 𝜋2)𝛾+ (9 −2𝜋2)√𝛾−6)(√𝛾−2)
3(√𝛾−1)(√𝛾+ 2)(3√𝛾−2) .
Finally, for 𝑐𝑘with 𝑘≥3we get
𝑐𝑘=|𝐴𝑘|=2
𝜋
𝛾2(√𝛾−2)
√𝛾−1⃒⃒sin 𝑘𝜋
√𝛾⃒⃒
(𝑘2−𝛾)(𝑘2−𝛾+𝑘2√𝛾(√𝛾−2)) ≤2
𝜋
𝛾2(√𝛾−2)
√𝛾−1
1
(𝑘2−𝛾)2.
Using the estimates above, we deduce that
∞

𝑘=1
𝑐𝑘‖𝑇𝑘‖*≤
4

𝑘=1
𝑐𝑘‖𝑇𝑘‖*+6
5·2
𝜋
𝛾2(√𝛾−2)
√𝛾−1
∞

𝑘=5
1
(𝑘2−9)2
≤√2·2
𝜋
𝛾2(√𝛾−2)
(√𝛾−1)2(√𝛾+ 1)(2√𝛾−1)
+((3 + 𝜋2)𝛾+ (9 −2𝜋2)√𝛾−6)(√𝛾−2)
3(√𝛾−1)(√𝛾+ 2)(3√𝛾−2)
+4
3·2
𝜋
𝛾2(√𝛾−2)
√𝛾−1
1
(32−𝛾)2+2
𝜋
𝛾2(√𝛾−2)
√𝛾−1
1
(42−𝛾)2
+6
5·2
𝜋
𝛾2(√𝛾−2)
√𝛾−1𝜋2
108 −536741
6350400=: 𝐸(𝛾).
By straightforward calculations, it is not hard to show that each summand in 𝐸(𝛾)is strictly
increasing with respect to 𝛾≥4and 𝐸(4) = 0. At the same time, we have 𝐸(5.682) =
0.9992..., which shows that 𝐸(𝛾)<1for any 𝛾∈[4,5.682]. This completes the proof of
Theorem 1.9.
6. Final remarks
1. The quadratic nearness assumption (1.7) used in the proof of Theorem 1.4 can be
weakened to the inequality
∞

𝑛=1 ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2−|⟨𝑓𝑛
𝛼,𝛽 −𝜑𝑛, 𝑓 𝑛
𝛼,𝛽⟩|2
‖𝑓𝑛
𝛼,𝛽‖2<𝜋
2(6.1)

18 F. BAUSTIAN AND V. BOBKOV
and still guarantee that the Fučík system 𝐹𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋), see [21, Theorem
V-2.21]. Noting that each summand in (6.1) can be written as
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2−‖𝑓𝑛
𝛼,𝛽‖2− ‖𝜑𝑛‖2+‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖22
4‖𝑓𝑛
𝛼,𝛽‖2,(6.2)
one can apply the formulas from Section 2to derive the explicit expression for (6.2) and
estimate it from above in the same way as in Section 3. However, this does not improve the
asymptotic behaviour of the function 𝐶𝑛(𝛼(𝑛), 𝛽(𝑛)) as 𝑛→ ∞ in Theorem 1.4, but only
slightly improves the constants.
2. We anticipate that the 𝜔-linear independence is satisfied for general Fučík systems.
Provided this claim is true, the assumptions of Theorem 1.4 can be significantly weakened in
the sense that the sum in (1.6) only has to converge.
3. In the proof of Theorem 1.9 we used the fact that normalized Fučík eigenfunctions 𝑓𝑛
𝛼,𝛽,
with (𝛼(𝑛), 𝛽(𝑛)) for even 𝑛on a straight line through the origin, form a dilated sequence in
the sense of (5.1). It is not hard to show that the normalized Fučík eigenfunctions for odd 𝑛
with (𝛼(𝑛), 𝛽(𝑛)) on the same line satisfy
𝑓𝑛
𝛼(𝑛),𝛽(𝑛)(𝑥) = 𝑓2
𝛼(2),𝛽(2) 𝑛−1
2+1
𝛼(2)𝑥.
Nevertheless, this property is less suitable for application of [15, Theorem D] since the opera-
tors 𝑇𝑘in (5.2) have to be independent of 𝑛.
4. Theorem 1.9 might suggest that the basisness of a Fučík system 𝐹𝛼,𝛽 also holds when
each point (𝛼(𝑛), 𝛽(𝑛)) ∈Γ𝑛,𝑛≥2, belongs to the angular sector in between the line (1.14)
and its reflection with respect to the main diagonal 𝛼=𝛽.
5. Recall that the constants in the function 𝐶𝑛in Theorem 1.4, as well as the upper
bound for 𝛾in Theorem 1.9, are not optimal due to the employed methods and the estimation
procedure.
Appendix A.
We start by providing several useful formulas. For the derivation of expressions in Section
2, we need to calculate several integrals of the general form
𝑐sin(√𝛿(𝑥−𝑥0)) ±sin(𝑛𝑥)2d𝑥
with constants 𝑐∈1,√𝛼/√𝛽, √𝛽/√𝛼,𝛿∈ {𝛼, 𝛽}, and certain 𝑥0∈[0, 𝜋]. The antideriva-
tive of this integral can be expressed through the following formulas:
sin2(𝑛𝑥) d𝑥=1
2𝑥−1
𝑛sin(𝑛𝑥) cos(𝑛𝑥)+𝐶,
sin2(√𝛿(𝑥−𝑥0)) d𝑥=1
2𝑥−𝑥0−1
√𝛿sin(√𝛿(𝑥−𝑥0)) cos(√𝛿(𝑥−𝑥0))+𝐶,
sin(√𝛿(𝑥−𝑥0)) sin(𝑛𝑥) d𝑥=√𝛿
𝑛2−𝛿cos(√𝛿(𝑥−𝑥0)) sin(𝑛𝑥)

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 19
−𝑛
𝑛2−𝛿sin(√𝛿(𝑥−𝑥0)) cos(𝑛𝑥) + 𝐶.
We observe that sin(√𝛿(𝑥−𝑥0)) vanishes for 𝑥=𝑥0and 𝑥=𝑥0+𝜋/√𝛿, and cos(√𝛿(𝑥−𝑥0))
evaluated in these points becomes either 1or −1, which significantly simplifies corresponding
definite integrals occurring in the sections above.
Let us now provide a more detailed derivation of the expression ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2for odd 𝑛≥3
and 𝛼≥𝑛2≥𝛽, see (2.3). Recall that each normalized Fučík eigenfunction 𝑓𝑛
𝛼,𝛽 with odd 𝑛
has 𝑛+1
2positive bumps and 𝑛−1
2negative bumps. Therefore, using the formulas from above,
we deduce that
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2=
𝑛+1
2−1

𝑖=0 𝑖𝑙+𝑙1
𝑖𝑙 √𝛽
√𝛼sin(√𝛼(𝑥−𝑖𝑙)) −sin(𝑛𝑥)2
d𝑥
+
𝑛−1
2−1

𝑖=0 (𝑖+1)𝑙
𝑖𝑙+𝑙1sin(𝛽(𝑥−𝑖𝑙 −𝑙1)) + sin(𝑛𝑥)2d𝑥
=𝜋
2+𝑛+ 1
4
𝛽
𝛼𝑙1+𝑛−1
4𝑙2
+ 4𝛽𝛼−𝛽
(𝑛2−𝛼)(𝑛2−𝛽)
𝑛−1
2−1

𝑖=0
[sin(𝑛(𝑖𝑙 +𝑙1)) + sin(𝑛(𝑖+ 1)𝑙)]
=𝜋
2+𝑛+ 1
4
𝛽
𝛼𝑙1+𝑛−1
4𝑙2
+ 8𝛽𝛼−𝛽
(𝑛2−𝛼)(𝑛2−𝛽)
sin (𝑛−1)𝑛𝑙
8cos 𝑛𝑙2
2
sin 𝑛𝑙
2sin (𝑛−1)𝑛𝑙
8+𝑛𝑙1
2
=𝜋−𝜋(𝑛+ 1)(√𝛼−1)
√𝛼(2√𝛼−(𝑛+ 1))2(√𝛼−𝑛)
+16(𝑛−1)√𝛼3
(2√𝛼−(𝑛+ 1))
(√𝛼−1)
(𝑛+√𝛼)(𝑛+ 1)(𝑛2−3𝑛√𝛼+𝑛+√𝛼)×
×
cos 𝜋
2
𝑛
√𝛼cos 𝜋
2
𝑛(2√𝛼−𝑛−1)
(𝑛−1)√𝛼
(√𝛼−𝑛) sin 𝜋𝑛(√𝛼−1)
(𝑛−1)√𝛼.(A.1)
Here, we used the summation formula
𝑘−1

𝑖=0
sin(𝑐𝑖 +𝑑) = sin 𝑘𝑐
2sin (𝑘−1) 𝑐
2+𝑑
sin 𝑐
2.(A.2)
Notice that the arguments of the last cosine and sine in (A.1) satisfy
𝜋
2≤𝜋
2
𝑛(2√𝛼−𝑛−1)
(𝑛−1)√𝛼≤3𝜋
2and 𝜋≤𝜋𝑛(√𝛼−1)
(𝑛−1)√𝛼≤3𝜋
2.

20 F. BAUSTIAN AND V. BOBKOV
That is, the cosine and sine of these arguments are negative. To make it easier to control the
total sign in (A.1), we apply the formulas
cos(𝑥) = −cos(𝜋−𝑥)and sin(𝑥) = −sin(𝑥−𝜋).
This gives the expression (2.3).
The derivation of the expression (2.6) for the scalar product ⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑚⟩for even 𝑛and 𝑚
with 𝑛>𝑚follows similar steps as above:
⟨𝑓𝑛
𝛼,𝛽, 𝜑𝑚⟩=
𝑛
2−1

𝑘=0
√𝛽
√𝛼𝑘𝑙+𝑙1
𝑘𝑙
sin(√𝛼(𝑥−𝑘𝑙)) sin(𝑚𝑥) d𝑥
−
𝑛
2−1

𝑘=0 (𝑘+1)𝑙
𝑘𝑙+𝑙1
sin(𝛽(𝑥−𝑘𝑙 −𝑙1)) sin(𝑚𝑥) d𝑥
=𝛽1
𝛼−𝑚2−1
𝛽−𝑚2𝑛
2−1

𝑘=0
sin(𝑚𝑘𝑙) + sin(𝑚(𝑘𝑙 +𝑙1)),
and we conclude by applying the summation formula (A.2).
Let us now provide details on the derivation of the upper bound for ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2with odd
𝑛≥3and 𝛼≥𝑛2≥𝛽, given by (3.4). Reducing the terms on the right-hand side of the first
inequality in (3.4) to a common denominator, we arrive at
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤𝜋(√𝛼−𝑛)2
144(𝑛−1)𝛼(2√𝛼−𝑛−1)2(𝑛+√𝛼)((3𝑛−1)√𝛼−𝑛(𝑛+ 1))
×144√𝛼(𝑛2−1)(2√𝛼+ 4√𝛼3+𝑛+𝑛2−2√𝛼𝑛(𝑛+ 2) + 𝛼(5𝑛−7))
+ 48𝜋2𝛼(√𝛼−1)(2√𝛼−𝑛−1)(𝑛2+ 1)
−𝜋4(√𝛼−1)(√𝛼−𝑛)2(2√𝛼−𝑛−1)(𝑛+ 1)2.
Using now the following simple estimates:
(2√𝛼−𝑛−1)2≥(𝑛−1)2
𝑛2𝛼, (𝑛+√𝛼)((3𝑛−1)√𝛼−𝑛(𝑛+ 1)) ≥4𝑛2(𝑛−1),
𝑛2−1≤𝑛2+ 1, 𝑛 +𝑛2−2√𝛼𝑛(𝑛+ 2) + 𝛼(5𝑛−7) ≤𝛼(5𝑛−7) ≤5√𝛼3,
(√𝛼−1)(2√𝛼−𝑛−1) ≤2𝛼, −𝜋4(√𝛼−1)(√𝛼−𝑛)2(2√𝛼−𝑛−1)(𝑛+ 1)2≤0,
we get
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤𝜋(√𝛼−𝑛)2(𝑛2+ 1)(288 + (1296 + 96𝜋2)𝛼)
576(𝑛−1)4𝛼.
Finally, recalling that √𝛼≥𝑛≥3and roughly estimating
288 + (1296 + 96𝜋2)𝛼
576 ≤4𝛼,
we obtain
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤4𝜋(𝑛2+ 1)(√𝛼−𝑛)2
(𝑛−1)4≤4𝜋𝑛2(𝑛2+ 1)
(𝑛−1)4√𝛼
𝑛−12
≤23𝜋√𝛼
𝑛−12
.

BASIS PROPERTIES OF FUČÍK EIGENFUNCTIONS 21
For the case 𝛽 > 𝑛2> 𝛼, we use a similar procedure as above to estimate the upper bound
for ‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2with odd 𝑛≥3given by (3.6). We convert the right-hand side of the first
inequality in (3.6) to a common denominator, to get
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤𝜋(√𝛽−𝑛)2
144(𝑛+ 1)𝛽(2√𝛽−𝑛+ 1)2(𝑛+√𝛽)((3𝑛+ 1)√𝛽−𝑛(𝑛−1))
×144𝛽(𝑛2−1)(2𝛽+ 4𝛽3+𝑛−𝑛2−2𝛽𝑛(𝑛−2) + 𝛽(5𝑛+ 7))
+ 48𝜋2𝛽(𝛽+ 1)(2𝛽−𝑛+ 1)(𝑛2+ 1)
−𝜋4(𝛽+ 1)(𝛽−𝑛)2(2𝛽−𝑛+ 1)(𝑛−1)2.
With the simple estimates
(2𝛽−𝑛+ 1)2≥(𝑛+ 1)2
𝑛2𝛽, (𝑛+𝛽)((3𝑛+ 1)𝛽−𝑛(𝑛−1)) ≥4𝑛2(𝑛+ 1),
𝑛2−1≤𝑛2+ 1, 𝑛 −𝑛2−2𝛽𝑛(𝑛−2) + 𝛽(5𝑛+ 7) ≤𝛽(5𝑛+ 7) ≤8𝛽3,
(𝛽+ 1)(2𝛽−𝑛+ 1) ≤2𝛽, −𝜋4(𝛽+ 1)(𝛽−𝑛)2(2𝛽−𝑛+ 1)(𝑛−1)2≤0,
we obtain
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤𝜋(√𝛽−𝑛)2(𝑛2+ 1)(288 + (1728 + 96𝜋2)𝛽)
576(𝑛+ 1)4𝛽.
Finally, estimating
288 + (1728 + 96𝜋2)𝛽
576 ≤5𝛽,
we obtain
‖𝑓𝑛
𝛼,𝛽 −𝜑𝑛‖2≤5𝜋(𝑛2+ 1)(√𝛽−𝑛)2
(𝑛+ 1)4≤5𝜋𝑛2(𝑛2+ 1)
(𝑛+ 1)4√𝛽
𝑛−12
≤5𝜋√𝛽
𝑛−12
.
Appendix B.
In this section, we calculate the norms of the operators 𝑇𝑘defined by (5.5). First, let us
show that each 𝑇𝑘is linear. Indeed, taking any 𝑔, ℎ ∈𝐿2(0, 𝜋), we have
𝑇𝑘(𝑔+ℎ)(𝑥)=(𝑔+ℎ)*𝑘𝑥
2= (−1)𝜅(𝑔+ℎ)𝑘𝑥
2−𝜋𝜅
for 𝜋𝜅 ≤𝑘𝑥
2≤𝜋(𝜅+ 1),𝜅∈N0. At the same time, for such 𝑥we get
(−1)𝜅(𝑔+ℎ)𝑘𝑥
2−𝜋𝜅= (−1)𝜅𝑔𝑘𝑥
2−𝜋𝜅+ (−1)𝜅ℎ𝑘𝑥
2−𝜋𝜅
=𝑔*𝑘𝑥
2+ℎ*𝑘𝑥
2=𝑇𝑘𝑔(𝑥) + 𝑇𝑘ℎ(𝑥),
which concludes the linearity.
Assume now that 𝑘is even, i.e., 𝑘= 2𝑚,𝑚≥1. For any 𝑔∈𝐿2(0, 𝜋)we obtain
𝜋
0
(𝑇2𝑚𝑔(𝑥))2d𝑥=𝜋
0
(𝑔*(𝑚𝑥))2d𝑥=1
𝑚𝜋𝑚
0
(𝑔*(𝑥))2d𝑥

22 F. BAUSTIAN AND V. BOBKOV
=1
𝑚
𝑚−1

𝜅=0 𝜋(𝜅+1)
𝜋𝜅
𝑔2(𝑥−𝜋𝜅) d𝑥=1
𝑚
𝑚−1

𝜅=0 𝜋
0
𝑔2(𝑥) d𝑥=𝜋
0
𝑔2(𝑥) d𝑥.
Therefore, we get
‖𝑇2𝑚‖*= sup
𝑔∈𝐿2(0,𝜋)∖{0}
‖𝑇2𝑚𝑔‖
‖𝑔‖= 1.
Assume that 𝑘is odd, i.e., 𝑘= 2𝑚+ 1,𝑚≥0. We have
𝜋
0
(𝑇2𝑚+1𝑔(𝑥))2d𝑥=𝜋
0𝑔*(2𝑚+ 1)𝑥
22
d𝑥=2
2𝑚+ 1 𝜋(𝑚+1
2)
0
(𝑔*(𝑥))2d𝑥
=2
2𝑚+ 1
𝑚−1

𝜅=0 𝜋(𝜅+1)
𝜋𝜅
𝑔2(𝑥−𝜋𝜅) d𝑥+2
2𝑚+ 1 𝜋(𝑚+1
2)
𝜋𝑚
𝑔2(𝑥−𝜋𝑚) d𝑥
=2𝑚
2𝑚+ 1 𝜋
0
𝑔2(𝑥) d𝑥+2
2𝑚+ 1 𝜋
2
0
𝑔2(𝑥) d𝑥≤2𝑚+ 2
2𝑚+ 1 𝜋
0
𝑔2(𝑥) d𝑥.
Notice that this estimate is sharp since equality holds for any 𝑔with the support on (0, 𝜋/2).
Thus, we deduce that
‖𝑇2𝑚+1‖*= sup
𝑔∈𝐿2(0,𝜋)∖{0}
‖𝑇2𝑚+1𝑔‖
‖𝑔‖=2𝑚+ 2
2𝑚+ 1.
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(F. Baustian)
Department of Mathematics, University of Rostock,
Ulmenstraße 69, 18057 Rostock, Germany
Email address:falko.baustian@uni-rostock.de
(V. Bobkov)
Department of Mathematics and NTIS, Faculty of Applied Sciences,
University of West Bohemia, Univerzitní 8, 301 00 Plzeň, Czech Republic,
Institute of Mathematics, Ufa Federal Research Centre, RAS,
Chernyshevsky str. 112, 450008 Ufa, Russia
Email address:bobkov@kma.zcu.cz