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# Approximation of Non -Interpolatory Complex Parabolic Spline on the Unit Circle

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## Abstract

In this paper we have constructed a non-interpolatory spline on the unit circle. The rate of convergence and the error in approximation corresponding to the complex valued function has been considered.
Filomat 35:10 (2021), 3549–3556
https://doi.org/10.2298/FIL2110549V
University of Niˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Approximation of Non - Interpolatory Complex Parabolic Spline on the
Unit Circle
Varuna, Neha Mathurb, Swarnima Bahadura, Pankaj Mathura
aDepartment of Mathematics and Astronomy, University of Lucknow, Lucknow.
bDepartment of Mathematics, Career Convent Degree College, Lucknow, India
Abstract. In this paper we have constructed a non-interpolatory spline on the unit circle. The rate
of convergence and the error in approximation corresponding to the complex valued function has been
considered.
1. INTRODUCTION
Let Kdenote the unit circle |z|=1 of the complex plane and let mand nbe integers, m1,n2.
Furthermore, let ∆ = {z1,z2,· · · ,zn}be a mesh of ndistinct points of Karranged in cyclic counter-clockwise
order. A complex valued function S(z) deﬁned on Kis called a polynomial spline function of degree m1,
if it satisﬁes the conditions:
1. S(z)Cm2(K),
2. S(z) agrees in values with a polynomial of degree at most m1, on each arc in which the points zj
divide the circle K.
If S1(z),S2(z)· · · ,Sn(z) denote the polynomial components of S(z) on the arcs Kj={(zj,zj+1),j=1,2,· · · ,n}
respectively, where zn+1=z1, then the condition (1) or more explicitly S(eiθ)Cm2(K), is equivalent to the
conditions:
S(ν)
j(zj+1)=S(ν)
j+1(zj+1), ν =0,1,2,· · · ,m2,j=1,2,· · · ,n(1)
where Sn+1(z)=S1(z).
In 1971, the problem of complex spline interpolation was initiated by Schoenberg  and Ahlberg, Nil-
son and Walsh in a sequence of papers [1–3]. The solutions were completely dierent. A related problem
on the trigonometric spline interpolation was beautifully studied by Schoenberg , connecting the study
to the dierential operators m=D(D2+12)· · · (D2+m2),(D=d/dx). Micchelli  exploiting Schoenberg’s
2020 Mathematics Subject Classiﬁcation. Primary 41A10 ; Secondary 97N50, 41A05, 30E10
Keywords. Spline Interpolation, Rate of Convergence, Non-Interpolatory Spline, Convergence on unit circle, Splines on unit circle
Received: 11 September 2020; Accepted: 22 November 2020
Communicated by Miodrag Spalevi´
c
Corresponding author: Pankaj Mathur
Email addresses: varun.kanaujia.1992@gmail.com (Varun), neha_mathur13@yahoo.com (Neha Mathur),
Varun et al. /Filomat 35:10 (2021), 3549–3556 3550
idea and using the cardinal L-splines related to the dierential operator L=Qn
j=0(Dγj) with γjas real
numbers, gave a complete and systematic treatment to the interpolation problem. The works of Shevaldin
, , Subbotin and Chernykh  also deserve a mention.
Schoenberg  revisited Micchelli’s theory and extended it to the operator Lwith imaginary γj’s.
Sharma and Tzimbalario  and Tzimbalario  further extended the study for cardinal splines related
to the operators mand L=Qn
j=0(Di(j+)η) for some η > 0 and real, respectively.
Kvasov , Subbotin  (with dierent conditions) and Shevaldin  (in a more general statement)
constructed local parabolic splines for functions deﬁned on the axis or on the segment of the axis that
preserve linear functions with an arbitrary distinct setting of nodes with good approximative property
and their own local preservation of the sign, monotonicity and convexity of approximate functions .
Recently in a joint paper, Subbotin and Shevaldin  developed a general scheme of constructing such
structures, special cases of which are the splines of [17, 23]. These splines and their generalizations are
widely used in computational mathematics. In other papers, Kostosov and Shevaldin , Shevaldin 
and Strelkova  have extended the study to trigonometric, exponential and average interpolation splines
respectively. Article  gave rise to a whole series of works by Subbotin and Telyakovskii [21, 22] on
estimates of Lebesgue constants of interpolatory splines and trigonometric polynomials and Konovalov’s
diameters of dierentiable classes of functions.
The aim of this paper is to construct a non - interpolatory complex parabolic spline S(z) on a unit
circle K, study its rate of convergence and error in approximation corresponding to an analytic function
f(z)W2
K={f: max |f00(z)| ≤ 1}on K.
2. CONSTRUCTION OF COMPLEX PARABOLIC SPLINE
We are interested to construct a non-interpolatory spline S(z) for the subdivision , on the unit circle K,
composed of complex quadratics Sj(z) on the arc Kjfrom zjto zj+1, where zj=exp 2jπi
n. For this purpose,
we follow the scheme of works [17, 23]. Obviously,
zj+1=exp 2(j+1)πi
n!=exp (ih)zj,
where h=2π
n. Let f:CCand yj=f(zj). Associate operator Λon the space of sequences {yj}, as
Λ(yj1) :=yj+1(eih +1)yj+eih yj1.
For zKj, the spline Sj(z), can be represented in the form
Sj(z)=C(j)
0+C(j)
1zzj+C(j)
2zzj2+C(j)
3zzj+1
22
+,(2)
where
zzj+1
2+=
zzj+1
2,arg z>arg zj+1
2
0,arg zarg zj+1
2
(3)
and C(j)
0,C(j)
1,C(j)
2,C(j)
3are complex constants, given by
C(j)
0=yj+eih
2(eih
21)Λ(yj1)
2(e2ih 1) ,(4)
Varun et al. /Filomat 35:10 (2021), 3549–3556 3551
C(j)
1=eih(yj+1yj1)
(e2ih 1)zj
,(5)
C(j)
2=Λ(yj1)
(eih 1)(e2ih 1)z2
j
(6)
and
C(j)
3=Λ(yj)Λ(yj1)
eih
2(eih
21)(e2ih 1)z2
j
.(7)
Theorem 2.1. For z Kj, the spline Sj(z), satisﬁes the following properties:
1. Sj(zj+1)=yj+1+bΛ(yj), where
b=eih
2(eih
21)
2(e2ih 1) .
2. Sj(z)has a continuous derivative on Kj, such that
S0
j(zj)=eih(yj+1yj1)
(e2ih 1)zj
.
3. For arg zarg zj+1
2
S00
j(zj)=2Λ(yj1)
(eih 1)(e2ih 1)z2
j
and for arg z>arg zj+1
2
S00
j(zj+1)=2(eih
2+1)Λ(yj)2Λ(yj1)
eih
2(eih 1)(e2ih 1)z2
j
.
Proof. 1. Let zKj, then putting z=zjin (2), we have
Sj(zj)=C(j)
0=yj+eih
2(eih
21)Λ(yj1)
2(e2ih 1)
and
Sj(zj+1)=C(j)
0+C(j)
1(zj+1zj)+C(j)
2(zj+1zj)2+C(j)
3(zj+1zj+1
2)2
+
=C(j)
0+C(j)
1(eih 1)zj+C(j)
2(eih 1)2z2
j+C(j)
3eih(eih
21)2z2
j,
which due to (4), (5), (6) and (7) implies
Sj(zj+1)=yj+1+eih
2(eih
21)Λ(yj)
2(e2ih 1) .
2. The continuity of S0
j(z) is obvious on Kexcept at the points zjof the spline. On dierentiating (2) w.r.t
z, we get
S0
j(z)=C(j)
1+2C(j)
2(zzj)+2C(j)
3(zzj+1
2)+,(8)
which on substituting z=zj+1, due to (5), (6) and (7), gives
S0
j(zj+1)=C(j)
1+2C(j)
2(eih 1)zj+2C(j)
3eih
2(eih
21)zj
=eih(yj+2yj)
(e2ih 1)zj+1
.
Varun et al. /Filomat 35:10 (2021), 3549–3556 3552
Also for zKj+1, due to (5), we have
S0
j+1(zj+1)=C(j+1)
1=eih(yj+2yj)
(e2ih 1)zj+1
,
which implies the continuity of S0
j(z) at the grid points zj+1.
3. Lastly on dierentiating (8) w.r.t zand putting z=zj, due to (6), we get
S00
j(zj)=2C(j)
2=2Λ(yj1)
(eih 1)(e2ih 1)z2
j
.
Similarly, dierentiating (8) w.r.t zand putting z=zj+1, due to (6) and (7), we have
S00
j(zj+1)=2C(j)
2+2C(j)
3
=2eih
2Λ(yj)+2(Λ(yj)Λ(yj1))
eih
2(eih 1)(e2ih 1)z2
j
,
which proves the theorem.
3. RATE OF CONVERGENCE
Convergence on the boundary. To study the convergence properties of the complex spline S(z), we
follow the ideas of Ahlberg, Nilson and Walsh . We consider the convergence of {Sk(t)}for the sequence
of meshes k={zk,1,zk,2,· · · ,zk,n}with kkk=maxj|zk,j+1zk,j| → 0,as k→ ∞. Let {Sk,j(z)}n
j=1be the complex
quadratic splines on the arcs Kk,jfrom zk,jto zk,j+11). Then, we shall prove the following:
Theorem 3.1. Let f (z)be continuous on K. Let {k}be a sequence of subdivisions of K with limk→∞ kkk=0. Let
Sk(z)be the complex quadratic spline on k, then nSk(z)of(z)uniformly as kkk → 0. Further, if f (z)satisﬁes a
older’s condition of order α(0 < α 1) on K, then
|Sk(z)f(z)|=O(kkkα).
Proof. Let f(z) be continuous on K. Then on Kj, by setting z=(zj+zj+1)/2+, where is a complex number
such that 0 <|/h| ≤ 1/2, we have
arg(z)arg(zj+1
2)=arg zj+1+zj+2
2!arg(zj+1
2)<0
and
(zzj)=zj+zj+1
2+zj=zj(eih 1)
2+.
1)For the sake of convenience we shall drop the index “k” from the subscript
Varun et al. /Filomat 35:10 (2021), 3549–3556 3553
Due to (3), for zKj, it follows that
|Sj(z)f(z)|≤|f(zj)f(z)|+
eih
2(eih
21)
2(e2ih 1)
[|f(zj+1)f(zj)|+|eih| | f(zj)f(zj1)|] (9)
+"|eih|(|f(zj+1)f(zj)|+|f(zj)f(zj1)|)
|(e2ih 1)| |zj|#
zj(eih 1)
2+
+"|f(zj+1)f(zj)|+|eih||(f(zj)f(zj1))|
|(eih 1)||(e2ih 1)||z2
j|#
zj(eih 1)
2+
2
ω(f,kkk)"1+
eih
2(eih
21)
2(e2ih 1)
(2) +2
|e2ih 1|
zj(eih 1)
2+
+2
|eih 1||e2ih 1|
zj(eih 1)
2+
2#.
where ω(f,kkk) is the modulus of continuity of fon K. Further, we need |eih|=1 and |eih 1|=
q(cos h1)2+sin2h=2 sin(h/2). From , we have for 0 ≤ |h| ≤ π/2
|eih 1| ≥ 2|h|(10)
and for h0
|eih 1| ≤ h.(11)
Using (10) and (11) in the last inequality of (9), we get
|Sj(z)f(z)| ≤ ω(f,kkk)h1+5π
8+3π
4
h+π2
4
2
h2i.
Since 0 <|/h|1/2, therefore
|Sj(z)f(z)|=Cω(f,kkk),(12)
where C is a constant, from which the Theorem follows.
In order to obtain the convergence properties of the complex spline S(z), it is necessary to show that
S(t)f(t) or its derivatives satisfy suitable H¨
older’s conditions.
We shall prove the following:
Corollary 3.2. Under the conditions of Theorem 3.1 with f (z)satisfying a H¨older condition of order α(0 < α 1),
the function [Sk(z)f(z)]/kkkαδsatisﬁes a H¨older’s condition of order δ,0< δ α, uniformly with respect to k.
Proof. For zand τon Kj, we have
Sj(z)Sj(τ)=
eih[f(zj+1)f(zj)+f(zj)f(zj1)]
(e2ih 1)zj
(zzj(τzj))
+"[f(zj+1)f(zj)eih(f(zj)f(zj1))]
(eih 1)(e2ih 1)z2
j#h(zzj)2(τzj)2i
+1
2"[f(zj+2)f(zj+1)eih(f(zj+1)f(zj))]
eih
2(eih
21)(e2ih 1)z2
j
+[f(zj+1)f(zj)eih(f(zj)f(zj1))]
eih
2(1 eih
2)(e2ih 1)z2
j#h(zzj+1
2)2
+(τzj+1
2)2
+i.
Varun et al. /Filomat 35:10 (2021), 3549–3556 3554
Let us consider two cases-:
Case(i) If arg(z)arg(zj+1
2) and arg(τ)arg(zj+1
2),
Case(ii) If arg(z)>arg(zj+1
2) and arg(τ)>arg(zj+1
2).
Case (i) implies that (zzj+1
2)2
+=(τzj+1
2)2
+=0, then
Sj(z)Sj(τ)=(zτ)(
eih[f(zj+1)f(zj)+f(zj)f(zj1)]
(e2ih 1)zj
+"[f(zj+1)f(zj)eih(f(zj)f(zj1))]
(eih 1)(e2ih 1)z2
j#(z+τ2zj)).
If f(z) satisﬁes H¨
older’s condition of order αand if aδsuch that 0 < δ α, then
Sj(z)Sj(τ)+f(τ)f(z)≤ |zτ|("|f(zj+1)f(zj)|+|f(zj)f(zj1)|
|e2ih 1|#
+"|f(zj+1)f(zj)|+|f(zj)f(zj1)|
|(eih 1)(e2ih 1)|#|zτ|+2|zjτ|)+|f(τ)f(z)|
≤ |zτ|("|zj+1zj|α+|zjzj1|α
|e2ih 1|#+"|zj+1zj|α+|zjzj1|α
|(eih 1)(e2ih 1)|#|zτ|+2|zjτ|)
+|τz|α
≤ |zτ|("2|eih 1|α
|e2ih 1|#+"2|eih 1|α1
|(e2ih 1)|#|zτ|+2|zjτ|)+|τz|α.
Since z, τ Kj, therefore, owing to (10) and (11), we have |zτ|≤|zj+1zj|≤|eih1| ≤ hand |zjτ|≤|eih 1|,
Sj(z)Sj(τ)+f(τ)f(z)≤ |zτ|δkkkαδ|zτ|αδ
kkkαδ(8|zτ||eih 1|α
|e2ih 1||zτ|α+1)
(2π+1)|zτ|δkkkαδ|zτ|
kkkαδ
(2π+1)|zτ|δkkkαδ.
Thus, we deduce that (Sj(z)f(z))/kkkαδsatisﬁes uniformly H¨
older’s condition of order δ. Working
corresponding to Case (ii) has been omitted as a mutatis-mutandis approach leads to the above conclu-
sion.
For the proof of the following theorem, we adopt the scheme of works [17, 23].
Theorem 3.3. Let f Cbe analytic on K and f W2
K. Let kbe a sequence of subdivisions of K with limk→∞ kkk=
0. Let Sj(z)be the complex quadratic spline on Kj, then
sup
fW2
K
|f(z)Sj(z)|K=O1
n2.(13)
Proof. Without violating generality, taking a periodic case, we can accept that zK1, where K1is the arc
joining the points z1and z2. Moreover, we can accept that zlies in the arc joining z1and z3/2, that is
Varun et al. /Filomat 35:10 (2021), 3549–3556 3555
where arg(z)arg(z3/2)<0. Otherwise we can make a change in variable z=z2v. Also, we can take
z1=eih,z3/2=e3ih/2, where h=2π
n. Consider z=z1eiθ, where 0 θh, hence
f(z)S1(z)=(z1f0(z1)(eiθ1) +Zz
z1
(z1eiθτ)f00(z1τ)z1dτ)eih
2(eih
21)Λ(y0)
2(e2ih 1)
"eih(y2y0)
(e2ih 1) #(eiθ1) +"Λ(y0)
(eih 1)(e2ih 1)#(eiθ1)2
=nz1f0(z1)(eiθ1) +Zz
z1
(z1eiθτ)f00(z1τ)dτo
+(f(z2)f(z1))
eih
2(eih
21)
2(e2ih 1) eih(eiθ1)
(e2ih 1) (eiθ1)2
(eih 1)(e2ih 1)
+(f(z1)f(z0))
eih
2eih(eih
21)
2(e2ih 1) eih(eiθ1)
(e2ih 1) +eih(eiθ1)2
(eih 1)(e2ih 1)
.
As kkk → 0, we can use Taylor’s theorem with integral form of the remainder, to get
f(z)S1(z)=nz1f0(z1)(eiθ1) +Zz
z1
(z1eiθτ)f00(τ)dτo
+
eih
2(eih
21)
2(e2ih 1) eih(eiθ1)
(e2ih 1) (eiθ1)2
(eih 1)(e2ih 1)
((z2z1)f0(z1)+Zz2
z1
(z2τ)f00(τ)dτ)
+
eih
2eih(eih
21)
2(e2ih 1) eih(eiθ1)
(e2ih 1) +eih(eiθ1)2
(eih 1)(e2ih 1)
((z1z0)f0(z0)+Zz1
z0
(z1τ)f00(τ)dτ)
f(z)S1(z)=
eih
2eih(eih
21)
2(e2ih 1) eih(eiθ1)
(e2ih 1) +eih(eiθ1)2
(eih 1)(e2ih 1)
Zz1
z0
(z0τ)f00(τ)dτ
Zz
z1
eih
2(eih
21)(z2τ)
2(e2ih 1) +eih(eiθ1)(z2τ)
(e2ih 1) +(z2τ)(eiθ1)2
(eih 1)(e2ih 1) (z1eiθτ)
f00(τ)dτ
eih
2(eih
21)
2(e2ih 1) +eih(eiθ1)
(e2ih 1) +(eiθ1)2
(eih 1)(e2ih 1)
Zz2
z
(z2τ)f00(τ)dτ.
Since fW2
K, thus due to (10) and (11), we have
f(z)S1(z)
eih
2eih(eih
21)
2(e2ih 1) eih(eiθ1)
(e2ih 1) +eih(eiθ1)2
(eih 1)(e2ih 1)
|z1z0|2
2
+
eih
2(eih
21)(z2τ)2
4(e2ih 1) +eih(eiθ1)(z2τ)2
2(e2ih 1) +(z2τ)2(eiθ1)2
2(eih 1)(e2ih 1) (z1eiθτ)2
2
z
z1
+
eih
2(eih
21)
2(e2ih 1) +eih(eiθ1)
(e2ih 1) +(eiθ1)2
(eih 1)(e2ih 1)
|z2z|2
2
h2 500π+13π2
256 +1
2!,
from which the theorem follows.
4. Acknowledgement
Authors are thankful to the referee for his constructive suggestions.
Varun et al. /Filomat 35:10 (2021), 3549–3556 3556
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