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Filomat 35:10 (2021), 3549–3556

https://doi.org/10.2298/FIL2110549V

Published by Faculty of Sciences and Mathematics,

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, m≥1,n≥2.

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 m−1,

if it satisﬁes the conditions:

1. S∆(z)∈Cm−2(K),

2. S∆(z) agrees in values with a polynomial of degree at most m−1, 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)∈Cm−2(K), is equivalent to the

conditions:

S(ν)

j(zj+1)=S(ν)

j+1(zj+1), ν =0,1,2,· · · ,m−2,j=1,2,· · · ,n(1)

where Sn+1(z)=S1(z).

In 1971, the problem of complex spline interpolation was initiated by Schoenberg [10] and Ahlberg, Nil-

son and Walsh in a sequence of papers [1–3]. The solutions were completely diﬀerent. A related problem

on the trigonometric spline interpolation was beautifully studied by Schoenberg [11], connecting the study

to the diﬀerential operators ∆m=D(D2+12)· · · (D2+m2),(D=d/dx). Micchelli [7] 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),

swarnimabahadur@ymail.com (Swarnima Bahadur), pankaj_mathur14@yahoo.co.in (Pankaj Mathur)

Varun et al. /Filomat 35:10 (2021), 3549–3556 3550

idea and using the cardinal L-splines related to the diﬀerential operator L=Qn

j=0(D−γj) with γjas real

numbers, gave a complete and systematic treatment to the interpolation problem. The works of Shevaldin

[14], [15], Subbotin and Chernykh [24] also deserve a mention.

Schoenberg [12] revisited Micchelli’s theory and extended it to the operator Lwith imaginary γj’s.

Sharma and Tzimbalario [13] and Tzimbalario [25] further extended the study for cardinal splines related

to the operators ∆mand L=Qn

j=0(D−i(j+`)η) for some η > 0 and `real, respectively.

Kvasov [6], Subbotin [23] (with diﬀerent conditions) and Shevaldin [17] (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 [16].

Recently in a joint paper, Subbotin and Shevaldin [20] 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 [5], Shevaldin [18]

and Strelkova [19] have extended the study to trigonometric, exponential and average interpolation splines

respectively. Article [23] 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 diﬀerentiable 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(yj−1) :=yj+1−(eih +1)yj+eih yj−1.

For z∈Kj, the spline Sj(z), can be represented in the form

Sj(z)=C(j)

0+C(j)

1z−zj+C(j)

2z−zj2+C(j)

3z−zj+1

22

+,(2)

where

z−zj+1

2+=

z−zj+1

2,arg z>arg zj+1

2

0,arg z≤arg 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

2−1)Λ(yj−1)

2(e2ih −1) ,(4)

Varun et al. /Filomat 35:10 (2021), 3549–3556 3551

C(j)

1=eih(yj+1−yj−1)

(e2ih −1)zj

,(5)

C(j)

2=Λ(yj−1)

(eih −1)(e2ih −1)z2

j

(6)

and

C(j)

3=Λ(yj)−Λ(yj−1)

eih

2(eih

2−1)(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

2−1)

2(e2ih −1) .

2. Sj(z)has a continuous derivative on Kj, such that

S0

j(zj)=eih(yj+1−yj−1)

(e2ih −1)zj

.

3. For arg z≤arg zj+1

2

S00

j(zj)=2Λ(yj−1)

(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(yj−1)

eih

2(eih −1)(e2ih −1)z2

j

.

Proof. 1. Let z∈Kj, then putting z=zjin (2), we have

Sj(zj)=C(j)

0=yj+eih

2(eih

2−1)Λ(yj−1)

2(e2ih −1)

and

Sj(zj+1)=C(j)

0+C(j)

1(zj+1−zj)+C(j)

2(zj+1−zj)2+C(j)

3(zj+1−zj+1

2)2

+

=C(j)

0+C(j)

1(eih −1)zj+C(j)

2(eih −1)2z2

j+C(j)

3eih(eih

2−1)2z2

j,

which due to (4), (5), (6) and (7) implies

Sj(zj+1)=yj+1+eih

2(eih

2−1)Λ(yj)

2(e2ih −1) .

2. The continuity of S0

j(z) is obvious on Kexcept at the points zjof the spline. On diﬀerentiating (2) w.r.t

z, we get

S0

j(z)=C(j)

1+2C(j)

2(z−zj)+2C(j)

3(z−zj+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

2−1)zj

=eih(yj+2−yj)

(e2ih −1)zj+1

.

Varun et al. /Filomat 35:10 (2021), 3549–3556 3552

Also for z∈Kj+1, due to (5), we have

S0

j+1(zj+1)=C(j+1)

1=eih(yj+2−yj)

(e2ih −1)zj+1

,

which implies the continuity of S0

j(z) at the grid points zj+1.

3. Lastly on diﬀerentiating (8) w.r.t zand putting z=zj, due to (6), we get

S00

j(zj)=2C(j)

2=2Λ(yj−1)

(eih −1)(e2ih −1)z2

j

.

Similarly, diﬀerentiating (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(yj−1))

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 [2]. We consider the convergence of {S∆k(t)}for the sequence

of meshes ∆k={zk,1,zk,2,· · · ,zk,n}with k∆kk=maxj|zk,j+1−zk,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→∞ k∆kk=0. Let

S∆k(z)be the complex quadratic spline on ∆k, then nS∆k(z)o→f(z)uniformly as k∆kk → 0. Further, if f (z)satisﬁes a

H¨older’s condition of order α(0 < α ≤1) on K, then

|S∆k(z)−f(z)|=O(k∆kkα).

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

(z−zj)=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 z∈Kj, it follows that

|Sj(z)−f(z)|≤|f(zj)−f(z)|+

eih

2(eih

2−1)

2(e2ih −1)

[|f(zj+1)−f(zj)|+|eih| | f(zj)−f(zj−1)|] (9)

+"|eih|(|f(zj+1)−f(zj)|+|f(zj)−f(zj−1)|)

|(e2ih −1)| |zj|#

zj(eih −1)

2+

+"|f(zj+1)−f(zj)|+|eih||(f(zj)−f(zj−1))|

|(eih −1)||(e2ih −1)||z2

j|#

zj(eih −1)

2+

2

≤ω(f,k∆kk)"1+

eih

2(eih

2−1)

2(e2ih −1)

(2) +2

|e2ih −1|

zj(eih −1)

2+

+2

|eih −1||e2ih −1|

zj(eih −1)

2+

2#.

where ω(f,k∆kk) is the modulus of continuity of fon K. Further, we need |eih|=1 and |eih −1|=

q(cos h−1)2+sin2h=2 sin(h/2). From [9], we have for 0 ≤ |h| ≤ π/2

|eih −1| ≥ 2|h|/π (10)

and for h≥0

|eih −1| ≤ h.(11)

Using (10) and (11) in the last inequality of (9), we get

|Sj(z)−f(z)| ≤ ω(f,k∆kk)h1+5π

8+3π

4

h+π2

4

2

h2i.

Since 0 <|/h|≤1/2, therefore

|Sj(z)−f(z)|=Cω(f,k∆kk),(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 [S∆k(z)−f(z)]/k∆kkα−δsatisﬁes 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(zj−1)]

(e2ih −1)zj

(z−zj−(τ−zj))

+"[f(zj+1)−f(zj)−eih(f(zj)−f(zj−1))]

(eih −1)(e2ih −1)z2

j#h(z−zj)2−(τ−zj)2i

+1

2"[f(zj+2)−f(zj+1)−eih(f(zj+1)−f(zj))]

eih

2(eih

2−1)(e2ih −1)z2

j

+[f(zj+1)−f(zj)−eih(f(zj)−f(zj−1))]

eih

2(1 −eih

2)(e2ih −1)z2

j#h(z−zj+1

2)2

+−(τ−zj+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 (z−zj+1

2)2

+=(τ−zj+1

2)2

+=0, then

Sj(z)−Sj(τ)=(z−τ)(

eih[f(zj+1)−f(zj)+f(zj)−f(zj−1)]

(e2ih −1)zj

+"[f(zj+1)−f(zj)−eih(f(zj)−f(zj−1))]

(eih −1)(e2ih −1)z2

j#(z+τ−2zj)).

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)≤ |z−τ|("|f(zj+1)−f(zj)|+|f(zj)−f(zj−1)|

|e2ih −1|#

+"|f(zj+1)−f(zj)|+|f(zj)−f(zj−1)|

|(eih −1)(e2ih −1)|#|z−τ|+2|zj−τ|)+|f(τ)−f(z)|

≤ |z−τ|("|zj+1−zj|α+|zj−zj−1|α

|e2ih −1|#+"|zj+1−zj|α+|zj−zj−1|α

|(eih −1)(e2ih −1)|#|z−τ|+2|zj−τ|)

+|τ−z|α

≤ |z−τ|("2|eih −1|α

|e2ih −1|#+"2|eih −1|α−1

|(e2ih −1)|#|z−τ|+2|zj−τ|)+|τ−z|α.

Since z, τ ∈Kj, therefore, owing to (10) and (11), we have |z−τ|≤|zj+1−zj|≤|eih−1| ≤ hand |zj−τ|≤|eih −1|,

which leads to

Sj(z)−Sj(τ)+f(τ)−f(z)≤ |z−τ|δk∆kkα−δ|z−τ|α−δ

k∆kkα−δ(8|z−τ||eih −1|α

|e2ih −1||z−τ|α+1)

≤(2π+1)|z−τ|δk∆kkα−δ|z−τ|

k∆kkα−δ

≤(2π+1)|z−τ|δk∆kkα−δ.

Thus, we deduce that (Sj(z)−f(z))/k∆kkα−δsatisﬁes 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→∞ k∆kk=

0. Let Sj(z)be the complex quadratic spline on Kj, then

sup

f∈W2

K

|f(z)−Sj(z)|K=O1

n2.(13)

Proof. Without violating generality, taking a periodic case, we can accept that z∈K1, 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=z2−v. Also, we can take

z1=eih,z3/2=e3ih/2, where h=2π

n. Consider z=z1eiθ, where 0 ≤θ≤h, hence

f(z)−S1(z)=(z1f0(z1)(eiθ−1) +Zz

z1

(z1eiθ−τ)f00(z1τ)z1dτ)−eih

2(eih

2−1)Λ(y0)

2(e2ih −1)

−"eih(y2−y0)

(e2ih −1) #(eiθ−1) +"Λ(y0)

(eih −1)(e2ih −1)#(eiθ−1)2

=nz1f0(z1)(eiθ−1) +Zz

z1

(z1eiθ−τ)f00(z1τ)dτo

+(f(z2)−f(z1))

−eih

2(eih

2−1)

2(e2ih −1) −eih(eiθ−1)

(e2ih −1) −(eiθ−1)2

(eih −1)(e2ih −1)

+(f(z1)−f(z0))

eih

2eih(eih

2−1)

2(e2ih −1) −eih(eiθ−1)

(e2ih −1) +eih(eiθ−1)2

(eih −1)(e2ih −1)

.

As k∆kk → 0, we can use Taylor’s theorem with integral form of the remainder, to get

f(z)−S1(z)=nz1f0(z1)(eiθ−1) +Zz

z1

(z1eiθ−τ)f00(τ)dτo

+

−eih

2(eih

2−1)

2(e2ih −1) −eih(eiθ−1)

(e2ih −1) −(eiθ−1)2

(eih −1)(e2ih −1)

((z2−z1)f0(z1)+Zz2

z1

(z2−τ)f00(τ)dτ)

+

eih

2eih(eih

2−1)

2(e2ih −1) −eih(eiθ−1)

(e2ih −1) +eih(eiθ−1)2

(eih −1)(e2ih −1)

((z1−z0)f0(z0)+Zz1

z0

(z1−τ)f00(τ)dτ)

f(z)−S1(z)=

eih

2eih(eih

2−1)

2(e2ih −1) −eih(eiθ−1)

(e2ih −1) +eih(eiθ−1)2

(eih −1)(e2ih −1)

Zz1

z0

(z0−τ)f00(τ)dτ

−Zz

z1

eih

2(eih

2−1)(z2−τ)

2(e2ih −1) +eih(eiθ−1)(z2−τ)

(e2ih −1) +(z2−τ)(eiθ−1)2

(eih −1)(e2ih −1) −(z1eiθ−τ)

f00(τ)dτ

−

eih

2(eih

2−1)

2(e2ih −1) +eih(eiθ−1)

(e2ih −1) +(eiθ−1)2

(eih −1)(e2ih −1)

Zz2

z

(z2−τ)f00(τ)dτ.

Since f∈W2

K, thus due to (10) and (11), we have

f(z)−S1(z)≤

eih

2eih(eih

2−1)

2(e2ih −1) −eih(eiθ−1)

(e2ih −1) +eih(eiθ−1)2

(eih −1)(e2ih −1)

|z1−z0|2

2

+

eih

2(eih

2−1)(z2−τ)2

4(e2ih −1) +eih(eiθ−1)(z2−τ)2

2(e2ih −1) +(z2−τ)2(eiθ−1)2

2(eih −1)(e2ih −1) −(z1eiθ−τ)2

2

z

z1

+

eih

2(eih

2−1)

2(e2ih −1) +eih(eiθ−1)

(e2ih −1) +(eiθ−1)2

(eih −1)(e2ih −1)

|z2−z|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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