# Quasi-Positive Delta Sequences and Their Applications in Wavelet Approximation

Authors: Shyam Lal; Susheel Kumar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2016)

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## Abstract

A sufficient literature is available for the wavelet error of approximation of certain functions in theL2-norm. There is no work in context of multiresolution approximation of a function in the sense of sup-error. In this paper, for the first time, wavelet estimator for the approximation of a function f belonging to Lipα[a,b] class under supremum norm has been obtained. Working in this direction, four new theorems on the wavelet approximation of a function f belonging to Lipα,0<α≤1 class using the projection Pmf of its wavelet expansions have been estimated. The calculated estimator is best possible in wavelet analysis.


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## Body

## 1. Introduction

Orthogonal wavelet is a new development in analysis but very useful in engineering and technology, especially in high-resolution images and signal processing due to their localization properties in time and frequency. Wavelet expansions are superior to classical orthogonal series like Fourier series. Some properties of wavelet expansions have been studied by Chui [1], Daubechies and Lagarias [2], Meyer [3], Walter [4, 5], Islam et al. [6], and so forth. The idea of approximation of various functional spaces under different norms is obtained by Lal and Kumar [7, 8], Abu-Sirhan [9], Coskun [10], and Shiri and Azadi Kenary [11] which gives the inspiration for the present work. But till now no work seems to have been done to obtain the wavelet approximation of a function f∈Lipα[a,b],0<α≤1 using the projection (Pmf) of its wavelet expansion and to discuss its convergence. In an attempt to make an advance study in this direction, in this paper, the best possible wavelet sup-error Emf of a function f∈Lipα[a,b] by Pmf of its wavelet expansion has been determined. The convergence of wavelet expansions has been also discussed.


## 2. Definitions and Preliminaries

### 2.1. Multiresolution Analysis

LetZ be the set of all integers. A multiresolution analysis of L2(R) is defined as a sequence of closed subspaces Vj of L2(R),j∈Z, with the following properties:(1)
Vj⊂Vj+1.(2)
fx∈Vj⇔f2x∈Vj+1.(3)
fx∈V0⇔fx+1∈V0.(4)
⋃j=-∞∞Vj is dense in L2(R) and ⋂j=-∞∞Vj=0.(5)
Suppose a functionϕ∈V0 exists such that the collection ϕ(x-k);k∈Z is a Riesz basis of V0.Letψ∈L2(R), and(1)ψj,k≔2j/2ψ2j.-k,Wj≔closψj,k:k∈Z.Then this family of subspaces of L2(R) gives direct sum decomposition of L2(R) in the sense that every f∈L2(R) has unique decomposition: (2)fx=⋯+g-2x+g-1x+g0x+g1x+g2x+⋯,where gj∈Wj for all j∈Z and we describe this by writing (3)L2R=Vj⊕i=j∞Wi,where (4)Vj≔⊕k=-∞j-1Wk.ψj,k;k∈Z is a Riesz basis of Wj.A functionϕ∈L2(R) is called a scaling function, if the subspaces Vj of L2R defined by (5)Vj≔closL2Rϕj,k:k∈Z,j∈Zsatisfy properties (1) to (5) stated above in this section. It is important to note that the scaling function ϕ generates a multiresolution analysis Vj of L2(R) (Debnath [12]).Following Walter [4], each f∈L2(R) has two representations (6)ft=∑m=-∞∞∑n=-∞∞f,ψm,nψm,nt,(7)ft=∑n=-∞∞f,ϕm,nϕm,nt+∑k=m∞∑n=-∞∞f,ψk,nψk,nt=Pmft+rmft,where convergence is in sense of L2(R). The function (Pmf)∈Vm and in fact is the projection of f onto Vm. It is also given in terms of the kernel Km(x,y) of Vm as (8)Pmfx=∫-∞∞Kmx,tftdt. The function Km is given by (9)Kmx,y=2mK2mx,2ny, where (10)Kx,y=∑n=-∞∞ϕx-nϕ¯y-n,1=∫-∞∞ϕxdx=∑n=-∞∞ϕx-n.It is remarkable to note that(11)Kx,y=∑n=-∞∞ϕx-nϕ¯y-n(12)Kmx,y=2mK2mx,2my=2m∑n=-∞∞ϕ2mx-nϕ¯2my-n=∑n=-∞∞2m/2ϕ2mx-n2m/2ϕ¯2my-n.Then (13)∫-∞∞Kmx,y=∑n=-∞∞2m∫-∞∞ϕ2mx-ndxϕ¯2my-n=∑n=-∞∞∫-∞∞ϕtdtϕ¯2my-n=∑n=-∞∞ϕ¯y-n=1.


### 2.2. Quasi-Positive Delta Sequences

If(1)
there isC>0 such that ∫-∞∞|δm(x,y)|dx≤C,y∈R,m∈N,(2)
∫y-Cy+Cδm(x,y)dx→1 uniformly on compact subset of R, as m→∞,(3)
for eachγ>0, (14)supx-y≥γδmx,y⟶0asm⟶∞then {δm(·,y)} is known as quasi-positive delta sequence of function.An example of a quasi-positive delta sequence is the Fejer kernel,(15)Fmx,y=sin2⁡m+1/2x-y2m+1πsin2⁡x-y/2χ-π,πx-y,while a delta sequence that is not quasi-positive is the Dirichlet kernel of Fourier series, (16)Dmx,y=sin⁡m+1/2x-y2πsin⁡x-y/2χ-π,πx-ybecause ∫-∞∞Dm(x,y)dx is not absolutely convergent.


### 2.3. Function ofLipα[a,b] Class

A functionf∈Lipα[a,b] if(17)fx-fy=Ox-yα,for0<α≤1,(Titchmarsh [13], p. 406).Examples(1)
I f   f ( x ) = x   ∀ x ∈ [ 0,1 ] ,   f ∈ L i p α . (2)
Iff(x)=1/x∀x∈(0,1],f∉Lip1(0,1]. LetN be a positive integer. Let x=1/N,y=1/(N+1)x,y∈(0,1]. Thenx-y=1/N-1/(1+N)<1/N and 1/(x-y)>N. Now,fx-fy/x-y=N-N+1/x-y=1/x-y≥N | f ( x ) - f ( y ) | > N | x - y | ∴ f ∉ L i p 1 ( 0,1 ] .Remarks (1)
Iff∈Lipα[a,b]α>0, then f is continuous, indeed, uniformly continuous on [a,b]. (2)
L i p α class is a linear space over R or C. (3)
Iff∈Lipαα>1, then f is constant function.


### 2.4. ProjectionPmf

P m f, the orthogonal projection of L2(R) onto Vm, is defined by (18)Pmf=∑k=-∞∞f,ϕm,kϕm,k,m∈Z,(Sweldens and Piessens [14]).


### 2.5. Wavelet Approximation

The wavelet approximationEm(f) of a function f under supremum norm is defined by(19)Emf=f-Pmf∞=supx⁡fx-Pmfx (Zygmund [15], p. 114).IfEm(f)→0 as m→∞ then Em(f) is called the best approximation of f of order m (Zygmund [15], p. 115).


## 2.1. Multiresolution Analysis

LetZ be the set of all integers. A multiresolution analysis of L2(R) is defined as a sequence of closed subspaces Vj of L2(R),j∈Z, with the following properties:(1)
Vj⊂Vj+1.(2)
fx∈Vj⇔f2x∈Vj+1.(3)
fx∈V0⇔fx+1∈V0.(4)
⋃j=-∞∞Vj is dense in L2(R) and ⋂j=-∞∞Vj=0.(5)
Suppose a functionϕ∈V0 exists such that the collection ϕ(x-k);k∈Z is a Riesz basis of V0.Letψ∈L2(R), and(1)ψj,k≔2j/2ψ2j.-k,Wj≔closψj,k:k∈Z.Then this family of subspaces of L2(R) gives direct sum decomposition of L2(R) in the sense that every f∈L2(R) has unique decomposition: (2)fx=⋯+g-2x+g-1x+g0x+g1x+g2x+⋯,where gj∈Wj for all j∈Z and we describe this by writing (3)L2R=Vj⊕i=j∞Wi,where (4)Vj≔⊕k=-∞j-1Wk.ψj,k;k∈Z is a Riesz basis of Wj.A functionϕ∈L2(R) is called a scaling function, if the subspaces Vj of L2R defined by (5)Vj≔closL2Rϕj,k:k∈Z,j∈Zsatisfy properties (1) to (5) stated above in this section. It is important to note that the scaling function ϕ generates a multiresolution analysis Vj of L2(R) (Debnath [12]).Following Walter [4], each f∈L2(R) has two representations (6)ft=∑m=-∞∞∑n=-∞∞f,ψm,nψm,nt,(7)ft=∑n=-∞∞f,ϕm,nϕm,nt+∑k=m∞∑n=-∞∞f,ψk,nψk,nt=Pmft+rmft,where convergence is in sense of L2(R). The function (Pmf)∈Vm and in fact is the projection of f onto Vm. It is also given in terms of the kernel Km(x,y) of Vm as (8)Pmfx=∫-∞∞Kmx,tftdt. The function Km is given by (9)Kmx,y=2mK2mx,2ny, where (10)Kx,y=∑n=-∞∞ϕx-nϕ¯y-n,1=∫-∞∞ϕxdx=∑n=-∞∞ϕx-n.It is remarkable to note that(11)Kx,y=∑n=-∞∞ϕx-nϕ¯y-n(12)Kmx,y=2mK2mx,2my=2m∑n=-∞∞ϕ2mx-nϕ¯2my-n=∑n=-∞∞2m/2ϕ2mx-n2m/2ϕ¯2my-n.Then (13)∫-∞∞Kmx,y=∑n=-∞∞2m∫-∞∞ϕ2mx-ndxϕ¯2my-n=∑n=-∞∞∫-∞∞ϕtdtϕ¯2my-n=∑n=-∞∞ϕ¯y-n=1.


## 2.2. Quasi-Positive Delta Sequences

If(1)
there isC>0 such that ∫-∞∞|δm(x,y)|dx≤C,y∈R,m∈N,(2)
∫y-Cy+Cδm(x,y)dx→1 uniformly on compact subset of R, as m→∞,(3)
for eachγ>0, (14)supx-y≥γδmx,y⟶0asm⟶∞then {δm(·,y)} is known as quasi-positive delta sequence of function.An example of a quasi-positive delta sequence is the Fejer kernel,(15)Fmx,y=sin2⁡m+1/2x-y2m+1πsin2⁡x-y/2χ-π,πx-y,while a delta sequence that is not quasi-positive is the Dirichlet kernel of Fourier series, (16)Dmx,y=sin⁡m+1/2x-y2πsin⁡x-y/2χ-π,πx-ybecause ∫-∞∞Dm(x,y)dx is not absolutely convergent.


## 2.3. Function ofLipα[a,b] Class

A functionf∈Lipα[a,b] if(17)fx-fy=Ox-yα,for0<α≤1,(Titchmarsh [13], p. 406).Examples(1)
I f   f ( x ) = x   ∀ x ∈ [ 0,1 ] ,   f ∈ L i p α . (2)
Iff(x)=1/x∀x∈(0,1],f∉Lip1(0,1]. LetN be a positive integer. Let x=1/N,y=1/(N+1)x,y∈(0,1]. Thenx-y=1/N-1/(1+N)<1/N and 1/(x-y)>N. Now,fx-fy/x-y=N-N+1/x-y=1/x-y≥N | f ( x ) - f ( y ) | > N | x - y | ∴ f ∉ L i p 1 ( 0,1 ] .Remarks (1)
Iff∈Lipα[a,b]α>0, then f is continuous, indeed, uniformly continuous on [a,b]. (2)
L i p α class is a linear space over R or C. (3)
Iff∈Lipαα>1, then f is constant function.


## 2.4. ProjectionPmf

P m f, the orthogonal projection of L2(R) onto Vm, is defined by (18)Pmf=∑k=-∞∞f,ϕm,kϕm,k,m∈Z,(Sweldens and Piessens [14]).


## 2.5. Wavelet Approximation

The wavelet approximationEm(f) of a function f under supremum norm is defined by(19)Emf=f-Pmf∞=supx⁡fx-Pmfx (Zygmund [15], p. 114).IfEm(f)→0 as m→∞ then Em(f) is called the best approximation of f of order m (Zygmund [15], p. 115).


## 3. Theorems

In this paper, we prove the following theorems.Theorem 11.
Letϕ∈L2(R) be a scaling function and ψ∈L2(R) be a basic wavelet satisfying the admissibility condition (20)Cψ=∫-∞∞ψ^ω2ωdω<∞.If a function f∈Lipα[a,b] is represented by its wavelet expansion as(21)f·=∑n=-∞∞f,ϕm,nϕm,n·+∑k=m∞∑n=-∞∞f,ψm,nψm,n·=Pmf·+rmf·, then the wavelet approximation Em(f) of f by (Pmf) satisfies (22)Emf=f-Pmf∞=supy∈a,b⁡fy-Pmfy=O1m+1α.Theorem 12.
If a functionf∈Lipα[a,b] and Pmf is the projection of f onto Vm then Pmf→f as m→∞uniformly on[a,b].Theorem 13.
Iff∈Lipα[a,b] and (23)f·=Pmf·+rmf· then (24)rmf∞=O1m+1α,0<α≤1.Theorem 14.
Iff∈Lipα[a,b], (25)f·=Pmf·+rmf·, then rmf→0 as m→∞ uniformly on [a,b].


## 4. Lemma

For the proof of our theorems, the following lemma is required.Lemma 15.
IfKm(x,y)=2mK(2mx,2my),x,y∈[a,b], then Km(·,y)-∞∞ is a quasi-positive delta sequence of function in L1(R).


### 4.1. Proof of Lemma15

Following the proof of Walter ([4] p. 113), (26)∫-∞∞Kmx,ydx=C.Let c>0; then (27)∫y-cy+cKmx,ydx=1-∫-∞y-2mcKt,ydt-∫y+2mc∞Kt,ydt∫-∞y-2mcK(t,y)dt→0 as m→∞ and ∫y+2mc∞K(t,y)dt→0 as m→∞.Thus,∫y-cy+cKm(x,y)dx→1 as m→∞ uniformly on [a,b]⊂R and sup|x-y|≥1/m+1⁡|Km(x,y)|→0 as m→∞, [4].ThereforeKm(x,y)-∞∞ is a quasi-positive delta sequence of functions.


## 4.1. Proof of Lemma15

Following the proof of Walter ([4] p. 113), (26)∫-∞∞Kmx,ydx=C.Let c>0; then (27)∫y-cy+cKmx,ydx=1-∫-∞y-2mcKt,ydt-∫y+2mc∞Kt,ydt∫-∞y-2mcK(t,y)dt→0 as m→∞ and ∫y+2mc∞K(t,y)dt→0 as m→∞.Thus,∫y-cy+cKm(x,y)dx→1 as m→∞ uniformly on [a,b]⊂R and sup|x-y|≥1/m+1⁡|Km(x,y)|→0 as m→∞, [4].ThereforeKm(x,y)-∞∞ is a quasi-positive delta sequence of functions.


## 5. Proofs

### 5.1. Proof of Theorem11

( P m f ) is given by (28)Pmfy=∑n=-∞∞f,ϕm,nϕm,ny=∑n=-∞∞∫-∞∞fxϕ¯m,nxdxϕm,ny=∫-∞∞fx∑n=-∞∞ϕm,nyϕm,nx¯dx=∫-∞∞fx∑n=-∞∞2m/2ϕ2my-n2m/2ϕ2mx-n¯dx=∫-∞∞fx∑n=-∞∞2mϕ2my-nϕ2mx-n¯dx=∫-∞∞fx2mK2mx,2mydx=∫-∞∞Kmx,yfxdxfy-Pmfy=fy-∫-∞∞Kmx,yfxdx=fy∫-∞∞Kmx,ydx-∫-∞∞Kmx,yfxdx,∫-∞∞Kmx,ydx=1=∫-∞∞Kmx,yfy-fxdx=∫-∞y-1/m+1Kmx,yfy-fx+∫y-1/m+1y+1/m+1Kmx,yfy-fxdx+∫y+1/m+1∞Kmx,yfy-fxfy-Pmfy≤∫-∞y-1/m+1Kmx,yfy-fx+∫y-1/m+1y+1/m+1Kmx,yfy-fxdx+∫y+1/m+1∞Kmx,yfy-fxdx=I1+I2+I3say.Let x,y∈[a,b]. Select m such that (29)y+1m+1<b,x-1m+1>a.Since it is known that (30)supx∈y-1/m+1,y+1/m+1⁡fx-fy≤fy+1m+1-fy+fy-1m+1-fy=M11m+1α+M21m+1α,f∈Lipα=M1+M21m+1αwhereM1>0,M2>0 therefore (31)I2=∫y-1/m+1y+1/m+1Kmx,yfx-fydx≤M1+M2m+1α∫y-1/m+1y+1/m+1Kmx,ydx≤M1+M2m+1α∫-∞∞Kmx,ydx=M1+M2Cm+1α,C>0by  Lemma15.Next,(32)I1+I3=∫-∞y-1/m+1Kmx,yfy-fxdx+∫y+1/m+1∞Kmx,yfy-fxdx=∫ay-1/m+1Kmx,yfy-fxdx+∫y+1/m+1bKmx,yfy-fxdx≤supx-y≥1/m+1Kmx,yM3m+1α∫ay-1/m+1dx+∫y+1/m+1bdx,M3>0≤supx-y≥1/m+1⁡Kmx,yM3m+1αb-a-2m+1,≤M3b-aC1m+1α,C1>0.{Km(x,y)} is a quasi-positive delta sequence of function.Now, collecting (50), (31), and (32), we have (33)fy-Pmfy≤M3b-aC1m+1α+M1+M2Cm+1α=M3b-aC1+M1+M2Cm+1α=O1m+1α. Thus (34)f-Pmf∞=supy∈a,b⁡fy-Pmfy=O1m+1α.Remark 16 (the converse of Theorem11 is also true).
Choosex,y∈[a,b] such that x-1/(|m|+1),y+1/(|m|+1)∈[a,b] and |y-x|=d/(|m|+1),d>0.(35)fx-fy=fx-Pmfx+Pmfx-Pmfy+Pmfy-fy≤fx-Pmfx+Pmfx-Pmfy+Pmfy-fy≤supx∈a,b⁡fx-Pmfx+supx,y∈a,b⁡Pmfx-Pmfy+supy∈a,b⁡Pmfy-fy=f-Pmf∞+supx,y∈a,b⁡Pmfx-y+Pmf-f∞=O1m+1α+Ox-yα+O1m+1α=O1m+1α+O1m+1α+O1m+1α=Ox-yα. Since |f(x)-f(y)|=O|x-y|α, therefore f∈Lipα[a,b].
Hence, Theorem11 is completely established.


### 5.2. Proof of Theorem12

By the proof of Theorem11(36)Pmf-f∞=O1m+1α.Since,(37)1m+1α⟶0asm⟶∞,therefore,(38)Pmf-f∞⟶0asm⟶∞.Thus,(39)Pmf⟶fasm⟶∞uniformly  ona,b.Hence, Theorem 12 is completely established.


### 5.3. Proof of Theorem13

Since,(40)fx=Pmfx+rmfxtherefore,(41)rmfx=fx-Pmfx,rmf∞=f-Pmf∞=O1m+1α,by  Theorem11.Hence, Theorem 13 is completely established.


### 5.4. Proof of Theorem14

Following the proof of Theorem13, (42)rmf∞=O1m+1α.Since,(43)rmf∞⟶0asm⟶∞,therefore,(44)rmf⟶0asm⟶∞uniformly  ona,b.Hence, Theorem 14 is completely established.


## 5.1. Proof of Theorem11

( P m f ) is given by (28)Pmfy=∑n=-∞∞f,ϕm,nϕm,ny=∑n=-∞∞∫-∞∞fxϕ¯m,nxdxϕm,ny=∫-∞∞fx∑n=-∞∞ϕm,nyϕm,nx¯dx=∫-∞∞fx∑n=-∞∞2m/2ϕ2my-n2m/2ϕ2mx-n¯dx=∫-∞∞fx∑n=-∞∞2mϕ2my-nϕ2mx-n¯dx=∫-∞∞fx2mK2mx,2mydx=∫-∞∞Kmx,yfxdxfy-Pmfy=fy-∫-∞∞Kmx,yfxdx=fy∫-∞∞Kmx,ydx-∫-∞∞Kmx,yfxdx,∫-∞∞Kmx,ydx=1=∫-∞∞Kmx,yfy-fxdx=∫-∞y-1/m+1Kmx,yfy-fx+∫y-1/m+1y+1/m+1Kmx,yfy-fxdx+∫y+1/m+1∞Kmx,yfy-fxfy-Pmfy≤∫-∞y-1/m+1Kmx,yfy-fx+∫y-1/m+1y+1/m+1Kmx,yfy-fxdx+∫y+1/m+1∞Kmx,yfy-fxdx=I1+I2+I3say.Let x,y∈[a,b]. Select m such that (29)y+1m+1<b,x-1m+1>a.Since it is known that (30)supx∈y-1/m+1,y+1/m+1⁡fx-fy≤fy+1m+1-fy+fy-1m+1-fy=M11m+1α+M21m+1α,f∈Lipα=M1+M21m+1αwhereM1>0,M2>0 therefore (31)I2=∫y-1/m+1y+1/m+1Kmx,yfx-fydx≤M1+M2m+1α∫y-1/m+1y+1/m+1Kmx,ydx≤M1+M2m+1α∫-∞∞Kmx,ydx=M1+M2Cm+1α,C>0by  Lemma15.Next,(32)I1+I3=∫-∞y-1/m+1Kmx,yfy-fxdx+∫y+1/m+1∞Kmx,yfy-fxdx=∫ay-1/m+1Kmx,yfy-fxdx+∫y+1/m+1bKmx,yfy-fxdx≤supx-y≥1/m+1Kmx,yM3m+1α∫ay-1/m+1dx+∫y+1/m+1bdx,M3>0≤supx-y≥1/m+1⁡Kmx,yM3m+1αb-a-2m+1,≤M3b-aC1m+1α,C1>0.{Km(x,y)} is a quasi-positive delta sequence of function.Now, collecting (50), (31), and (32), we have (33)fy-Pmfy≤M3b-aC1m+1α+M1+M2Cm+1α=M3b-aC1+M1+M2Cm+1α=O1m+1α. Thus (34)f-Pmf∞=supy∈a,b⁡fy-Pmfy=O1m+1α.Remark 16 (the converse of Theorem11 is also true).
Choosex,y∈[a,b] such that x-1/(|m|+1),y+1/(|m|+1)∈[a,b] and |y-x|=d/(|m|+1),d>0.(35)fx-fy=fx-Pmfx+Pmfx-Pmfy+Pmfy-fy≤fx-Pmfx+Pmfx-Pmfy+Pmfy-fy≤supx∈a,b⁡fx-Pmfx+supx,y∈a,b⁡Pmfx-Pmfy+supy∈a,b⁡Pmfy-fy=f-Pmf∞+supx,y∈a,b⁡Pmfx-y+Pmf-f∞=O1m+1α+Ox-yα+O1m+1α=O1m+1α+O1m+1α+O1m+1α=Ox-yα. Since |f(x)-f(y)|=O|x-y|α, therefore f∈Lipα[a,b].
Hence, Theorem11 is completely established.


## 5.2. Proof of Theorem12

By the proof of Theorem11(36)Pmf-f∞=O1m+1α.Since,(37)1m+1α⟶0asm⟶∞,therefore,(38)Pmf-f∞⟶0asm⟶∞.Thus,(39)Pmf⟶fasm⟶∞uniformly  ona,b.Hence, Theorem 12 is completely established.


## 5.3. Proof of Theorem13

Since,(40)fx=Pmfx+rmfxtherefore,(41)rmfx=fx-Pmfx,rmf∞=f-Pmf∞=O1m+1α,by  Theorem11.Hence, Theorem 13 is completely established.


## 5.4. Proof of Theorem14

Following the proof of Theorem13, (42)rmf∞=O1m+1α.Since,(43)rmf∞⟶0asm⟶∞,therefore,(44)rmf⟶0asm⟶∞uniformly  ona,b.Hence, Theorem 14 is completely established.


## 6. Notes

(1)Em(f)→0 as m→∞ in Theorem 11; the wavelet approximations determined in this theorem are best possible in wavelet analysis (Zygmund, [15], p. 115).(2) Define a functionϕ:[0,1]→R by (45)ϕt=t-1α-tα-1∀t∈0,1,0<α<1=1-tα-1-tα=1-tα+tα-1∀t∈0,1.∴ϕ′t=-α1-tα-1+αtα-1∀t∈0,1=α1t1-α-11-t1-α∴ϕ′t>0,∀t∈0,12=0,t=12<0,∀t∈12,1.Therefore ϕ is monotonic increasing in (0,1/2) and monotonic decreasing in (1/2,1)(46)∴ϕt≥ϕ0=ϕ1=0∀t∈0,1∴t-1α-tα-1≥0∴t-1α≥tα-1.Thus,(47)xy-1α≥xyα-1,t=xy,0<x,y<1.Hence x-yα≥xα-yα(48)that  isxα-yα≤x-yα0<x,y<1. A function f:[a,b]→R is defined by fx=(1-x2)α, 0<α<1,∀x∈[a,b]⊂[0,1].(49)fx-fy=1-x2α-1-y2α≤1-x2-1-y2α,xα-yα≤x-yα=y2-x2α=x-yαx+yα≤x-yαxα+yα≤max⁡2aα,2bαx-yα≤2αaα+bαx-yα=Ox-yα∴f∈Lipα[a,b],0<α<1.For this function,(50)fy-Pmfy≤∫-∞y-1/m+1Kmx,yfy-fxdx+∫y-1/m+1y+1/m+1Kmx,yfy-fxdx+∫y+1/m+1∞Kmx,yfy-fxdx=I1+I2+I3say.Then(51)I2=∫y-1/m+1y+1/m+1Kmx,yx-yαx+yαdx≤2αaα+bα∫y-1/m+1y+1/m+1Kmx,yx-yαdx,x+yα≤2αaα+bα≤22αaα+bαm+1α∫y-1/m+1y+1/m+1Kmx,ydx,x-y≤2m+1≤22αaα+bαm+1α∫-∞∞Kmx,ydx=22αaα+bαm+1α,∫-∞∞Kmx,ydx=1=O1m+1α(52)I1+I3=∫ay-1/m+1Kmx,yfy-fxdx+∫y+1/m+1bKmx,yfy-fxdx≤supx-y≥1/m+1⁡Kmx,y2αaα+bαm+1α∫ay-1/m+1dx+∫y+1/m+1bdx≤supx-y≥1/m+1⁡Kmx,y2αaα+bαm+1αb-a=O1m+1α.Hence (53)supy∈a,b⁡fy-Pmfy=O1m+1α.f(y)-Pmf(y)∞=O1/(|m|+1)α.Thus, the result of Theorem11 is verified for a function f defined by f(x)=(1-x2)α,0<α<1.(3)Hölder class: Let C2π denote the Banach spaces of all 2π periodic continuous functions under sup-norm. For each 0<α≤1 and some positive constant k, the function space Hα is defined by(54)Hα=f∈C2π:fx-fy≤kx-yα(Das et al. [16] p. 83).Letf∈Hα. Then(55)fx-fy=kx-yα=Ox-yα∴f∈Lipα.The functionf(x)=(1-x3)α,0<α<1,∀x∈[0,1]. It is continuous and belonging to Lipα but f(x+2π)≠f(x). Then f∉Hα.ThusHα⫋Lipα.Theorem11 is also valid for a function f∈Hα.



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*Source: 9121249-2016-10-25.xml*