# Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators

Authors: M. L. Mittal; Mradul Veer Singh
Journal: International Journal of Analysis
 (2015)

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## Abstract

Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of orderα: Cα, Euler Eq, and Nörlund Np) means of its Fourier-Laguerre series at the point x=0 after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by (T·Eq)-transform of its Fourier-Laguerre series at frontier point x=0, where T is a general lower triangular regular matrix. A particular case, when T is a Cesáro matrix of order 1, that is, C1, has also been discussed as a corollary of main result.


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## Body

## 1. Introduction

Let∑un≔∑n=0∞un be given infinite series with the sequence of its (n+1)th partial sums {sn}. Define Enq=(1+q)-n∑k=0nnkqksk, ∀q>0. If limn→∞Enq=S1, then the series ∑‍un is said to be Eq-summable to S1.LetT≡(an,k) be an infinite triangular matrix with real constants. The sequence-to-sequence transformation tn=∑k=0n‍an,ksk,∀n≥0, defines the T-transform of the sequence {sn}. The series ∑un is said to be T-summable to S2 if limn→∞tn=S2. Throughout this paper, T≡(an,k) has nonnegative entries with row sums one. T is said to be regular if it is limit preserving over the space of convergent sequences. Thus, T behaves as a linear operator.The(T·Eq)-transform of {sn}, denoted by (TE)n,q, are defined by (1)TEn,q=∑r=0nan,rErq=∑r=0nan,r11+qr∑k=0rrkqksk.If (TE)n,q→S as n→∞, then the series ∑n=0∞un is said to be (T·Eq)-summable to S. The regularity of (T·Eq) method follows from the regularity of Eq method as well as T-method and thus the matrix (T·Eq) behaves as a linear operator. Some important particular cases of the matrix-Euler operator are as follows:(i)
Ifan,k=1/n-k+1log⁡(n+1), then we get (H1·Eq) operator.(ii)
Let{pn} be a sequence of real, nonnegative numbers such that Pn=∑k=0n‍pk≠0, P-1=0=p-1, and p0≠0. If an,k=pn-k/Pn, then we get (Np·Eq) operator. A special case in which pn=n+α-1α-1, α>0; then (Np·Eq) operator further reduces to (Cα·Eq) operator.(iii)
Ifan,k=pn-kqk/Rn, where Rn=∑k=0npn-kqk≠0, then we get (N,pn,qn)·Eq operator.(iv)
Ifq=1 in above cases, then we get (H1·E1), (Np·E1), (Cα·E1), and (N,pn,qn)·E1 operators, respectively.(v)
If we take identity matrixI instead of Euler matrix Eq, then (T·Eq) operators reduce to T-operators which further reduce to Cesáro Cα, Euler Eq, Harmonic H1, and Nörlund Np operators with suitable choice of an,k as above.Remark 1.
The product summability methods are more powerful than the individual summability methods; for example, the infinite series1-4∑n=1∞‍(-3)n-1 is neither C1-summable nor E1-summable. However, it can be shown easily that the above series is (C1·E1)-summable [1, page 11]. Thus, the product summability methods give an approximation for wider class of functions than the individual methods. Some more examples and recent results on product summability methods can be seen in [2, 3].Remark 2.
As in [4], for operator (T·C1), the product summability operator (T·Eq) behaves as double digital filter and thus plays an important role in signal theory.The Fourier-Laguerre expansion of a functionf(x)∈L1[0,∞) is given by (2)fx~∑n=0∞anfLnαx,where (3)anf=Γα+1n+αα-1∫0∞e-yyαfyLnαydyand Lnα(x) denotes the nth degree Laguerre polynomial of order α>-1, defined by the generating function (4)∑n=0∞Lnαxωn=1-ω-α-1e-xω/ω-1,provided the integral in (3) exists. The elementary properties of Laguerre polynomials can be seen in [5, 6]. Let sn(f;x)=∑k=0nak(f)Lkα(x) denote the partial sums, called Fourier-Laguerre polynomials of degree n, of the first (n+1) terms of the Fourier-Laguerre series of f in (2). At the point x=0, (5)snf;0=∑k=0nakfLkα0=1Γα+1∫0∞e-yyαfy∑k=0nLkαydy=1Γα+1∫0∞e-yyαfyLnα+1ydy,since Lnα(0)=n+αα and ∑k=0nLkα(x)=Lnα+1(x). Thus, using sn(f;0) and (1), we get (6)TEn,qf;0=∑r=0nan,r1+q-r∑k=0rrkqkΓα+1∫0∞e-yyαfyLkα+1ydy.We write (7)ϕy=e-yyαfy-f0Γα+1.


## 2. Main Results

Various investigators such as Gupta [7], Singh [8], Beohar and Jadiya [9], Lal and Nigam [10], and Nigam and Sharma [11] have studied the degree of approximation of a function using different summability (Cα,Eq, and Np) methods of series (2) at the point x=0 after replacing the continuity condition in Szegö theorem [12] by much lighter conditions. The main aim of this paper is to generalize these earlier results in view of Remark 1. We prove the following.Theorem 3.
LetT≡(an,k) be an infinite lower triangular regular matrix with nonnegative entries. Then, the degree of approximation of a function f by its Fourier-Laguerre expansion (2) at the point x=0 using matrix-Euler operators is given by (8)TEn,qf;0-f0=oξn,provided that (9)Φt=∫0tϕydy=otα+1ξ1t,t⟶0,(10)∫δney/2y-2α+3/4ϕydy=on-2α+1/4ξn,(11)∫n∞ey/2y-1/3ϕydy=oξn,n⟶∞,where δ is a fixed positive constant, α∈(-1,-1/2), and ξ(t) is a positive monotonic increasing function of t such that ξ(n)→∞ as n→∞.Corollary 4 (see [13]).
The degree of approximation of a functionf by its Fourier-Laguerre expansion (2) at the point x=0 using [C1Eq](q≥1)-means is given: (12)C1Eqnf;0-f0=oξn,provided (9), (10), and (11) and supplementary conditions on δ, α, and ξ(t) hold as in Theorem 3.Proof.
Ifan,k=(n+1)-1, then (T·Eq) operator reduces to (C1·Eq) operator. Hence, the proof is completed.


## 3. Lemmas

Lemma 5 (see [6, page 177]).
Letα be arbitrary and real and c and ω fixed positive constants and let n→∞. Then, (13)Lnαx=Onα,0≤x≤cn-1;x-2α+1/4On2α-1/4,cn-1≤x≤ω.Lemma 6 (see [6, page 241]).
Letα and λ be arbitrary and real, a>0, and 0<η<4. Then, for n→∞, (14)maxx⁡e-x/2xλLnαx~nQ,where (15)Q=maxλ-12,α2-14,α≤x≤4-ηn;maxλ-13,α2-14,x≥a.


## 4. Proof of the Main Results

In view of the orthogonality of Laguerre polynomials [6, page 100] and (6) and (7), (16)TEn,qf;0-f0=∑r=0nan,r1+q-r∑k=0rrkqk∫0∞ϕyLkα+1ydy=∑r=0nan,r1+q-r∑k=0rrkqk∫0c/n​+∫c/nδ​+∫δn​+∫n∞​ϕyLkα+1ydy=∑j=14Ij,say,where (17)I1≤∑r=0nan,r1+q-r∑k=0rrkqk∫0c/nϕyLkα+1ydy=∑r=0nan,r1+q-r∑k=0rrkqkOkα+1ocα+1ξn/cnα+1=Onα+1oξn/cnα+1∑r=0nan,r1+q-r∑k=0rrkqk=oξnc=oξn,in view of Lemma 5 (first part) and condition (9), and(18)I2≤∑r=0nan,r1+q-r∑k=0rrkqk∫c/nδϕyLkα+1ydy=∑r=0nan,r1+q-r∑k=0rrkqkOk2α+1/4∫c/nδϕyy-2α+3/4dy=On2α+1/4∫c/nδϕyy-2α+3/4dy=oξn,in view of Lemma 5 (second part) and condition (9), integrating by parts and using the argument as in [11, page 6]. Alternatively, using (19)1+q-r∑k=0rrkqkk2α+1/4≤1+2kr2α+1/4,0<q≤K-1,α≥-12,for I2, (18) can be proved as in [13, pages 37-38]. Now (20)I3≤∑r=0nan,r1+q-r∑k=0rrkqk∫δney/2y-2α+3/4ϕye-y/2y2α+3/4Lkα+1ydy=∑r=0nan,r1+q-r∑k=0rrkqkOk2α+1/4∫δney/2y-2α+3/4ϕydy=On2α+1/4on-2α+1/4ξn=oξn,using Lemma 6 and condition (10), and (21)I4≤∑r=0nan,r1+q-r∑k=0rrkqk∫n∞ey/2y-3α+5/6ϕye-y/2y3α+5/6Lkα+1ydy=∑r=0nan,r1+q-r∑k=0rrkqkOkα+1/2∫n∞ey/2y-3α+5/6ϕydy=oξn,using Lemma 6 and condition (11). Combining (17)–(21) and putting them into (16), this completes the proof of Theorem 3.



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