Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators
Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of order : , Euler , and Nörlund ) means of its Fourier-Laguerre series at the point 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 -transform of its Fourier-Laguerre series at frontier point , where is a general lower triangular regular matrix. A particular case, when is a Cesáro matrix of order 1, that is, , has also been discussed as a corollary of main result.
Dedicated to Professor Bani Singh
Let be given infinite series with the sequence of its th partial sums . Define , . If , then the series is said to be -summable to .
Let be an infinite triangular matrix with real constants. The sequence-to-sequence transformation , defines the -transform of the sequence . The series is said to be -summable to if . Throughout this paper, has nonnegative entries with row sums one. is said to be regular if it is limit preserving over the space of convergent sequences. Thus, behaves as a linear operator.
The -transform of , denoted by , are defined by If as , then the series is said to be -summable to . The regularity of method follows from the regularity of method as well as -method and thus the matrix behaves as a linear operator. Some important particular cases of the matrix-Euler operator are as follows:(i)If , then we get operator.(ii)Let be a sequence of real, nonnegative numbers such that , , and . If , then we get operator. A special case in which , ; then operator further reduces to operator.(iii)If , where , then we get operator.(iv)If in above cases, then we get , , , and operators, respectively.(v)If we take identity matrix instead of Euler matrix , then operators reduce to -operators which further reduce to Cesáro , Euler , Harmonic , and Nörlund operators with suitable choice of as above.
Remark 1. The product summability methods are more powerful than the individual summability methods; for example, the infinite series is neither -summable nor -summable. However, it can be shown easily that the above series is -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 , for operator , the product summability operator behaves as double digital filter and thus plays an important role in signal theory.
The Fourier-Laguerre expansion of a function is given by where and denotes the th degree Laguerre polynomial of order , defined by the generating function provided the integral in (3) exists. The elementary properties of Laguerre polynomials can be seen in [5, 6]. Let denote the partial sums, called Fourier-Laguerre polynomials of degree , of the first terms of the Fourier-Laguerre series of in (2). At the point , since and . Thus, using and (1), we get We write
2. Main Results
Various investigators such as Gupta , Singh , Beohar and Jadiya , Lal and Nigam , and Nigam and Sharma  have studied the degree of approximation of a function using different summability , and methods of series (2) at the point after replacing the continuity condition in Szegö theorem  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. Let be an infinite lower triangular regular matrix with nonnegative entries. Then, the degree of approximation of a function by its Fourier-Laguerre expansion (2) at the point using matrix-Euler operators is given by provided that where is a fixed positive constant, , and is a positive monotonic increasing function of such that as .
Corollary 4 (see ). The degree of approximation of a function by its Fourier-Laguerre expansion (2) at the point using -means is given: provided (9), (10), and (11) and supplementary conditions on , , and hold as in Theorem 3.
Proof. If , then operator reduces to operator. Hence, the proof is completed.
Lemma 5 (see [6, page 177]). Let be arbitrary and real and and fixed positive constants and let . Then,
Lemma 6 (see [6, page 241]). Let and be arbitrary and real, , and . Then, for , where
4. Proof of the Main Results
In view of the orthogonality of Laguerre polynomials [6, page 100] and (6) and (7), where in view of Lemma 5 (first part) and condition (9), andin view of Lemma 5 (second part) and condition (9), integrating by parts and using the argument as in [11, page 6]. Alternatively, using for , (18) can be proved as in [13, pages 37-38]. Now using Lemma 6 and condition (10), and using Lemma 6 and condition (11). Combining (17)–(21) and putting them into (16), this completes the proof of Theorem 3.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The second author is thankful to the Ministry of Human Resource Development, India, for financial support to carry out this research work.
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