On necessary and sufficient conditions of the regularity of summation methods for Laguerre-Fourier series

Summary The problem of convergence of linear means is considered for the Laguerre-Fourier series of continuous functions. An upper estimate is obtained for the Laguerre-Lebesgue function in terms of the entries of the matrix which determines the linear summability method in question. This allows us to prove for such series an analogue of the well-known theorem by S. M. Nikol'skii which provides necessary and sufficient conditions for the summability of trigonometric Fourier series. A theorem on the regularity of the summability methods is also established.     

On the Lebesgue summability of double trigonometric series

We recall that the Lebesgue summability of a single trigonometric series is defined in terms of the symmetric differentiability of the sum of the formally integrated trigonometric series in question. In this paper, we present another proof of the theorem given in Zygmund's monograph. Then we define the notion of Lebesgue summability of a double trigonometric series and extend the theorem of Fatou and Zygmund from single to double trigonometric series.     

On application of matrix summability to Fourier series

Bor has recently obtained a main theorem dealing with absolute weighted mean summability of Fourier series. In this paper, we generalized that theorem for summability method. Also, some new and known results are obtained dealing with some basic summability methods.     

Distributional versions of littlewood’s Tauberian theorem

We provide several general versions of Littlewood’s Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.     

A New Application of Quasimonotone Sequences

We prove a general theorem dealing with the generalized absolute Cesàro summability factors of infinite series. This theorem also includes some new and known results.     

Summability of Laguerre series at the pointx=0

In this paper, the authors prove a theorem on matrix summability of Laguerre series at the point x=0. Various results on Casàro, Nörlund and generalized Nörlund summability method have been deduced.     

On strong Nörlund summability of Fourier series

In this note, a sufficient condition for summability of Fourier series has been obtained which in conjunction with the author's Tauberian theorem [M.L. Mittal, A Tauberian theorem on strong Nörlund summability, J. Indian Math. Soc. 44 (1980) 369-377] on strong Nörlund summability gives a sufficient condition for summability [C,1,2] of a Fourier series. This generalizes results due to Prasad [G. Prasad, On strong Nörlund summability of Fourier series, Univ. Roorkee Res. J. 9 (1966-1967) 1-10] and Varshney [O.P. Varshney, Note on H₂ summability of Fourier series, Boll. Un. Mat. Ital. 16 (1961) 383-385].     

A characterization of conjugate WCG banach spaces

It is proved that a WCG Banach space X is isomorphic to a conjugate Banach space if and only if there exists a closed subspace V of its conjugate space X' with positive characteristic such that X possesses the following summability property with respect to V: For every bounded sequence in X there exists a regular essentially positive summability method A such that the A-means of the sequence are σ(X,V)-convergent in X. This extends a well-known theorem of Nishiura-Waterman [8] and yields an analogous characterization of quasi-reflexive spaces. Conjugate spaces of smooth Banach spaces can also be characterized by the above summability condition.     

On absolute summability factors of infinite series

In this paper using δ-quasi-monotone sequences a theorem on summability factors of infinite series, which generalizes a theorem of Bor [4] on summability factors of infinite series, is proved. Also, in the special case this theorem includes a result of Mazhar [8] on |C, 1|_k summability factors.     

Multipliers for ¦N,p n;δ¦ k summability of infinite series

In this paper a theorem on ¦N,p _n;δ¦_ksummability factors, which generalizes a theorem of Bor [3] on ¦N,p _n¦_ksummability factors, has been proved.     

Some theorems on the general summability methods

In this paper a new theorem which covers many methods of summability is proved. Several results are also deduced.     

A Tauberian theorem for Cesàro summability of integrals

In this paper we give a proof of the generalized Littlewood Tauberian theorem for Cesàro summability of improper integrals.     

On the uniform Nörlund summability of Fourier series

A general theorem on uniform Nörlund summability of Fourier series has been derived. Some known results become its special cases.     

Note On The Utility Of A Theorem Of Leindler–Meir–Totik

Our purpose is to generalize and to extend a theorem of S. Sharma and S. K. Varma [15] concerning the order of approximation by Abel means in the Lipschitz norm. The proof is basically based on a simple extension of a general theorem of L. Leindler, A. Meir and V. Totik [6] related to approximation by finite summability methods.     

On Fourier–Haar coefficients of the function from the Marcinkiewicz space

The main objective of the paper is to describe asymptotic behaviour of Fourier–Haar coefficients of functions from Marcinkiewicz spaces. We also discuss the Cesàro summability and the almost convergence of sequences related to Fourier–Haar coefficients and generalise a result from [17]. Some analogues of Mercer's theorem for Fourier coefficients are proved.     

Compactness in the spaces of variable integrability and summability

We study totally bounded sets in the spaces of variable integrability and summability. The full characterization of these sets is given. Furthermore, the Sudakov theorem in the setting of the mixed Lebesgue sequence spaces is proven.     

A remark about the statistical Cesàro summability and the Orlicz-Pettis theorem

We prove a version of the Orlicz-Pettìs theorem within the frame of the Statistical Cesàro summability.     

A generalisation of a theorem of Pati on the Nörlund summability of Fourier series

Pati generalised a theorem by Siddiqi on the harmonic summability of Fourier series, by replacing the special sequence by a more gererral sequence. Dikshit proved an analogue of it in the case of conjugate series of Fourier series and further he improved his theorem by introducing a functional factor. The problem which was unsolved as to the same improvement and generalisation can be done in case of Fourier series is tackled and proved by me in the paper attached herewith.     

Hausdorff-summability of power series II

Let H _x a regular Hausdorff method and P(w)=∑ a_k w^k a power series with positive radius of convergence. A theorem of Okada states that P(w) is summable (H _x ) for w in a certain starshaped region G(H _x ,P). We call G=G(H _x ,P) the exact region of summability for P if summability cannot hold for any w \( \in \bar G\) Okada's theorem is said to be sharp for H_x if G(H_x,P) is the exact region of summability for any P. Three items are treated: 1. Criteria for Okada's theorem to be sharp are given in terms of the distribution function X (t) and the Mellin transform \(D(z) = \int\limits_0^1 {t^z d\chi (t)} \) . 2. When is Okada's theorem sharp for product methods? 3. Special classes of functions P(w) are indicated such that G(H_x, P) is the exact region of summability for any H_x. We use the notations of “Hausdorff-Summability of Power Series I” referred as “I”.     

On the convergence of Padé-type approximants to analytic functions

For a given summability method, the Okada theorem describes a domain, into which an arbitrary power series can be analytically continued, if such a domain is known for the geometric series. In this paper, Okada's theorem is extended to more general methods of analytic continuation. This results is applied to a special rational approximation, the so-called Padé-type approximation.