In this paper, we prove a simple inequality which plays important role in the summability theory, matrix operators theory, approximation theory, and also provides great convenience in computations. As a corollary, we give the well known results of [1,2,5] under some simpler conditions, and a very short and different.proofs of results in [6,7] .

In this note the almost sure convergence of stationary, '-mixing sequences of random variables with values in real, separable Banach spaces according to summability methods is linked to the fullllment of a certain integrability condition generalizing and extending the results for i.i.d. sequences. Furthermore we give via Baum-Katz type results an estimate for the rate of convergence in these laws.

In a previous paper [9], some classes of triangular matrix transformations between the series spaces summable by absolute weighted summability methods were characterized. present paper, we extend these to four dimensional matrices and double methods.

In this paper, an application to the approximation by wavelets has been obtained by using matrix-Cesàro (Λ · C1) method of Jacobi polynomials. The rapid rate of convergence of matrix-Cesàro method of Jacobi polynomials are estimated. The result of Theorem (6.1) of this research paper is applicable for avoiding the Gibbs phenomenon in intermediate levels of wavelet approximations. There are majo...

In the classical summability setting rates of summation have been introduced in several ways (see, e.g., [10], [21], [22]). The concept of statistical rates of convergence, for nonvanishing two null sequences, is studied in [13]. Unfortunately no single de...nition seems to have become the “standard” for the comparison of rates of summability transforms. The situation becomes even more uncharte...

The degree of approximation of a function f belonging to Lipschitz class by the Cesàro mean and f ∈ Hα by the Fejér means has been studied by Alexits [4] and Prössdorf [7] respectively. But till now no work seems to have been done to obtain best approximation of functions belonging to generalized Zygmund class, Z (w) r , (r ≥ 1) by product summability means of the form (∆.E1). Z (w) r class is ...

k=1 |ak|, in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤ ...

In this article, we generalize results of Lutz, Miyake and Sch afke concerning summability of formal solutions of the Cauchy problem for the complex heat equation. In particular , we show that the type of summability depends on the given initial condition. 0. Introduction. In detail, we will be concerned with the following problem for the complex heat equation (in two complex variables and z): ...

k=1 |ak| , in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤...

A good amount of work has been done on degree of approximation of functions belonging to Lipα, Lip(α, r), Lip(ξ(t), r) and W (L, ξ(t)) classes using Cesàro and (generalized) Nörlund single summability methods by a number of researchers like Alexits [1], Sahney and Goel [11], Qureshi and Neha [9], Quershi [7, 8], Chandra [2], Khan [4], Leindler [5] and Rhoades [10]. But till now no work seems to...