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 ≤...

Various investigators such as Khan 1974 , Chandra 2002 , and Liendler 2005 have determined the degree of approximation of 2π-periodic signals functions belonging to Lip α, r class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al. 2007 and 2011 have obtained the degree of approximation of signals belonging to...

Let V be the intertwining operator with respect to the reflection invariant measure hαdω on the unit sphere S d−1 in Dunkl’s theory on spherical h-harmonics associated with reflection groups. Although a closed form of V is unknown in general, we prove that ∫ Sd−1 V f(y)hα(y)dω = Aα ∫ Bd f(x)(1 − |x|2)|α|1−1dx, where Bd is the unit ball of Rd and Aα is a constant. The result is used to show that...

p . It follows that inequality (1.2) holds for any a ∈ lp when U1/p ≥ ||C||p,p and fails to hold for some a ∈ lp when U1/p < ||C||p,p. Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cn,k = 1/n, k ≤ n and 0 otherwise, is bounded on l p and has norm ≤ p/(p−1). (The norm is in fact p/(p− 1).) We say a matrix A = (an,k) is a lower triangular matrix if an,k = 0 for n < k...

‎let $x,y$ be normed spaces with $l(x,y)$ the space of continuous‎ ‎linear operators from $x$ into $y$‎. ‎if ${t_{j}}$ is a sequence in $l(x,y)$,‎ ‎the (bounded) multiplier space for the series $sum t_{j}$ is defined to be‎ [ ‎m^{infty}(sum t_{j})={{x_{j}}in l^{infty}(x):sum_{j=1}^{infty}%‎ ‎t_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $s:m^{infty}(sum t_{j})rightarrow y$ associat...