Some Theorems on a Generalized Laplace Transform of Generalized Functions

Abstract

In this paper we extend the generalized Laplace transform \[F(s)=\frac{\Gamma(\beta+\eta+1)}{\Gamma(\alpha+\beta+\eta+1)}\int_0^\infty (st)^\beta\ _1F_1(\beta+\eta+1, \alpha+\beta+\eta+1; -st)f(t) dt\] where $f(t)\in L(0,\infty)$, $\beta\ge 0$, $\eta > 0$; to a class of generalized functions. We will extend the above transform to a class of generalized functions as a special case of the convolution transform and prove an inversion formula for it.

Approximation numbers of matrix transformations and inclusion maps

“…Since each component operator of I is the identity operator on X , using the Property [5] of approximation numbers and Theorem 2.5, we conclude that a n…”

“…To show that a n (I ) = 1, 1 ≤ n ≤ k, note that a 1 (I ) = 1. If 1 < n ≤ k, we can conclude that a k (I ) = 1, making use of Property [5] of approximation numbers, the fact that each component operator of I is the identity operator on X and Theorem 2.5. Since approximation numbers are decreasing in nature, we get the required result.…”

“…Proof. When p = ∞, the map under consideration is the identity map on ℓ ∞ (X ) and in this case we get a n (I ) = 1, ∀ n ∈ N using the Property [5] of approximation numbers. For 1 < p < ∞, we always have a n (I ) ≤ 1.…”

“…A function s associating each linear operator T with a sequence {s n (T )} of non-negative reals satisfying the properties analogous to [1]- [5] has been termed in the literature as an sfunction. The map defined by the approximation numbers is indeed the largest s-function [14,15].…”

“…Coming to the study of domain and range spaces of the matrix transformations to be studied in this paper, we refer to [4,5] and [11], for detailed theory of these spaces, namely the vector valued sequence spaces. We denote by Ω(X ), the class of all sequences from X and by Φ(X ), the subspace of Ω(X ) consisting of all finitely non-zero sequences.…”