Functional Sequential and Trigonometric Summability of Real and Complex Functions

M H Hooshmand

doi:10.28924/2291-8639-15-2017-222

Abstract

Limit summability of functions was introduced as a new approach to extensions of the summation of real and complex functions, and also evaluating antidifferences. Also, limit summand functions generalize the (logarithm of) Gamma-type functions satisfying the functional equation F(x + 1) = f(x)F(x). Recently, another approach to the topic entitled analytic summability of functions, has been introduced and studied by the author. Since some functions are neither limit nor analytic summable, several types of summabilities are needed for improving the problem. Here, I introduce and study functional sequential summability of real and complex functions for obtaining multiple approaches to them. We not only show that the analytic summability is a type of functional sequential summability but also obtain trigonometric summability (for functions with a fourier series) as another its type. Hence, we arrive at a class of real and complex function spaces with various properties. Thereafter, we prove several properties of functional sequential, and also many criteria for trigonometric summability. Finally, we state many problems and future directions for the researches.

Highlights

In the theory of indefinite sum, antidifference and finite calculus, obtaining some special solutions of the difference functional equation∇F (x) := F (x) − F (x − 1) = f (x) ; x ∈ E, (1.1)is very important, where E is the domain of a real or complex function f or a subset of C (analogously for F (x) := F (x + 1) − F (x) = f (x), where is the forward difference operator, e.g., see [1])
The paper [2] states a framework for studying the limit summable functions including basic conditions, many criteria for limit summability, uniqueness conditions for its summand function and so on
We prove many equivalent conditions for trigonometric summability and several criteria for trigonometric summability of functions with a Fourier series

Summary

Introduction

In the theory of indefinite sum, antidifference and finite calculus, obtaining some special solutions of the difference functional equation. The function f is called limit summable at x0 ∈ Σf if the functional sequence {fσn (x)} is convergent at x = x0. The paper [2] states a framework for studying the limit summable functions including basic conditions, many criteria for limit summability, uniqueness conditions for its summand function and so on. Since some of the important special functions are not limit summable, recently, another type of summability entitled ”analytic summability” has been introduced in [4]. We obtain it again in this paper (as a type of functional sequential summability)

Functional sequential summability

Trigonometric summability