# Functional Sequential and Trigonometric Summability of Real and Complex Functions

## Main Article Content

### 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.

## Article Details

### References

- P.M. Batchelder, Introduction to Linear Difference Equations, Dover Publications Inc., 1968.
- M.H. Hooshmand, Limit Summability of Real Functions, Real Anal. Exch. 27(2001), 463-472.
- M.H. Hooshmand, Another Look at the Limit Summability of Real Functions, J. Math. Ext. 4(2009), 73-89.
- M.H. Hooshmand, Analytic Summability of Real and complex Functions, J. Contemp. Math. Anal., 5(2016), 262-268.
- R. J. Webster, Log-Convex Solutions to the Functional Equation f(x + 1) = g(x)f(x): Γ -Type Functions, J. Math. Anal. Appl., 209 (1997), 605-623.