Addendum to my paper `The Lebesgue summability of trigonometric integrals" (J. Math. Anal. Appl. 390 (2012), 188-196)

TL;DR

This addendum extends a previous theorem on Lebesgue summability of trigonometric integrals, showing that a key condition can be weakened to a local integrability requirement.

Contribution

It demonstrates that the main result holds under a weaker condition, broadening the applicability of the original theorem.

Findings

Theorem 2 remains valid with the weaker condition (2.7').

The condition f(t) ∈ L^1(-T, T) for all T>0 suffices.

The original statement (2.9) is preserved under this relaxation.

Abstract

We observed that statement (2.9) in Theorem 2 remains valid if condition (2.7) is replaced by the weaker condition that $$f(t) \in L^1(-T, T) \quad {\rm for\ all} \quad T>0.\eqno(2.7')$$