Analytic Representation of the Sequence of Functions on \(L^1\mathbb{R}\) Space

Abstract

In this paper studies the convergence of the functional sequence, of functions belonging to \(L^1\mathbb{R}\) space and their analytic representations. In the first theorem using Parseval’s theorem and a result that describes the inverse Fourier transform between the Heaviside function, we prove the analytic representation of the functional sequence, of the functions belonging to the \(L^1\mathbb{R}\) space, and uniform convergence of the sequence of their analytic representation. Using Fourier transform and the Cauchy representation we show that the sequence of the analytic representation, converges uniformly to the functions belonging to the same space on the compact subset. In the last part, we applied Fubini’s theorem to the functional sequence \(\theta_n(t)\) , and if we have a sequence of functions on \(L^1\mathbb{R}\) and another function from the same space, then the sequence of convolutions converges also on \(L^1\mathbb{R}\) space.