Pointwise behavior of Fourier integrals of functions of bounded variation over R✩

We investigate the uniform boundedness and convergence of the partial (also called Dirichlet) integral of the Fourier integral of a function that is Lebesgue integrable and of bounded variation over R. Our theorems are formulated and proved in a sharper form than the ones in the literature. Our methods do not rely on the localization principle of the convergence of a Fourier integral and on the second mean value theorem involving a monotone function. Instead, we use integration by parts extended to improper Riemann–Stieltjes integral. The periodic analogues of our theorems were proved by Telyakovskii in a slightly weaker form.  2004 Elsevier Inc. All rights reserved