Pointwise convergence of double Fourier integrals of functions of bounded variation over R2

We investigate the pointwise convergence and uniform boundedness of the symmetric rectangular partial (also called Dirichlet) integrals of the double Fourier integral of a function that is Lebesgue integrable and of bounded variation over R2. Our theorems are the two-dimensional extensions of those proved in [7] in the case of single Fourier integral. Our methods do not rely on the localization principle of the convergence of the 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, and the reduction of such integrals to Lebesgue integrals. As corollaries of our main theorems, we obtain two-dimensional extensions of such results that are known for single Fourier integrals.