Uniform convergence of double sine transforms of general monotone functions (Q1677614)

Let \(\int_0^\infty\int_0^\infty f(x,y)\sin (ux)\sin (vy)\, dx\,dy =F(u,v)\). The author proves several results concerning uniform convergence in the regular sense of this double integral. One of the main results gives sufficient conditions for such convergence and for a strong estimate of the difference \(| F(u,v)-S_{M,N} (u,v)|\), where \(S_{M,N} (u,v)=\int_0^M\int_0^Nf(x,y)\sin (ux)\sin (vy)\, dx\,dy\). Another of the main results provides necessary and sufficient conditions on nonnegative functions \(f(x,y)\) for uniform convergence in the regular sense.