ACTA issues

Lebesgue integrability of double Fourier transforms

Dang Vu Giang, Ferenc Móricz

Acta Sci. Math. (Szeged) 58:1-4(1993), 299-328

Abstract. First, we consider a complex-valued function $f$ defined and absolutely continuous on the positive quadrant ${\msbm R}_+^2$ of the plane, and study the double cosine $F_c,$ double sine $F_s$, and cosine-sine Fourier transform $F_{cs}$ of $f.$ We give sufficient conditions, under which $F_c, F_s,$ and $F_{cs}$ are Lebesgue integrable on ${\msbm R}_+^2,$ respectively; and the inversion formula holds. Our basic tools are Sidon type inequalities, which we elaborate also in this paper. Second, we deduce sufficient conditions for Lebesgue integrability of double cosine, double sine, and cosine-sine series on the two-dimensional torus ${\msbm T}^2.$ Third, we extend these results to double complex Fourier transform of functions defined and absolutely continuous on the whole plane ${\msbm R}^2$ as well as to double complex trigonometric series. Fourth, as a by-product, we obtain sufficient conditions for an absolutely continuous function to be the double complex Fourier transform of a Lebesgue integrable function on ${\msbm R}^2.$

AMS Subject Classification (1991): 42B99, 42A38, 26A46

Keyword(s): double cosine, double sine, cosine-sine Fourier transforms, double complex Fourier transform, absolute continuity of function in two variables, Lebesgue integrability, inversion formula, double cosine, double sine, cosine-sine series, double complex trigonometric series, double null sequence of bounded variation, Fourier series, Hausdorff-Young inequality, Sidon type inequalities

Received February 2, 1993. (Registered under 5550/2009.)