One-dimensional Fourier series of a function of many variables

Abstract

It is well known that to each summable in the $n$-dimensional cube ${[-\pi ,\pi ]}^n$ function $f$ of variables $x_1,\dots ,x_n$ there corresponds one $n$-multiple trigonometric Fourier series $S\left[f\right]$ with constant coefficients.

In the present paper, with the function $f$ we associate $n$ one-dimensional Fourier series $S{\left[f\right]}_1,\dots ,S{\left[f\right]}_n$, with respect to variables $x_1,\dots ,x_n$, respectively, with nonconstant coefficients and announce the preliminary results. In particular, if a continuous function $f$ is differentiable at some point $x=(x_1,\dots ,x_n)$, then all one-dimensional Fourier series $S{\left[f\right]}_1,\dots ,S{\left[f\right]}_n$ converge at $x$ to the value $f\left(x\right)$.

For illustration we consider the well known example of Ch. Fefferman’s function $F(x,y)$ whose double trigonometric Fourier series $S\left[F\right]$ diverges everywhere in the sense of Prinsheim. Namely, we establish the simultaneous convergence of the one-dimensional Fourier series $S{\left[F\right]}_1$ and $S{\left[F\right]}_2$ at almost all points $(x,y)\in {[-\pi ,\pi ]}^2$ to the values $F(x,y)$.