Multiple Fourier integral and multiple Fourier series under square summation
The author studies a correlation between the convergence of multiple Fourier series and corresponding multiple Fourier integral. For a \(2\pi\)- periodic function f(x), \(x\in R^ N\) with the Fourier decomposition \(\sum_{k}c_ ke^{ikx}\sim f(x),\) let us denote \[ S_ n(x;f)=\sum_{| k_ 1| \leq n_ 1}...\sum_{| k_ N| \leq n_ N}c_ ke^{i(k_ 1x_ 1+...+k_ Nx_ N)},\quad n=(n_ 1,n_ 2,...,n_ N)\in Z^ N, \] and for g(x)\(\sim \int_{R^ N}\hat g(y)e^{i(y,x)}dy\) let us denote \[ J_{\alpha}(x;g)=(2\pi)^{- N}\int^{\alpha_ 1}_{-\alpha_ 1}...\int^{\alpha_ N}_{- \alpha_ N}\hat g(y)e^{i(x,y)}dy,\quad \alpha \in R^ N_+. \] Theorem. There exist functions \(f\in L_ 1([-\pi,\pi]^ N)\), \(g\in L_ 1(R^ N)\), \(N\geq 2\) and for almost every \(x\in [-\pi,\pi]^ N\) there are sequences \(\{n(k,x)\}^{\infty}_{k=1}\) such that 1) \(f(x)=g(x)\), \(x\in [-\pi,\pi]^ N\); 2) (a) \(n(k,x)\in Z^ N\), (b) \(n_ 1(k,x)=n_ 2(k,x)=...=n_ N(k,x)=n_ 0(k,x)\), (c) \(0<n_ 0(1,x)<n_ 0(2,x)<...\); 3) (a) \(\lim_{k\to \infty}| S_{n_ 0(k,x)}(x;f)| =+\infty\), (b) \(\lim_{k\to \infty}J_{n_ 0(k,x)}(x;g)=g(x)\).
scientific article; zbMATH DE number 4163480