Lebesgue constants of conjugate Fourier sums in the multidimensional case
We introduce the following notation: \(R^ m\), m-dimensional Euclidean space; (x,y), scalar product in \(R^ m\); \(A^*\) adjoint operator of an operator A; \(e_ j\), coordinate unit vectors; \(Z^ m\) integral lattice in \(R^ m\); \(\bar T^ m=[-1/2,1/2]^ m\); \(c_ k(f)=\int_{\bar T^ m}f(t)e^{-2\pi i(k,t)} dt,\) Fourier coefficients of the function f; \(C^ m\), space of the functions that are 1-periodic with respect to each argument and continuous in \(R^ m\), \(\xi \Lambda =\{x\in R^ m| \xi^{-1}x\in \Lambda \}\) \((\xi >0)\), and \[ s_{n\Lambda}^{[\alpha]}(f;x)=-i\sum_{k\in n\Lambda \cap Z^ m}sign (\alpha,k)c_ k(f)e^{2\pi i(k,x)}. \] Let \(A=\{a_{i,j}\}^ m_{i,j=1}\) be a nonsingular linear transformation with rational coefficients, \(\xi >0\), \(\Lambda =\xi A\bar T^ m\), \(\alpha_ j=(A^*)^{-1}e_ j\) \((j=1,...,\ell\); \(0\leq \ell \leq m)\), \(\alpha =(\alpha_ 1,...,\alpha_{\ell},0,...,0)\). Then \[ \| s_{n\Lambda}^{[\alpha]}\|_{C^ m\to C^ m}=(2/\pi)^{2m- \ell}\ell n^ m n+O(\ell n^{m-1} n)\quad (n\to \infty). \]
scientific article; zbMATH DE number 3919020