Approximation of functions of several variables by trigonometric polynomials with harmonics from hyperbolic crosses
Denote by \(T(Q_ n)\) the set of trigonometric polynomials \(t(x)=\sum_{| k| \in Q_ n}c_ ke^{i(k,x)},\) where \(x=(x_ 1,...,x_ d)\in {\mathbb{R}}^ d,\quad k=(k_ 1,...,k_ d)\) are vectors with integral coordinates, \(| k| =(| k_ 1|,...,| k_ d|);\) \[ Q_ n=\cup_{s\in \theta_ n}\{k:[2^{s_ j-1}]\leq k_ j\leq 2^{s_ j},\quad j=1,...,d\}, \] \[ \theta_ n=\{s=(s_ 1,...,s_ d):s_ j\quad are\quad nonnegative\quad integers,\quad | s_ 1| +...+| s_ d| \leq n\}. \] Polynomials of this form are called polynomials with harmonics of hyperbolic crosses. The author obtains an analogue of Jackson's theorem for approximation of continuous functions by trigonometric polynomials from hyperbolic crosses.
approximation of continuous functions by trigonometric polynomials from hyperbolic crosses.
scientific article; zbMATH DE number 4175665