On local smoothness classes of periodic functions (Q818495)

The authors consider bi-infinite matrices of the form \( H = (h_{k,2^n})_{k \in \mathbb Z, \; n = 0, 1, 2 \ldots}\) where \( h_{k,2^n} = 0\), if \( |k| > 2^n\). One objective of the paper is to consider the relationship between local Besov spaces and the behavior of the operators \[ \tau_n (H,f) := \sum_{k\in \mathbb Z }h_{k,2^n} \widehat f (k) \exp(ikx) - \sum_{k\in \mathbb Z}h_{k,2^{n-1}} \widehat f (k) \exp (ikx) \] near the point in question. This relationship can be represented by \( L^p\)-norms and weighted sequence norms. The values of the operator \( \tau_n (H,f) \) are trigonometric polynomials being frequency localized as well as time localized. It is shown that the global behavior of the operator, constructed only from values of \( f \) near a point, is determined entirely by the smoothness of \( f \) near that point. The considered operators preserve trigonometric polynomials of degree commensurate with the degree of polynomials given by the operators. Several numerical examples are presented.