Smoothness and function spaces generated by homogeneous multipliers (Q416332)

Summary: Differential operators generated by homogeneous functions \(\psi\) of an arbitrary real order \(s > 0 ~(\psi\)-derivatives) and related spaces of \(s\)-smooth periodic functions of \(d\) variables are introduced and systematically studied. The obtained scale is compared with the scales of Besov and Triebel-Lizorkin spaces. Explicit representation formulas for \(\psi\)-derivatives are obtained in terms of the Fourier transform of their generators. Some applications to approximation theory are discussed.