Hans-Jürgen Schmeisser

Spaces of functions of mixed smoothness and approximation from hyperbolic crosses

We compare function spaces of dominating mixed smoothness and spaces of best approximation with respect to hyperbolic crosses.

Imbeddings of Brezis-Wainger type. The case of missing derivatives

We prove limiting imbeddings of spaces with dominating mixed derivatives into the spaces of almost Lipschitz continuous functions.

On the Method of Approximation by Families of Linear Polynomial Operators

It is shown that best approximation by trigonometric polynomials is achieved in average by families of linear polynomial operators in the Lp-metric for all p, 0 < p ≤ ∞. This is compared with approximation by Fourier means and interpolation means which is restricted to 1 ≤ p ≤ ∞ and p = ∞, respectively.

Smoothness and Function Spaces Generated by Homogeneous Multipliers

Differential operators generated by homogeneous functions 𝜓 of an arbitrary real order 𝑠>0 (𝜓-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 𝜓-derivatives are obtained in terms of the Fourier transform of their generators. Some applications to approximation theory are discussed.