István Mezö

Modulus of Continuity and Best Approximation with Respect to Vilenkin-Like
Systems in Some Function Spaces

Abstract:
We rephrase Fridli's result on the modulus of continuity with respect to a Vilenkin group in the Lebesgue space. We show that this result is valid in the logarithm space and for Vilenkin-like systems. In addition, we prove that there is a strong connection between the best approximation of Fourier series and the modulus of continuity, not only in the Lebesgue space (G\'{a}t, 2001) but in the logarithm space too. We formulate two variable generalizations of the obtained results, which have not been known till now even in the Walsh case.

Keywords:
Modulus of continuity, best approximation, Vilenkin-like systems, logarithm space.

MSC 2000: 42C10