Spaces of functions of generalized bounded variation. I. ~Embedding theorems. Estimates for Lebesgue constants (Q1975804)

The article depends on the author's paper [Math. Notes 56, No.5, 1105-1112 (1994; Zbl 0899.26002)] in which a unified approach is proposed to defining various classes of functions of generalized bounded variation and an exact refinement theorem is proven for these classes. The author continues studying the general spaces of functions of bounded variation. In the first part, various embedding theorems are proven for spaces of functions of bounded variation and the behavior of the Lebesgue constants is examined for the trigonometric system. The general approach used by the author enables us to reveal the reasons for mutual embedding of classes of functions of generalized bounded variation and embedding of the Lipschitz spaces into these classes. Exact two-sided estimates are also obtained for the Lebesgue constants of the trigonometric system which provide a new form of convergence tests for the Fourier series in the available spaces of functions of generalized bounded variation. The second part of the article is devoted to the question of uniform convergence of Fourier series in spaces of functions of generalized bounded variation.