Separability of the space of continuous functions that are continuous in Λ-variation

Abstract

The Shao-Sablin index of a Λ-sequence Λ = (λ i ) is defined by S Λ := lim sup n→∞The main result of the paper states that the Banach space CΛBV of continuous functions of bounded Λ-variation with the standard Λ-variation norm is separable if and only if S Λ < 2. Also, ΛBV = ΛBV c if and only if S Λ < 2, where ΛBV c denotes the space of functions continuous in Λ-variation. A number of corollaries is drawn, and one of them being that the Garsia-Sawyer class GS is a dense subset of the Banach space HBV of functions of bounded harmonic variation.Functions of bounded Λ-variation are a natural generalization of functions of bounded variation and were introduced by D. Waterman in order to extend a number of results related to Fourier series of a function, in particular, to essentially generalize the Dirichlet-Jordan Test (see [17,18,20]).

On some nonlinear operators in $${\varvec{\Lambda }} {\varvec{BV}}$$ Λ B V -spaces

“…, t n−1 , t n }. Now we are going to rephrase the definition of Λ-variation (in a slightly different way than it was done before by Prus-Wiśniowski-see [16]). This, very simple, observation will play an essential role in our considerations.…”

“…An example of such function was given in the paper [8]. Moreover, Prus-Wiśniowski in [16] described the relation between the ShaoSablin index of a Waterman sequence Λ and the existence of functions that are not continuous in Λ-variation. …”

“…Moreover, for such functions a Helly-type theorem holds (see [11]). Let us add also that the space of functions of bounded Λ-variation from a functional point of view was investigated, for example, in the paper [16]. It has appeared that the so-called Shao-Sablin index plays an essential role in that approach.…”

Bernstein and Kantorovich polynomials diminish the Λ-variation

“…This observation is used in several papers to simplify a notation of proofs (for example [17][18][19], the proof of this fact in [7]). …”

“…Note that the symbol σ Λ was used in the papers of Prus-Wiśniowski but with a slightly different meaning (cf. [17][18][19]). For a fixed Λ-sequence Λ, the space of functions of bounded Λ-variation, denoted by ΛBV , is a linear space, which may be equipped with the norm f Λ = V Λ (f ) + |f (0)|.…”