Absolute convergence of Fourier series of functions of λBV(p) and ϕλBV

Estimates ofLpModulus of Continuity of Generalized Bounded Variation Classes

“…Functions of classes , , Λ , and Λ are considered in trigonometric Fourier series and some of them share good approximative properties (see [1][2][3][6][7][8][9][10][11], etc.). What we mention here is the following theorem proved by Shiba [12], Schramm and Waterman [5], and Wang [13]:…”

“…Combining the notion of Λbounded variation with that of -bounded variation, Leindler [4] introduced the class Λ of functions of Λ -bounded variation, and both classes of Λ-bounded variation andbounded variation are its special cases. Actually the class Λ first appeared in Schramm and Waterman's paper [5], and some restrictions are imposed on in their definition. Here we adopt Leindler's definition.…”

Relationships between ΦBV and ΛBV

“…The classes ΛBV and ΦBV were first introduced in [13] and [17] respectively, and had been studied mainly because of their applicability to the theory of Fourier series (see [1,[12][13][14][15][16][17]). …”

Locally Lipschitz composition operators in spaces of functions of bounded variation

“…For instance, Waterman proved in [27] that Helly's selection principle is valid in the space BV [a, b]. However, it is not known whether the local Lipschitz continuity of f is a necessary and sufficient condition for the composition operator Another space where Helly's selection principle holds is the space BV [a, b] of functions of -bounded variation in the sense of Schramm [21][22][23]. However, it is again not known whether or not the local Lipschitz continuity of f is a necessary and sufficient condition for the composition operator F generated by f to act in this space.…”