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Paper IPM / M / 16841  


Abstract:  
Let ν be a nondecreasing concave sequence of positive real numbers and 1 ≤ p < ∞. In this note, we introduce the notion of modulus of pvariation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain Kfunctionals. Using this new tool, we first define a Banach space, denoted V_{p}[ν], that is a natural unification of the Wiener class BV_{p} and the Chanturiya class V[ν]. Then we prove that V_{p}[ν] satisfies a Hellytype selection principle which enables us to characterize continuous functions in V_{p}[ν] in terms of their Fejér means. We also prove that a certain Kfunctional for the couple (C,BV_{p}) can be expressed in terms of the modulus of pvariation, where C denotes the space of continuous functions. Next, we obtain equivalent optimal conditions for the uniform convergence of the Fourier series of all functions in each of the classes C∩V_{p}[ν] and H^{ω}∩V_{p}[ν], where ω is a modulus of continuity and H^{ω} denotes its associated Lipschitz class. Finally, we establish sharp embeddings into V_{p}[ν] of various spaces of functions of generalized bounded variation. As a byproduct of these latter results, we infer embedding results for certain symmetric sequence spaces.
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