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Paper   IPM / M / 16841
School of Mathematics
  Title:   The modulus of p-variation and its applications
1.  Milad Moazami Goodarzi
2.  Mahdi Hormozi (Joint with G. H. Esslamzadeh and M. Lind)
  Status:   Published
  Journal: Journal of Fourier Analysis and Applications
  Vol.:  28
  Year:  2022
  Pages:   1-40
  Supported by:  IPM
Let ν be a nondecreasing concave sequence of positive real numbers and 1 ≤ p < ∞. In this note, we introduce the notion of modulus of p-variation 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 K-functionals. Using this new tool, we first define a Banach space, denoted Vp[ν], that is a natural unification of the Wiener class BVp and the Chanturiya class V[ν]. Then we prove that Vp[ν] satisfies a Helly-type selection principle which enables us to characterize continuous functions in Vp[ν] in terms of their Fejér means. We also prove that a certain K-functional for the couple (C,BVp) can be expressed in terms of the modulus of p-variation, 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 CVp[ν] and HωVp[ν], where ω is a modulus of continuity and Hω denotes its associated Lipschitz class. Finally, we establish sharp embeddings into Vp[ν] of various spaces of functions of generalized bounded variation. As a by-product of these latter results, we infer embedding results for certain symmetric sequence spaces.

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