The Modulus of p-Variation and Its Applications

Let $$\nu $$ be a nondecreasing concave sequence of positive real numbers and $$1\le p<\infty . In this note, we introduce the notion modulus p-variation for function variable, show that it serves in at least two important problems, namely, uniform convergence Fourier series computation certain K-functionals. Using new tool, first define Banach space, denoted $$V_p[\nu ]$$ , is natural unification Wiener class $$BV_p$$ Chanturiya $$V[\nu Then prove satisfies Helly-type selection principle which enables us to characterize continuous functions terms their Fejér means. We also K-functional couple $$(C,BV_p)$$ can expressed p-variation, where C denotes space functions. Next, obtain equivalent optimal conditions all each classes $$C\cap V_p[\nu $$H^\omega \cap $$\omega continuity its associated Lipschitz class. Finally, establish sharp embeddings into various spaces generalized bounded variation. As by-product these latter results, infer embedding results symmetric spaces.