“School of Mathematics”
Back to Papers HomeBack to Papers of School of Mathematics
| 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 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 C∩Vp[ν] and Hω∩Vp[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.
Download TeX format |
|||||
| back to top | |||||
