Embedding Schramm spaces into Chanturiya classes
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00093-8 ORIGINAL PAPER
Embedding Schramm spaces into Chanturiya classes Milad Moazami Goodarzi1 Received: 18 May 2020 / Accepted: 10 August 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract The main theorem of this paper establishes a necessary and sufficient condition for embedding Schramm spaces 𝛷BV into Chanturiya classes V[𝜈] . This result is new even for the classical spaces in the theory of Fourier series, namely, for the Wiener and the Salem classes. Furthermore, it provides a characterization of the embedding of Waterman classes 𝛬BV into V[𝜈] . As a by-product of the main result, we establish a convergence criterion for the Fourier series of functions of 𝛷BV ; this is an extension of a well-known result due to Salem. An estimate on the magnitude of the Fourier coefficients in the space 𝛷BV is also given, and finally it is shown that some of these results can be extended to a more general setting. Keywords Fourier series · Uniform convergence · Fourier coefficients · Generalized bounded variation · Embedding Mathematics Subject Classification 42A20 · 42A16 · 46E35 · 46E30 · 26A45
1 Introduction Embedding theory for Banach spaces in the theory of Fourier series has a long history, and is mainly applied in the convergence problem of Fourier series. This subject was originated in the work of Hardy and Littlewood [14], where, among other things, a connection was established between the Jordan class BV of functions of bounded variation and the generalized Lipschitz class Hp𝜔 , i.e., the class of all functions in Lp such that 𝜔p (𝛿, f ) = O(𝜔(𝛿)) as 𝛿 → 0+ , where 𝜔p (𝛿, f ) is the Lp-modulus of continuity of f and 𝜔 is a modulus of continuity in the classical sense. The modulus of variation of a function f on an interval [a, b] was originally ∑n defined by Lagrange [17] to be the sequence v(n, f ) ∶= sup j=1 �f (Ij )� , where the Communicated by Dachun Yang. * Milad Moazami Goodarzi [email protected] 1
Department of Mathematics, Faculty of Sciences, Shiraz University, 71454 Shiraz, Iran Vol.:(0123456789)
M. M. Goodarzi
supremum is taken over all finite collections {Ij }nj=1 of nonoverlapping intervals in [a, b]. This is a nondecreasing concave sequence, and any sequence 𝜈 of positive numbers with such properties is called a modulus of variation. Chanturiya used this notion to define the function space V[𝜈] consisting of all functions f for which v(n, f ) = O(𝜈(n)) as n → ∞ , and studied it in connection with the Fourier series (see, e.g., [4, 5]). Later on, Chistyakov [6] utilized moduli of variation in unifying various selection principles (of Helly type). It is worth noting that modulus of variation may be viewed as a “variational” counterpart to modulus of continuity, and as such, V[𝜈] can be thought of as a counterpart to H1𝜔 . For more information on 𝜈 and V[𝜈] , see [1, 8]. On the other hand, the notion of bounded variation has been generalized in many ways and studied extensively in regard to the theory of Fourier ser
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