Fourier coefficients and generalized Lipschitz classes in uniform metric. (English) Zbl 1180.42003

Volosivets, S. S.

Real Anal. Exch. 34(2008-2009), No. 1, 219-226 (2009).

Absolutely convergent Fourier series \[ \sum_{k\in\mathbb{Z}} c_k e^{ikx}:= f(x),\quad\text{where }\sum_{k\in\mathbb{Z}} |c_k|< \infty, \] are considered, and equivalence relations are proved between the behavior of the Fourier coefficients \(c_k\) of a special kind and the smoothness properties of the sum function \(f\). The following Theorem 3 is typical: If \[ \sum_{|k|> n}|c_k|= o(n^{-m})\quad\text{for some }m\in\mathbb{N}, \] then the Schwarz derivative of \(f\) of order \(m\) exists at a given point \(x\) and equals \(A\) if and only if the formally differentiated series \[ \sum(ik)^m c_k e^{ikx}\tag{\(*\)} \] is convergent and its sum equals \(A\).
Reviewer’s remark: The reviewer proved in a joint paper with G. Brown and Z. Sáfár [Acta Sci. Math. (Szeged), 75, 161–173 (2009)] that an analogous characterization is valid for the existence of the ordinary derivative \(f^{((m)}(x)\) in place of the Schwarz derivative; and the following supplement is also valid: the derivative \(f^{(m)}\) is continuous on \(\mathbb{T}:= [-\pi,\pi)\) if and only if the formally differentiated series \((*)\) converges uniformly on \(\mathbb{T}\).

Reviewer: Ferenc Móricz (Szeged)

MSC:

42A10	Trigonometric approximation
41A25	Rate of convergence, degree of approximation

Keywords:

absolute convergence; moduli of smoothness; generalized Lipschitz classes