Approximation properties of trigonometric Fourier series in generalized variation classes

In this paper the approximation properties of the partial sums of trigonometric Fourier series for functions within the generalized variation classes \(BV(p(n)\uparrow \infty ,\varphi )\) and \(B\Lambda (p(n)\uparrow \infty ,\varphi )\) are investigated. The primary goal is to determine if these classes can provide better rates of uniform convergence compared to the classical Lebesgue estimate. The results show that under certain conditions, this classes offer improved convergence rates. Specifically, when the modulus of continuity \(\omega \) and the sequences p(n) and \(\varphi (n)\) satisfy particular growth conditions, the uniform convergence rate can surpass the classical Lebesgue estimate. The paper also demonstrates that the conditions required for these improved estimates are not mutually exclusive, allowing a wide range of acceptable rates for \(\omega \). Additionally, a function is constructed within the class \(H^\omega \cap B\Lambda (p(n) \uparrow \infty , \varphi )\) (but not in \(BV(p(n) \uparrow \infty , \varphi )\)) whose Fourier series converges uniformly, emphasizing the advantage of the \(B\Lambda (p(n) \uparrow \infty , \varphi )\) class.