Best approximations of infinitely differentiable functions in the space \(L_ s\) (Q1116026)

Let \(f\in L(2\pi)\) and \(a_ K\), \(b_ K(K=0,1,...)\) are its Fourier coefficients. Assume that the series \(\sum^{\infty}_{k=1}\frac{1}{\psi (k)}(a_ k\cos (kx+\frac{\beta \pi}{2})+b_ k\sin (Kx+\frac{\beta \pi}{2}))\) is the Fourier series for the function \(f^{\psi}_{\beta}\in L(0,2\pi)\) for some \(\beta\in R\) and sequence \(\psi\). Let \(L^{\psi}_{\beta}\) denote the set of functions satisfying this condition. Let also \({\mathcal R}\subset L(0,2\pi)\). The class of function f for which \(f\in L^{\psi}_{\beta}\) and \(f^{\psi}_{\beta}\in {\mathcal R}\) is denoted by \(L^{\psi}_{\beta}{\mathcal R}\). In the paper the set \({\mathcal R}\) is chosen so, that functions from \(L^{\psi}_{\beta}{\mathcal R}\) are infinitely differentiable. Some \(L_ s(1\leq s\leq \infty)\) estimates for the best approximation of functions \(f\in L^{\psi}_{\beta}{\mathcal R}\) by means of subspace \(T_{2n-1}\) of trigonometric polynomials of degree n-1 are given.