CLASSES OF CONVOLUTIONS WITH A SINGULAR FAMILY OF KERNELS: SHARP CONSTANTS FOR APPROXIMATION BY SPACES OF SHIFTS

Let sigma > 0, and let G, B is an element of L(R). The paper is devoted to approximation of classes of functions f for every epsilon > 0 representable as f(x) = F-epsilon(x) + 1/2 pi integral(R) phi(t)G(epsilon)(x - t) dt, where F-epsilon is an entire function of type not exceeding epsilon, G(epsilon) is an element of L(R), and phi is an element of L-p (R). The approximating space S-B consists of functions of the form s(x) = Sigma(j is an element of Z)beta B-j(x - j pi/sigma). Under some conditions on G = {G(epsilon)} and B, linear operators X-sigma,X-G,X-B with values in S-B are constructed for which parallel to f - X-sigma(,G,B)(f)parallel to p <= K-sigma(,G)parallel to phi parallel to(p). For p = 1, infinity the constant kappa(sigma,G) (it is an analog of the well-known Favard constant) cannot be reduced, even if one replaces the left-hand side by the best approximation by the space S-B. The results of the paper generalize classical inequalities for approximations by entire functions of exponential type and by splines.