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We introduce the cosine-type approximation processes in abstract Banach space setting. The historical roots of these processes go back to W. W. Rogosinski in 1926. The given new definitions use a cosine operator functions concept. We proved that in presented setting the cosine-type operators possess the order of approximation, which coincide with results known in trigonometric approximation. Moreover, a general method for factorization of certain linear combinations of cosine operator functions is presented. The given method allows to find the order of approximation using the higher order modulus of continuity. Also applications for the different type of approximations are given.

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For the Fredholm integral equation u=T u+f on the real line, fast solvers are designed on the basis of a discretized wavelet Galerkin method with the Sloan improvement of the Galerkin solution. The Galerkin system is solved by GMRES or by the Gauss elimination method. Our concept of the fast solver includes the requirements that the parameters of the approximate solution u n can be determined in O(n ) flops and the accuracy u − u n 0,b cn −m f (m) 0,a is achieved where n = n (n) is the number of sample points at which the values of f and K, the kernel of the integral operator, are involved; moreover, we require that, having determined the parameters of u n , the value of u n at any particular point x ∈ (−∞, ∞) is available with the same accuracy O(n −m ) at the cost of O(1) flops. Here · 0,a and · 0,b are certain weighted uniform norms. Using GMRES, the 2m-smoothness of K is sufficient; in case of Gauss method, K must be smoother.