Zastavnyi V. Positive definite functions and some problems of analysis

In this thesis we study positive-definite functions depending on a nonnegative homogeneous function defined on a real linear space. For finite-dimensional spaces of dimension three or higher, a general sufficient condition for the triviality of the classes of positive definite functions depending on the norm is obtained. Problems of Schoenberg, Kuttner and Leonenko-Yadrenko are solved. General sufficient conditions for positive definiteness are obtained. A criterion of positive definiteness of the Gneiting type functions is established. The four-parameter family of Buhmann radial functions is fully studied. Sufficient conditions for an entire function to possess only real zeros, which alternate with zeros of sine type, are obtained. Besides sufficient conditions for an entire function of exponential type to have no zeros in the lower half-plane are obtained. A connection of these functions with positive definite functions is established. A series of the proven results is concerned with approximation theory. In particular, asymptotic expansions for series and integrals which arises in the approximation of classes of differentiable functions by Riesz and Cesaro means, by Abel-Poisson means, by methods of Zhuk type are obtained. Asymptotic expansions for Mathieu series are established. For Mathieu series, sharp inequalities are also proved. General sufficient conditions for exact calculation of approximation of classes of differentiable functions by convolution operators are obtained. In the problem of estimating the sums of the modules of blocks of Fourier series, the case of an arbitrary set of indices in each block is considered. Key words: positive definite functions, multiple monotonic functions, problem of Schoenberg, problem of Kuttner, entire functions, asymptotic expansion, approximation of classes of functions by convolution operators, Fourier series.