"Topics in mathematics for statisticians". School of Mathematical Sciences. (Ya. Yakubov)

1. Introduction to functional analysis, linear operators and RKHS: linear vector spaces and linear normed spaces, examples: C, C^s, L_2, H^s, Hilbert spaces, Schwarz inequality and the parallelogram low, orthonormal sets, Gram-Schmidt process, Legendre orthonormal basis, Fourier orthonormal basis, Bessel inequality and Parseval identity, Haar systems and wavelets, Linear operators, Reproducing kernel Hilbert space (RKHS).
2. Fourier series: complex numbers and functions, Euler formula, real and complex forms of Fourier series, decay of the Fourier coefficients and dependence of the decay on the smoothness of a function, differentiation and integration of Fourier series, formulas for the Fourier coefficients via the inner product, Besel inequality and Parseval identity, L_2-convergence and uniform convergence, Weierstrass approximation theorem.
3. Fourier transform: definition, properties, the inverse Fourier transform, Plancherel theorem, convolution, characteristic functions (binomial and normal) and central limit theorem.
4. Linear ordinary differential equations: first order equations, second order equations with constant coefficients