"Mathematics for graduate statistics students". School of Mathematical Sciences. (Ya. Yakubov)

1. Linear algebra (finite-dimensional spaces): matrixes (transposed, triangular, diagonal, inverse), orthogonal and idempotent matrixes, determinants, the rank and the trace of a matrix, symmetric and positive definite matrixes, eigenvalues and eigenfunctions of a matrix, Gram-Schmidt process, QR-decomposition, least squares, application to linear regression.
2. Introduction to functional analysis (infinite-dimensional spaces): 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.
3. 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.
4. Fourier transform: definition, properties, the inverse Fourier transform, Plancherel theorem, convolution, characteristic functions (binomial and normal) and central limit theorem.
5. Linear ordinary differential equations: first order equations, second order equations with constant coefficients