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

Repeat by yourself: matrixes (transposed, triangular, diagonal), determinants and the rank of a matrix.
  1. Linear algebra (finite-dimensional spaces): matrixes (inverse, orthogonal, idempotent, symmetric and positive definite), 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
Books:
  • H. Anton, "Elementary Linear Algebra", John Wiley & Sons, 1994 (for the 1st part of the course).
  • E. Levin, V. Grinshtein, "Mavo leAnaliza Fynkzionalit", Open University, 2009 (in Hebrew; for the 2nd part of the course).
  • S. Zafrani, A. Pinkus, "Turei Fourier veHatmarot Integralijot", The Technion, 1997 (in Hebrew; for the 3rd and 4th parts of the course).
  • W. E. Boyce, R. C. DiPrima, "Elementary Differential Equations and Boundary Value Problems", John Wiley & Sons, 1992 (for the 5th part of the course).