"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.
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
- 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.
- 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.
- 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.
- Fourier transform: definition, properties, the inverse Fourier
transform, Plancherel theorem, convolution, characteristic functions
(binomial and normal) and central limit theorem.
- Linear ordinary differential equations: first order equations,
second order equations with constant coefficients