Fourier Analysis
Lectures for Fall 2006
Introduction
Thur (8/24):
Subject Motivation: Derivation of the
heat equation & motivation of Fourier series through separation of
variables. Questions of representation of the solution (and functions
general), sets of pt-wise convergence and divergence, continuity
with respect to data, and modes of convergence.
Elementary Hilbert Space Properties
Tues (8/29): Elementary separable Hilbert space: orthogonality,
generalized Pythagorean theorem, Schwartz's and Bessel's inequalities.
Fri (8/30): Continuity of inner product, complete orthonormal
systems, norm convergence of Fourier projection (N-th partial sums),
Parseval's equation & isometry of H onto l2(Z);
Gram-Schmidt & construction of Multiresolution Ladder of
spaces Vn as the orthogonal sum: Vn-1 + Wn-1.
Classical Fourier Series on the Circle
Classical Summability Kernels {kn}
Homogeneous Banach Spaces X
Norm-convergence in X of Summability Operators
Tues (9/5): Wrap up from Thursday: Characterizations of
complete orthonormal systems: uniqueness theorem for L2(T).
Classical Fourier series, trigonometric polynomials, and orthogonality.
Thurs (9/7): Elementary properties of the Fourier
transform on T. Absolutely continuous functions and improved rates of decay
of Fourier coefficients.
Tues (9/12): The convolution product and its
properties on L1(T) x L1(T); convolution as a
Fourier multiplier. Properties of the Dirichlet and Fejer kernels,
and relation to N-th Fourier partial sum projection.
Cesaro summability of Fourier series.
Thurs (9/14): Definition and examples of Homogeneous Banach
spaces: C(T), Lebesgue spaces for p finite. Continuity in the parameter t of
the family of translations Tt; the X-modulus of continuity.
Tues (9/19): The space
Lipa(T), where a>0
is less or equal 1. Additional examples of Homogeneous
Banach spaces: C(1)(T) and lipa(T).
Thurs (9/21): X-valued continuous functions and X-valued Riemann
integrals. Convergence of summability operators in homogeneous Banach spaces. Specific
examples of Fejer and de la Vallee Pouisson kernels.
The Fourier spectra of these kernels and the Dirichlet kernel. Their effect as
Fourier multipliers. The Riemann-Lebesgue lemma for L1
functions.