Fourier Analysis

Lectures for Fall 2002

Introduction

Elementary Hilbert Space

Classical Fourier Series

Homogeneous Banach Spaces & Summability Kernels

  • Week 3
      Wed (9/11): Definition and examples of Homogeneous Banach spaces, X-modulus of continuity, X-valued continuous functions and Riemann integrals.
      Fri (9/13): Lipa(T) and lipa(T), Besov spaces Bpa,q, summability kernels.

Classical Kernels

  • Week 3 (cont.)
      Mon (9/16): The Fejer kernel and Cesaro means of the N-th Fourier partial sums. Properties of X-valued integrals. Convergence of summability operators in homogeneous Banach spaces.
  • Week 4
      Mon afternoon (9/16): Properties the Dirichlet and Fejer kernels, and relation to N-th Fourier partial sum projection. Cesaro summability of Fourier series.

Decay of Fourier Coefficients

  • Week 4 (cont.)
      Wed (9/18): Completeness of the trigonometric system in L2. Riemann-Lebesgue lemma. Additional examples of summability kernels: de la Vallee Poussin kernel, Poisson kernel, Abel means & radial limits of harmonic functions. Rates of decay of Fourier coefficients for BV, absolutely continuous, and Sobolev functions.
      Fri (9/20): Improvement of the Riemann-Lebesgue theorem with rates for Lipa.Properties of the Poisson kernel.

Absolutely Convergent Fourier Series and Besov Spaces

  • Week 5
      Mon (9/23): The Banach algebra A of absolutely convergent Fourier series. Embeddings with Sobolev and Lipschitz spaces (Bernstein's theorem).
      Mon afternoon (9/23): Example of sawtooth function and an identity for p2/8. Extension of Bernstein's theorem to Besov spaces. Embeddings of Besov spaces.
      Wed (9/25): Higher order moduli of smoothness and Marchaud's inequality. Embeddings of higher order Besov spaces.

Tauberian Theorems and Convergence of Fourier Series

  • Week 6
      Fri (9/27): Fejer's Theorem on pointwise convergence of Fejer sums.
      Fri afternoon (9/27): A simple Tauberian theorem and pointwise convergence of Fourier series.
      Mon (9/30): Lebesgue criterion for pointwise convergence of Fejer sums.
  • Week 7
      Wed (10/2): Hardy's Tauberian theorem. Examples. Pointwise convergence of Fourier series for BV functions.
      Fri (10/4): Special examples of trigonometric series and when they may or may not be Fourier series of an integrable function.
      Mon (10/7): Summarize results to date. Examples.

Interpolation of Operators (Riesz-Thorin Theorem, the "complex method")

  • Week 8
      Wed (10/9): Statement of Riesz-Thorin theorem. Corollaries (Hausdorff-Young theorem, Young's convolution theorem).
      Fri (10/11): Proof of the Riesz-Thorin theorem.
      Mon (10/11): Fall Break.
      Wed (10/16): The Hadamard Three Lines theorem.

Interpolation of Operators (the "real method")

  • Week 9
      Fri (10/18): The Peetre K-functional and the Real Method of interpolation of operators; an elementary interpolation theorem for intermediate spaces (X1,X2)q,q.
      Mon (10/21): Equivalence of the K-functional for C and C1 with the modulus of continuity, and the interpolation spaces as particular Besov spaces (case q=¥).

Decreasing Rearrangements and the Hardy-Littlewood Maximal Operator

  • Week 9 (cont.)
      Wed (10/23): Distribution functions for measurable functions and their properties.
  • Week 10
      Fri (10/25): Decreasing rearrangements and their properties.
      Mon (10/28): The Hardy-Littlewood inequality; the subadditive operator f**. Identification of the K-functional for the pair (L1,L¥). The Lorentz spaces as interpolation spaces.
      Wed (10/30): The Hardy-Littlewood maximal operator M[f], an elementary covering lemma, weak type (1,1) estimate, and the strong version of Lebesgue's theorem on a.e. convergence of integral averages.
      Fri (11/1): No class. [MURI preparation with Aero @ Univ. Florida.]

Introduction to Signal Processing and the Fast Fourier Transform

  • Week 11
      Mon (11/4): The discrete Fourier transform (DFT), the Fourier matrix and elementary properties (orthogonality, operation count, ...). Introduction to the FFT.
      Wed (11/6) - Mon (11/11): NSF IDR postdoc marathon on wavelets and multiresolution analysis (schedule) - required attendance.
      Wed (11/13): Derivation of FFT recursive butterfly algorithm and operation count, bit reversal, implementation in Matlab, demonstrations.
  • Week 12
      Fri (11/15): Elementary properties of the Fourier transform on the line, the energy spectrum, average frequency and standard deviation, motivation for analytic signals and conjugation, filtering operations and causal filters.

Conjugation

  • Week 12 (cont.)
      Mon (11/18): Conjugate Fourier series, the conjugate Poisson kernel and its properties; relation to analytic functions.
      Wed (11/20): Norm convergence of Fourier series and relation to conjugation.
  • Week 13
      Fri (11/22): Proof of equivalence of operations of conjugation, analytic projection, and norm convergence; unboundedness of operator norms of SN on L1 and C.
      Mon (11/25): Summarize earlier lectures, motivate the principal value conjugate operator. Define various maximal operators to control convergence of radial limits and the principal values.
      Wed (11/27) - Mon (12/2): Thanksgiving holidays and students' foreign travel (makeup on Dec 10).

Weak-type Operators and the Marcinkiewicz Theorem

  • Week 14
      Wed (12/4): The Hardy inequalities and equivalent 'renorming' of the Lorentz spaces. Weak type and restricted weak type operators, the Marcinkiewicz interpolation theorem using Calderon's operator.
      Fri (12/6): No class (On-site ARO review. Makeup on Dec 10)

Riesz' Theorem on Norm Convergence of Fourier series

  • Week 14 (cont.)
      Tues 9-10:15 (12/10): Fatou's theorem for harmonic functions, existence of radial limits of conjugate harmonic functions.
      Tues 10:30-11:45 (12/10): Weak type (1,1) estimate for the conjugate operator, Riesz's theorem on conjugation in Lp(T), (1 < p < ). Summarize convergence of Fourier series and related operations in Lp(T).