Besov spaces (1 ≤ p ≤ ∞)

• Week 2 (Jan. 17-21)
    Mon: Dr. Martin Luther King, Jr. Service Day - no classes
    Lecture 4 - Wed: Intermediate multivariate derivative estimates, embedding theorems. Review Lectures 25-27 from Math 758L, but in multivariate context.
    Lecture 5 - Fri: Structure theorems for the K-method of interpolation. Reiteration and preliminaries.
   
• Week 3 (Jan. 24-28
    Lecture 6 - Mon: Holmstedt's weak type formula for the K-functional of two interpolation spaces, applications to reiteration.
    Wed: Classes dismissed - Snow Day
    Lecture 7 - Fri: Embeddings of Sobolev spaces between Besov spaces.
   
• Week 4 (Jan. 31 - Feb. 4)
    Mon: Classes dismissed due to weather (ice storm).
    Lecture 8 - Wed: Application of reiteration to reduction theorem and a weak type estimate between moduli of smoothness. An elementary multiresolution example.
    Friday: Class rescheduled due to professional travel.

Besov Spaces (0 < p < 1)

• Week 5 (Feb. 7-11)
    Monday: Class rescheduled due to professional travel.
    Lecture 9 - Wed: L^p and Besov spaces as complete metrizable (Frechet) spaces. Basic inequalities involving the norm. Properties of the modulus of smoothness.
    Lecture 10 - Fri: The integrated modulus of smoothness for L^p (0 < p ≤ ∞) and equivalence with the usual modulus.

Classical Approximation Theory with Trigonometric Polynomials

• Week 6 (Feb. 14-19)
    Lecture 11 - Mon: Marchaud's inequality for 0 < p < 1. Hardy inequalities: continuous and discrete versions. Introduction to the classical approximation theory.
    Lecture 12 - Wed: Approximation of continuous functions on the circle using trigonometric polynomials (Jackson's Direct Theorem). Error in L^p controlled by the L^p - modulus (all p > 0)
    Lecture 13- Fri: Bernstein's inequality in the uniform norm for trig polynomials. Bernstein's inverse theorem and multiresolution for the case p=∞. Characterization of Besov spaces in terms of their approximation errors in weighted l^α_q(Z).
   
• Week 7 (Feb. 22-26)
    Lecture 14 - Mon. morning: Bernstein's inequality for 1 ≤ p and the Hardy-Littlewood-Polya condition.

Local Approximation Theory

• Week 7 (continued)
    Lecture 15 - Mon. afternoon: Jensen's formula and subharmonic functions. Sketch of the proof of Bernstein's inequality for the case 0 < p < 1.
    Lecture 16 - Wed: Whitney's theorem for local (algebraic) polynomial approximation - cases of (1) p = ∞ and all r > 0 and (2) 0 < p < ∞ and r=1.
    Lecture 17 - Fri: Remaining cases of Whitney's theorem.
   
• Week 8 (Feb. 28 - March 3)
    Lecture 18 - Mon: Finish proof of Whitney's theorem. Proof of the fact that if r-th differences of an L^p function (0 < p) vanish almost everywhere, then up to a set of measure zero, the function is a polynomial of degree at most r-1 (from DeVore and Lorentz).
    Lecture 19 - Wed. morning: Compact embeddings of Besov spaces into the Lebesgue spaces (same p, p > 0).
    Lecture 20 - Wed. afternoon: Best and near-best local polynomial approximation. Properties of these polynomials as the index p and the cubes are varied.
    Lecture 21 - Fri: Additional properties of near-best approximates. Review of B-splines and introduction to piecewise polynomial and spline spaces.
   
• Spring Break (March 6-10)
   

Dyadic Spline Approximation

• Week 9 (March 13-17)
    Lecture 22 - Wed: Overview of direct and inverse theorems for dyadic spline approximation for functions in Besov space. Corresponding localized multiresolution analysis. Various representations of splines: e.g. deBoor-Fix representation of coefficient functionals.
    Lecture 23 - Fri. Morning: Quasi-interpolants "projections" onto the space of dyadic splines (proofs for all cases p > 0). Estimates for norms of quasi-interpolants in terms of the sequence space norms of the coefficient functionals.
    Lecture 24 - Fri. Afternoon: Direct (Jackson type) dyadic spline approximation estimates for Besov spaces (all p > 0).
   
• Week 10 (March 20-24)
    Lecture 25 - Mon: Bernstein estimate for dyadic splines (all p > 0). Inverse theorem and characterization of Besov spaces in terms of "linear approximation" spaces by dyadic splines.
    Lecture 26 - Wed. Morning: Continue with characterization. Multiresolution analysis with dyadic splines: rewrite condition, Fourier transform decay, ...

Rademacher Functions and Probability

• Week 10 (continued)
    Lecture 27 - Wed. Afternoon: Properties of Rademacher functions, Khinchin's inequality assuming probabilistic independence of system.
    Lecture 28 - Fri: Probabilistic independence, multiplicative property of integrals of independent functions, and Rademacher functions as an independent system. Introduction to unconditional convergence in Banach and Frechet spaces.
   

Unconditional Convergence

• Week 11 (March 27-31)
    Lecture 29 - Mon: Unconditional convergence of series, characterization in terms of rearranging the series, arbitrary changes of signs of the terms, and multipliers of the terms.
    Lecture 30 - Wed: Continue with characterizations of unconditional convergence.
    Lecture 31 - Fri: Bases in Banach spaces. Boundedness of coefficient linear functionals. Various characterizations of unconditional bases.
   

The Haar System and Paley's Square Function

• Week 12 (April 3-7)
    Lecture 32 - Mon: Unconditional bases and multipliers of coefficient functionals.
    Lecture 33 - Wed: Bessel's inequality and Parseval's equation in Hilbert space. The Haar basis as an unconditional basis for L^p (1) by establishing a uniform weak type (1,1) estimate for coefficient multipliers and using interpolation.
    Lecture 34 - Fri: Paley's square function for biorthogonal systems on L^p. Nonexistence of uniformly bounded unconditional bases for L^p (1 < p < ∞) if p ≠ 2.

Convergence of Rearrangements of Classical Fourier Series

• Week 13 (April 10-14)
    Lecture 35 - Mon: Convergence of Fourier series: review Dirichlet and Fejer kernels, Hilbert transform and convergence in norm. L²(T) functions have Fourier series which converge almost everywhere for almost all choices of signs.
    Lecture 36 - Wed: Summability methods. L^p(T) functions have Fourier series which diverge almost everywhere in all senses (i.e., for all summability methods) for almost all choices of signs (p ≠ 2).
    Lecture 37 - Fri: Finish proof of almost everywhere divergence of almost all rearrangements of Fourier series of functions not square integrable. Distinctions with wavelet series.
   

Maximal Entropy Encoders and Tree Approximation

• Week 14 (April 17-21)
    Lecture 38 - Mon: Introduction to maximal entropy encoders of Cohen, Daubechies, Dahmen, and DeVore. Multiresolution analysis in L². Statement of wavelet representation of Besov spaces as sequence spaces of coefficients. Proof of compact embeddings of Besov spaces   B_q^α(L^r[Q₀])  in the Lebesgue spaces L^p if   α/d   >   1/τ- 1/p   ≥  0.   Relationship of encoders with Kolmogorov entropy.
    Lecture 39 - Wed: Temlyakov's inequalities estimating norms of finite wavelet representations by the number of terms. Best n-term approximation by wavelets. Best n-term approximation by wavelets where the support set is a progressive collection of trees. Log-convex estimate of the direct approximation theorem.
    Lecture 40 - Fri: Direct Estimate for tree approximation and characterization of Besov spaces in terms of nonlinear tree approximation spaces.
   
• Week 15 (April 24-28)
    Monday - Easter Holiday
    Lecture 41 - Wed: Continue with proof.

Wavelets as Unconditional Bases

• Week 15 (continued)
    Lecture 42 - Thur: The Fefferman-Stein vector-valued maximal function inequalities and wavelet representations for L^p using the development of DeVore, Konyagin, and Temlyakov. Peetre maximal function and wavelets as unconditional bases for Besov spaces.