- Week 1
Mon (1/13): Course Preview and Motivation. Introduction of the Haar system and tree structure for dyadic cubes; the discrete Haar transform and comparison to the Fast Fourier Transform. Burn-in demonstration for images.
Wed (1/15): Orthogonality and completeness of the Haar system; the multiresolution definition for L2(R) in the context of Haar basis.
Multiresolution Structure
Stromberg's Wavelet
The L1 theory of the Fourier Transform on Rd
The L2 theory of the Fourier Transform on Rd; Brief PDE Applications
Relations between a function and its Fourier transform
MRA and Scaling Functions
MRA and Wavelets
Unconditional convergence in Banach Spaces
Wavelets and Nonlinear Approximation in Lp; compact embeddings of Besov spaces.
Additional applications of multiresolution analysis
Fri (1/17): Scale and wavelet projection operators for
the Haar system; the multiresolution ladder of subspaces for L2(R).
Mon (1/20): MLK in-service day. No classes.
Wed (1/22): Dilation and translation operators as isometric
isomorphisms of steps of the multiresolution scale; Boundedness of projection
operators on Lp(R)
(1 < p < 
) using interpolation.
The multiresolution analysis for Lp(R) using the Haar basis.
Fri (1/24): Definition of Stromberg's wavelet
and proof that the dilates and translates form an orthonormal
basis for L2(R).
Mon (1/27): Explicit evaluation of the Stromberg
wavelet. Growth and oscillation properties.
Wed (1/29): Projection operators for scale as bounded
operators on L.
Fri (1/31): Continuation of proof of boundedness of the projection
operators; the Stromberg multiresolution decomposition scale for Lp(R).
Mon (2/3): Convergence of the Stromberg wavelet projections
within a scale; Orthogonal collection of projections for Lp(R) and
multiresolution of Lp functions.
Wed (2/5): Definition and elementary properties of the Fourier
transform, in particular - dilations and translations; Convolution on L1
and the convolution theorem; multivariate differentiation as a Fourier multiplier.
Fri (2/7): The Gauss kernel as a summability kernel. Review of the
proof of the Summability Theorem but in the setting of Rd.
Mon (2/10): The radial and nontangential maximal operators for the
Gauss kernel. Convergence in Lp and almost everywhere of the Gauss means.
The Fourier inversion theorem on L1.
Wed (2/12): Catch-up: The Sinc function and a proof of the
Riemann-Lebesgue lemma. The Plancherel Theorem and the Hausdorff-Young inequality.
Fri (2/14): The Poisson and Gauss-Weierstrass kernels and derivation
of their Fourier transforms.
Mon (2/17): No classes. Lecture time previously made up.
Wed (2/19): Solutions of Elliptic and Parabolic PDE ( for
the upper half space in Rd+1 ) as convolution operators
with, respectively, the Poisson and Gauss-Weierstrass kernels. Integrated
maximum principles; convergence in norm and nontangentially a.e. as one
approaches the boundary using the weak type boundedness of the nontangential
maximal operators.
Fri (2/21): The Schwartz class as a complete, separable metric
space; basic properties and linear functionals (tempered distributions).
(Poisson summation, Paley-Weiner, and Shannon sampling)
Mon (2/24): Periodization of an L1(R) function;
The Poisson summation formula for f and its Fourier transform; Jacobi's
identity.
Wed (2/26): A simple Sobolev embedding
theorem: W1(L2) -> C(R); The Heisenberg inequality
and interpretations.
Fri (2/28): Entire functions of exponential type; easy
half of the Paley-Weiner theorem; the Phragmen-Lindelof maximum modulus
theorem.
Mon (3/03): Complete the Paley-Weiner Theorem, i.e.
suppt f is contained in [-T,T] implies its Fourier transform is
in E(2 \pi T).
Wed (3/05): The Shannon sampling theorem.
Fri (3/07): Formal definition of Multiresolution Analysis using
the Haar and Stromberg systems as a guide; Riesz systems and bases.
Mon (3/17): MRA's and the scaling function.
Necessary and sufficient conditions for a translates of a
function to be Riesz basis.
Wed (3/19): The scaling function for piecewise linear
splines; Construction of functions whose translates form an
orthonormal basis. The two scale relationship and properties of the
scaling multiplier.
Fri (3/21): Sufficient conditions for scaling functions
and MultiResolution Analyses: The Scaling Theorem.
Mon (3/24): Completion of the proof of the scaling theorem
(limj -> Vj = L2(R),
limj ->-
Vj = {0}).
Wed (3/26): Examples (cont.) of scaling functions, and
illustration of MRA properties of the scaling functions and its scaling
coefficients.
Fri (3/28): Construction of wavelets from an MRA; properties
of wavelet coefficients.
Mon (3/31): Fourier transform characterization of wavelets
derived from a multiresolution analysis; Battle-Lemarie spline family of
wavelets.
Wed (4/2): Verification of MRA properties
required of wavelets.
Fri (4/4): The Meyer wavelets.
Wavelets formulated directly in terms of discrete
equations relating coefficients; recursion to generate the scaling
function from the scaling equation.
(Independence; Rademacher functions and Khinchin's inequality.)
Mon (4/7): The Rademacher functions; orthonormality in
L2([0,1]); Probabilistic independence of functions; independent
sets; Khinchin's inequality.
Wed (4/9): Finish the proof of Khinchin's inequality; proof
of properties of independent collections of functions (integration of products, ...).
Fri (4/11): Unconditional convergence of series in a Banach space.
Begin proof of five equivalent conditions for unconditional convergence: defn,
convergence of rearranged series, bounded multipliers, arbitrary signs,
sums of arbitrary subsequences.
Mon (4/14): Continue with proof of five equivalent conditions for
unconditional convergence.
Wed (4/16): Unconditional bases in Banach spaces; a multiplier theorem.
Fri (4/18): A maximal operator for the Haar system; weak (1,1) estimate;
Wavelets as unconditional bases in Lp.
Mon (4/21): Easter Holiday, no classes.
Wed (4/23): Riesz' factorization theorem; Daubechies construction
of compactly supported smooth wavelets.
Fri (4/25): Nonlinear approximation in L2 and
compact embeddings of Besov spaces.
Mon (4/28): Nonlinear approximation in Lp,
1< p< .
Wed (4/30): Image compression:
Optimal entropy encoders, tree approximation, and image compression.
Multiresolution analysis and optimal approximation
of triangulated surfaces by redundant bases.
Classes End
Student lectures on applications and extensions of results.
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(sharpley@math.sc.edu)
and last updated April 14, 2003.
This page ©2002-2003, The Board of Trustees of the
University of South Carolina.
URL: http://www.math.sc.edu/~sharpley/math758S_sp03/lectures.html
