Topic

No. of hours

Lecture

Tutorial/Practical

FOURIER SERIES :-

Motivation and Heuristics.

Motivation from Physics.

The Vibrating String.

Heat Flow in Solids.

Absolutely Convergent Trigonometric Series.

Examples of Factorial and Bessel Functions.

Poisson Kernel Example.

Proof of Laplace's Method.

Non absolutely Convergent Trigonometric Series.

Formulation of Fourier Series.

J Fourier Coefficients and Their Basic Properties.

Fourier Series of Finite Measures.

*Rates of Decay of Fourier Coefficients.

Piecewise Smooth Functions.

Fourier Characterization of Analytic Functions.

Sine Integral.

Point wise Convergence Criteria.

*Integration of Fourier Series.

Convergence of Fourier Series of Measures.

Riemann Localization Principle.

Gibbs-Wilbraham Phenomenon.

The General Case.

Fourier Series in L 2.

Mean Square Approximation-Parseval's Theorem.

*Appljcation to the Isoperimetric Inequality.

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Norm Convergence and Summability.

Approximate Identities.

Almost-Everywhere Convergence of the Abel Means.

Summability Matrices.

Fejer Means of a Fourier Series.

Wiener's Closure Theorem on the Circle.

*Equjdistribution Modulo One.

*Hardy's Tauberian Theorem.

Improved Trigonometric Approximation.

Rates ofConvergence in C(1I').

Approximation with Fejer Means.

*Jackson's Theorem.

*Higher-Order Approximation.

*Converse Theorems of Bernstein.

Divergence of Fourier Series.

The Example ofdu Bois-Reymond.

Analysis via Lebesgue Constants..

Divergence in the Space L1

*Appendix: Complements on Laplace's Method.

First Variation on the Theme-Gaussian Approximation.

Second Variation on the Theme-Improved Error Estimate.

*Application to Bessel Functions.

*The Local Limit Theorem of DeMoivre-Laplace.

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FOURIER TRANSFORMS ON THE LINE AND SPACE :-

Basic Properties of the Fourier Transform.

Riemann-Lebesgue Lemma.

Approximate Identities and Gaussian Summability.

Improved Approximate Identities for Point wise Convergence.

Application to the Fourier Transform.

Then-Dimensional Poisson Kernel.

Fourier Transforms of Tempered Distributions.

*Characterization of the Gaussian Density.

*Wiener's Density Theorem.

Fourier Inversion in One Dimension.

Dirichlet Kernel and Symmetric Partial Sums.

Example of the Indicator Function.

Gibbs-Wilbraham Phenomenon.

Dini Convergence Theorem.

Extension to Fourier's Single Integral.

Smoothing Operations in JR 1-Averaging and Summability.

Averaging and Weak Convergence.

Cesaro Summability.

Approximation Properties of the Fejer Kernel.

Bernstein's Inequality.

*One-Sided Fourier Integral Representation.

Fourier Cosine Transform.

Fourier Sine Transform.

Generalized h-Transform.

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L2 Theory in Rn :-

Plancherel's Theorem.

*Bernstein's Theorem for Fourier Transforms.

The Uncertainty Principle.

Uncertainty Principle on the Circle.

Spectral Analysis of the Fourier Transform.

Hermite Polynomials.

Eigenfunction of the Fourier Transform.

Orthogonality Properties.

Completeness.

Spherical Fourier Inversion in Rn.

Bochner's Approach.

Piecewise Smooth Viewpoint.

Relations with the Wave Equation.

The Method of Brandolini and Colzani.

Bochner-Riesz Summability.

A General Theorem on Almost-Everywhere Summability.

Bessel Functions.

Fourier Transforms of Radial Functions.

L2-Restriction Theorems for the Fourier Transform.

An Improved Result.

Limitations on the Range ofp.

The Method of Stationary Phase.

Statement of the Result.

Application to Bessel Functions.

Proof of the Method of Stationary Phase.

Abel 's Lemma.

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FOURIER ANALYSIS IN Lp SPACES :-

The M. Riesz-Thorin Interpolation Theorem.

Generalized Young's Inequality.

The Hausdorff-Young Inequality.

Stein's Complex Interpolation Theorem.

The Conjugate Function or Discrete Hilbert Transform.

Lp Theory of the Conjugate Function.

L1 Theory of the Conjugate Function.

Identification as a Singular Integral.

The Hilbert Transfom1 on lR.

L2 Theory of the Hilbert Transform.

Lp Theory of the Hilbert Transform, J < p < oo.

Applications to Convergence of Fourier Integrals.

L1 Theory of the Hilbert Transform and Extensions.

Kolmogorov's Inequality for the Hilbert.

Transform.

Application to Singular Integrals with Odd Kernels.

Hardy-Littlewood Maximal Function.

Application to the Lebesgue Differentiation Theorem.

Application to Radial Convolution Operators.

M aximal Inequalities for Spherical Averages.

The Marcinkiewicz Interpolation Theorem.

CaJder6n-Zygmund Decomposition.

A Class of Singular Integrals.

Properties of Harmonic Functions.

General Properties.

Representation Theorems in the Disk.

Representation Theorems in the Upper Half-Plane.

Herglotz/Bochner Theorems and Positive Definite Functions.

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POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES :-

The Poisson Summation Formula in R1.

Periodization of a Function.

Statement and Proof.

Shannon Sampling.

Multiple Fourier Series.

Basic L1 Theory.

Point wise Convergence for Smooth Functions.

Representation of Spherical Partial Sums.

Basic L2 Theory.

Restriction Theorems for Fourier Coefficients.

Poisson Summation Formula in Rd.

*Simultaneous Non localization.

Application to Lattice Points.

Kendall's Mean Square Error.

Landau's Asymptotic Formula.

Application to Multiple Fourier Series.

Three-Dimensional Case.

Higher-Dimensional Case.

Schrodinger Equation and Gauss Sums.

Distributions on the Circle.

The Schrodinger Equation on the Circle.

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APPLICATIONS TO PROBABILITY THEORY :-

Motivation and Heuristics.

Basic Definitions.

The Central Limit Theorem.

Restatement in Terms of Independent.

Random Variables.

Extension to Gap Series.

Extension to Abel Sums.

Weak Convergence of Measures.

An Improved Continuity Theorem.

Another Proof of Bochner's Theorem.

Convolution Semigroups.

The Berry-Esseen Theorem.

Extension to Different Distributions.

The Law of the Iterated Logarithm.

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INTRODUCTION TO WAVELETS :-

Motivation and Heuristics.

Heuristic Treatment of the Wavelet Transform.

Wavelet Transform.

Wavelet Characterization of Smoothness.

Haar Wavelet Expansion.

Haar Functions and Haar Series.

Haar Sums and Dyadic Projections.

Completeness of the Haar Functions.

Haar Series in Co and Lp Spaces.

Point wise Convergence of Haar Series.

*Construction of Standard Brownian Motion.

Haar Function Representation of Brownian Motion.

*Proofof Continuity.

*Levy's Modulus of Continuity.

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Multi resolution Analysis :-

Orthonormal Systems and Riesz Systems.

Scaling Equations and Structure Constants .

From Scaling Function to MRA.

Additional Remarks.

Meyer Wavelets.

From Scaling Function to Orthonormal Wavelet.

Null Integrability of Wavelets Without Scaling Functions.

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Wavelets with Compact Support.

From Scaling Filter to Scaling Function.

Explicit Construction of Compact Wavelets.

Daubechies Recipe.

Hernandez-Weiss Recipe.

Smoothness of Wavelets.

A Negative Result.

Cohen's Extension of Theorem.

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Convergence Properties of Wavelet Expansions.

Wavelet Series in Lp Spaces.

Large Scale Analysis.

Almost-Everywhere Convergence.

Convergence at a Preassigned Point.

Jackson and Bernstein Approximation Theorems.

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Wavelets in Several Variables.

Two Important Examples.

Tensor Product of Wavelets.

General Formulation of MRA and Wavelets in Rd.

Notations for Subgroups and Cosets.

Riesz Systems and Orthonormal Systems in Rd.

Scaling Equation and Structure Constants.

Existence of the Wavelet Set.

Proof That the Wavelet Set Spans V1 e Vo.

Cohen's Theorem in Rd.

Examples of Wavelets in Rd.

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