Gerald B. Folland

The Uncertainty Principle: A Mathematical Survey

We survey various mathematical aspects of the uncertainty principle, including Heisenberg’s inequality and its variants, local uncertainty inequalities, logarithmic uncertainty inequalities, results relating to Wigner distributions, qualitative uncertainty principles, theorems on approximate concentration, and decompositions of phase space.

How to Integrate A Polynomial Over A Sphere

Several recent articles in the MONTHLY ([1], [2], [4]) have involved finding the area of n-dimensional balls or spheres or integrating polynomials over such sets. None of these articles, however, makes use of the most elegant and painless method for performing such calculations, which is to reverse the usual trick for computing flOe 2 dx. There is nothing new in this idea. It was shown to me in 1971 by V. Bargmann and E. Nelson, and I included it as an exercise in my book [3], whose first edition appeared in 1984. But the evidence suggests that it is not as universally known as it should be. First, some notation: For x = (xi, ... , xn) E Rn, we set Ix = (X2 + * + x2)1/2, and we denote the sphere and ball of radius r about the origin in Rn by Sn(r) and Bn (r), respectively:

Compact Heisenberg manifolds as CR manifolds

Let M be the quotient of the Heisenberg group by a discrete co-compact subgroup, with the natural strongly pseudoconvex CR structure. We identify the eigenvalues and eigenforms of the Kohn Laplacians on M and show how to realize M as the boundary of a bounded domain in a line bundle over an Abelian variety.

The abstruse meets the applicable: Some aspects of time-frequency analysis

The area of Fourier analysis connected to signal processing theory has undergone a rapid development in the last two decades. The aspect of this development that has received the most publicity is the theory of wavelets and their relatives, which involves expansions in terms of sets of functions generated from a single function by translations and dilations. However, there has also been much progress in the related area known astime-frequency analysis orGabor analysis, which involves expansions in terms of sets of functions generated from a single function by translations and modulations. In this area there are some questions of a concrete and practical nature whose study reveals connections with aspects of harmonic and functional analysis that were previously considered quite pure and perhaps rather exotic. In this expository paper, I give a survey of some of these interactions between the abstruse and the applicable. It is based on the thematic lectures which I gave at the Ninth Discussion Meeting on Harmonic Analysis at the Harish-Chandra Research Institute in Allahabad in October 2005.

From calculus to wavelets: A new mathematical technique

A ‘wavelet’ is a function that exhibits oscillatory behaviour in some interval and then decays rapidly to zero outside this interval. A remarkable discovery of recent years is that the translations and dilations of certain wavelets can be used to form sets of ‘basic’ functions for expanding general functions into infinite series, and these expansions have many theoretical and practical applications.

Hermite distributions associated to the group (

We calculate the tempered O(p, q)-invariant eigendistributions of the O(p, q)-invariant Hermite operator − 1 2 (∆x − ∆y) + 12 (|x|2 − |y|2) (x ∈ R, y ∈ R). They are singular on the cone |x| = |y| and are given elsewhere in terms of confluent hypergeometric functions. Suppose p and q are positive integers. We consider the quadratic form

Time-Frequency Analysis and Representations of the Discrete Heisenberg Group

The operators [ϱ ω (j, k, l)f](t) = e2πiωl e2πiωkt f(t + j) on \(L^{2}(\mathbb{R})\) constitute a representation of the discrete Heisenberg group. We investigate how this representation decomposes as a direct integral of irreducible representations. The answer is quite different depending on whether ω is rational or irrational, and in the latter case it provides illustrations of some interesting pathological phenomena.

Constantin Carathéodory: Mathematics and Politics in Turbulent Times

1. Origin and the formative years * 2. Academic Career in Germany * 3. The Asia Minor Project * 4. A Scholar of World Reputation * 5. National Socialism and War * 6. The Last years.

Ripples in Mathematics (Book)

1. Introduction.- 1.1 Prerequisites.- 1.2 Guide to the Book.- 1.3 Background Information.- 2. A First Example.- 2.1 The Example.- 2.2 Generalizations.- Exercises.- 3. The Discrete Wavelet Transform via Lifting.- 3.1 The First Example Again.- 3.2 Definition of Lifting.- 3.3 A Second Example.- 3.4 Lifting in General.- 3.5 DWT in General.- 3.6 Further Examples.- Exercises.- 4. Analysis of Synthetic Signals.- 4.1 The Haar Transform.- 4.2 The CDF(2,2) Transform.- Exercises.- 5. Interpretation.- 5.1 The First Example.- 5.2 Further Results on the Haar Transform.- 5.3 Interpretation of General DWT.- Exercises.- 6. Two Dimensional Transforms.- 6.1 One Scale DWT in Two Dimensions.- 6.2 Interpretation and Examples.- 6.3 A 2D Transform Based on Lifting.- Exercises.- 7. Lifting and Filters I.- 7.1 Fourier Series and the z-Transform.- 7.2 Lifting in the z-Transform Representation.- 7.3 Two Channel Filter Banks.- 7.4 Orthonormal and Biorthogonal Bases.- 7.5 Two Channel Filter Banks in the Time Domain.- 7.6 Summary of Results on Lifting and Filters.- 7.7 Properties of Orthogonal Filters.- 7.8 Some Examples.- Exercises.- 8. Wavelet Packets.- 8.1 From Wavelets to Wavelet Packets.- 8.2 Choice of Basis.- 8.3 Cost Functions.- Exercises.- 9. The Time-Frequency Plane.- 9.1 Sampling and Frequency Contents.- 9.2 Definition of the Time-Frequency Plane.- 9.3 Wavelet Packets and Frequency Contents.- 9.4 More about Time-Frequency Planes.- 9.5 More Fourier Analysis. The Spectrogram.- Exercises.- 10. Finite Signals.- 10.1 The Extent of the Boundary Problem.- 10.2 DWT in Matrix Form.- 10.3 Gram-Schmidt Boundary Filters.- 10.4 Periodization.- 10.5 Moment Preserving Boundary Filters.- Exercises.- 11. Implementation.- 11.1 Introduction to Software.- 11.2 Implementing the Haar Transform Through Lifting.- 11.3 Implementing the DWT Through Lifting.- 11.4 The Real Time Method.- 11.5 Filter Bank Implementation.- 11.6 Construction of Boundary Filters.- 11.7 Wavelet Packet Decomposition.- 11.8 Wavelet Packet Bases.- 11.9 Cost Functions.- Exercises.- 12. Lifting and Filters II.- 12.1 The Three Basic Representations.- 12.2 From Matrix to Equation Form.- 12.3 From Equation to Filter Form.- 12.4 From Filters to Lifting Steps.- 12.5 Factoring Daubechies 4 into Lifting Steps.- 12.6 Factorizing Coiflet 12 into Lifting Steps.- Exercises.- 13. Wavelets in Matlab.- 13.1 Multiresolution Analysis.- 13.2 Frequency Properties of the Wavelet Transform.- 13.3 Wavelet Packets Used for Denoising.- 13.4 Best Basis Algorithm.- 13.5 Some Commands in Uvi_Wave.- Exercises.- 14. Applications and Outlook.- 14.1 Applications.- 14.2 Outlook.- 14.3 Some Web Sites.- References.

The Neumann Problem for the Cauchy-Riemann Complex. (AM-75)

Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.