FOURIER TRANSFORMS In this section we define Fourier transforms and Fourier series, and illustrate, intuitively, that one can guess at the inversion formula in terms of the uncertainty principle.
We illustrate the uncertainty principle in a piano experiment; and, in this context, we describe temporal and frequency localization and how they are related. We also point out that the uncertainty principle is pervasive in the deepest problems of harmonic analysis.
The inversion formula has been a critical component of harmonic analysis from its origins in solving PDEs by transforming them to algebraic equations, and then inverting the algebraic solution. The associated uniqueness problems were initiated by Riemann, and have led to fundamental work by great analysts of the 19th and 20th century. The characterization of sets of uniqueness is still open, and the problem involves fundamental ideas from number theory and spectral synthesis. There are also algebraic geometric problems associated with applying Fourier inversion to the solution of PDEs, and spectral synthesis, ideal structure formulation of such problems in harmonic analysis is analogous to the Nullstellensatz in algebraic geometry.
There are also significant applications of harmonic analysis in signal processing. We illustrate the role of harmonic in signal processing in the next two transparencies by analyzing epileptic seizure data by means of spectrograms. Before going to these sections, we emphasize the algebraic structure and geometric invariants that allow for the effectiveness of modern harmonic analysis.
Details on the history of the subject, as well as fundamental examples are found in the speaker's book, HARMONIC ANALYSIS AND APPLICATIONS (CRC Press).
EPILEPTIC SEIZURE DATA The frequency or spectral content of this data is difficult to see by looking directly at the time series. As such we compute the short time Fourier transform of the data, and can analyze its spectral content by analyzing the following spectrogram.
SPECTROGRAM OF THIS SEIZURE DATA Although this is Fourier data, our eventual analysis of such data led us to wavelet periodicity detection. (With Goetz Pfander.)
POISSON SUMMATION FORMULA In this section we first define the Fourier transform in different settings. We note two directions in the case of locally compact abelian groups (LCAGs) G. If nondiscrete G contains open subgroups then we are in a setting appropriate for dealing with number theoretic ideas, e.g., involving the p-adics. If G contains discrete subgroups then it is possible to prove Poisson summation formulas (PSFs). There are even PSFs in the non-abelian LCG, e.g., the celebrated Selberg trace formula in number theory. (With Georg Zimmermann.)
For most computational signal processing and scientific applications the Fourier transform is used in the setting of finite groups. In this case the Fourier transform is called the discrete Fourier transform (DFT). The DFT is extraordinarily useful because of fast Fourier transform (FFT) algorithms going back to Gauss. In order to formulate a Fourier analysis problem in Euclidean space in such a way as to implement the DFT, sampling and periodization are required; and PSFs are essentially equivalent to sampling formulas. The PSF is also a method of dealing with Euler-Maclaurin expansions and various problems in numerical analysis. Further, the PSF is a staple in number theory, not only at the level of the Poisson summation formula, but in establishing analytic continuations of zeta functions. (With Dave Joyner.)
WAVELETS Spectral analysis (Fourier methods) is not sufficient to characterize complicated signals. Wavelet theory provides another set of tools, ultimately related to the affine group, with which to do signal analysis. In particular, wavelets are most effective in dealing with multiresolution (zooming-in) and localization (comparing local signal changes with coefficient changes) problems.
A 2D-HAAR WAVELET We saw that the Haar function on the real line (R) is a wavelet in the sense that its integer translates and dilates are an orthonormal basis for the square integrable functions on R. It is also the first as well as a primordial example of a so-called multiresolution analysis wavelet. Wavelet packets are a generalization of multiresolution analysis, but the essential example of wavelet packets is the family of Walsh functions, which are known to be generated from the Haar wavelet. Professor Walsh set out the properties of Walsh functions in 1921, but they were known to reduce channel crosstalk by engineers around 1900. Professor Walsh was a professor at U. of Maryland after retiring from Harvard. (With Sandra Saliani, Erica Bernstein, and Ioannis Konstantinidis.)
This twin dragon, if colored differently for each twin, is a 2-dimensional Haar function for a certain non-dyadic expansive matrix, giving rise to a non-separable filter.
A FRACTAL ASSOCIATED WITH A CLASS OF WAVELET SETS The jagged line is a fractal associated with a set of so-called wavelet sets. Each of these sets is the support of the Fourier transform of a 2-dimensional dyadic wavelet which is NOT a multiresolution wavelet. Generally, multidimensional and LCAG wavelet theory involves subtleand intricate geometric analysis, e.g., the construction of sets which are tiles for more than one group of operations. (With Manuel Leon and Songkiat Sumetkijakan.)
FRAMES AND SAMPLING The theory of frames has long history through 19th and 20th century analysis. Ideas of Riemann, Weber, and Dini come from the 19th century; and then the work of G.D. Birkhoff, Wiener, Paley, Levinson, and Beurling and Malliavin led the way in the 20th century. The topic evolved into the study of non-harmonic Fourier series from the time of Paley and Wiener, dovetailing with the deepest classical analysis of the century. Duffin and Schaeffer defined frames in a general context and characterized so-called Fourier frames in terms of density criteria. (With Jean-Pierre Gabardo, Bill Heller, and Joe Lakey.)
Simply, frames provide decompositions of functions into "harmonics" which are not necessarily bases. This generality not only involves significant tools from abstract analysis, but is a natural model in dealing with a host of applications, e.g., noise reduction, wavelet auditory modelling. (With Tony Teolis, Chris Heil, Dave Walnut, Shidong Li, Oliver Treiber.)
AN MRI PROBLEM AND FOURIER FRAMES Fourier frames are a natural tool for dealing with nonuniform sampling problems. Their difficulty is well-known to experts, their structure is tantalizing, and their applicability is extensive. In this example, Fourier frames are used to obtain fast acquisition of date in Magnetic Resonance Imaging problems. Other applications include synthetic aperture radar (SAR), functional mapping for the brain, as well as any uniform sampling problem whose data is contaminated by noises or communication processes. (With Hui-Chuan Wu, Rod Kerby, Alex Powell, Alfredo Nava-Tudela, Sherry Scott, and Jeff Sieracki.)
THE VERTICES OF THE PLATONIC SOLIDS ARE FINITE NORMALIZED TIGHT FRAMES IN 3D In order to implement the theory of frames, it is usually necessary to deal with the finite frame case directly since spurious eigenvalues limit truncation effectiveness in linear algebra modelling. (With Melissa Harrison and Anwar Saleh.) Here are surprising examples of finite normalized tight (good for fast convergence) frames in 3d. (With Matt Fickus.)
THE VERTICES OF A SOCCER BALL ARE A FINITE NORMALIZED TIGHT FRAME IN 3D
In this first lecture we have gone into nonuniform sampling and finite frames, whereas some would say that it is reasonable to start with uniform sampling. Well, let's finish with uniform sampling!
HISTORICAL BACKGROUND FOR UNIFORM SAMPLING The Classical (uniform) Sampling Theorem (CST) was proved by Cauchy in 1841. We trace its influence in 19th century analytic number theory, complex analysis, and interpolation theory. The form in which we shall state it was known at the end of the 19th century and was used in very sophisticated ways by Hadamard, Steffensen, Borel, de la Valle'e-Poussin, et al. early in the 20th century. The CST was rediscovered by E. T. Whittaker, although some of the previously mentioned authors published their results in major journals and well known books. The Russian engineer Kotel'nikov also rediscovered it in the context of "electrocommunications", and the first English translation of Kotel'nikov's paper appears in the speaker's edited volume, MODERN SAMPLING THEORY (Birkhauser), with Paulo Ferreira. The editors' Chapter 1 has a technical and historical overview of mathematical results in both uniform and nonuniform sampling. Whittaker's, Kotel'nikov's, and Claude Shannon's names are often used to designate the CST. Shannon was aware of the result, but did give another proof (similar to some previous ones) and most importantly applied it.
THE CLASSICAL SAMPLING THEOREM Our statement is for the real line, but both statement and proof can be generalized to d-dimensional Euclidean space as well as LCAGs (Kluvanek's theorem). The following figure provides the setting for the theorem in the spectral domain.
SPECTRAL SETTING FOR THE CLASSICAL SAMPLING THEOREM
THE SHANNON WAVELET We combine the statement of the Classical Sampling Theorem as well as the Spectral Setting with a remark about the Shannon Wavelet.
BACKGROUND FOR THE PROOF OF THE CLASSICAL SAMPLING THEOREM These definitions and results are required for "our" proof of the CST. The main idea in any proof evolves around the notion of periodization, which in turn depends on understanding the role of discrete subgroups of a given group.
PROOF OF THE CLASSICAL SAMPLING THEOREM This is a proof of the CST. PROOF CONTINUED
Here is the same proof, a bit more cleanly, and stated for the sinc. THE PROOF RE-EMPHASIZED
THE POISSON SUMMATION FORMULA We revisit the PSF since in certain settings it is essentially equivalent to the CST; and the CST can be proved by means of the PSF. Because of the importance of the PSF we close this exposition with the following transparency, which recalls two of its aforementioned applications, as well as several others.
SOME APPLICATIONS OF THE POISSON SUMMATION FORMULA