Fourier analysis on the hypercube, the coefficient problem, and applications

Abstract

In this dissertation, we primarily study some problems that revolve around Fourier analysis. More specifically we focus on the magnitudes of the frequency components. Firstly, we perform a study on the hypercube. It is well known that the Delsarte linear programming bounds provide rich information on the magnitudes of the Fourier coefficients, grouped by Hamming weight. Classically, such information is primarily used to attack coding problems, where the objective is to maximize cardinality of a subset of a metric space subject to a minimum distance constraint. Here, we use it to study anticoding problems, where the objective is to maximize cardinality of a subset of a metric space subject to a maximum distance (diameter) constraint. One motivation for such study is the problem of finding memories that are cheap to update, where the cost of an update is a function of the distance in the metric space.

Such a view naturally supports the study of different cost functions going beyond hard diameter constraints. We work accordingly with different cost functions, with a particular emphasis on completely monotonic functions. Our emphasis is on the phenomenon of "universal optimality", where the same subset (anticode) simultaneously optimizes a wide range of natural cost functions. Among other things, our work here gives some answers to a question in computer science, namely finding Boolean functions with maximal noise stability subjected to an expected value constraint. Secondly, we work with Fourier analysis on the integers modulo a number by draw-ing upon Nazarov's general solution to the "coefficient problem". Roughly speaking, the coefficient problem asks one to construct time domain signals with prescribed magnitudes of frequency components, subject to certain natural constraints on the signal. In particular, Nazarov's solution works with l[subscript p] constraints in time.

This solution to the coefficient problem allows us to give an essentially complete resolution to the mathematical problem of designing optimal coded apertures that arises in computational imaging. However, the resolution we provide is for an l[subscript infinity] constraint on the aperture, corresponding to partial occlusion. We believe it is important to also examine a binary valued ({0, 1}) constraint on the aperture as one does not need to synthesize partial occluders for such apertures. We therefore provide some preliminary results as well as directions for future research. Finally, inspired by the recent breakthroughs in understanding the d = 8, 24 cases of sphere packing and universal optimality in R[superscript d], we attempt to show that the associated lattices (E₈ and the Leech lattice for d = 8, 24 respectively) are also optimal for the problem of vector quantization in the sense of minimizing mean squared error.

Accordingly, we develop a dispersion and anticoding based approach to lower bounds on the mean squared error. We also generalize Tóth's method, which shows optimality of the hexagonal lattice quantizer for d = 2, to arbitrary d. To the best of our knowledge, these methods give the first rigorous improved lower bounds for the mean squared error for all large enough d since the work of Zador over 50 years ago.