Introduction

One of the most basic operations on signals in any domain, including 2-sphere, is linear filtering or convolution. Yet, unlike conventional time-domain signals, for signals on the 2-sphere this is not consistently well defined and there exist competing definitions.

So the first aim of this chapter is to study existing definitions of convolution on 2-sphere in the literature and identify their properties, advantages and shortcomings. We determine the relationship between various definitions and show that two seemingly different definitions are essentially the same (which stems from the azimuthally symmetric or isotropic convolution property inherent in those definitions) (Kennedy et al., 2011). Our framework reveals that none of the existing formulations are natural extensions of convolution in time domain. For example, they are not commutative. Changing the role of signal and filter would change the outcome of convolution. Moreover, in one definition, the domain of convolution output does not remain on the 2-sphere.

Recognizing that a well-posed general definition for convolution on 2-sphere which is anisotropic and commutative is more difficult to formulate and that a true parallel with Euclidean convolution may not exist, in the second part of this chapter we use the power of abstract techniques and the tools we have studied to show that a commutative anisotropic convolution on the 2-sphere can indeed be simply constructed. We discuss additional properties of the proposed convolution, especially in the spectral domain, and present some clarifying examples.

We begin with the familiar case of convolution in time domain or convolution on the real line, which shall guide our subsequent developments for convolution on the 2-sphere.

This is our book on the theory of Hilbert spaces, its methods and usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. There are many fine books on Hilbert spaces and our intention is not to generate another book to stick on the pile or to be used to level a desk. So from the onset, we have sought to synthesize the book with special goals in mind.

The needs and concerns of researchers in engineering differ from those of the pure sciences. It is difficult to put the finger on what distinguishes the engineering approach that we take. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a foundational concept that links deeply with other areas of mathematics or is part of a long-standing human intellectual endeavor — not that engineering, in comparison, concerns less intellectual pursuits. As an example, Carleson in 1966 proved a conjecture by Luzin in 1915 concerning the almost-everywhere convergence of Fourier series of continuous functions. Carleson's theorem, as it is called, has its roots in the questions Fourier asked himself, in French presumably, about the nature of convergence of the series named after him. As a result it is important for mathematics, but less clear for engineers.