Preface

The position variable and the derivative operator play eminent roles in quantum physics. They are not commutative, i.e., acting the composite operators and onto a test function is not the same (the order matters in profound ways). Rather, it is the case that as a simple calculation reveals. This motivates considering the commutator , which stands for as a realization of the identity operator. The same is true for the time variable and the derivative operator . The application of the identity matrix to vectors and the operation of the Dirac delta function to functions also reproduce the object onto which they act, leaving the objects unaltered.

However, the position variable and the derivative operator , respectively, the time variable and the derivative operator , and generalizations thereof, are not exclusive preoccupations of quantum physicists. Their rolls in mathematical physics and computational engineering is pervasive and wide ranging. Associating with the wave number and with the radian frequency is embodied in the planewave propagating along the -axis. Planewaves are (computationally nonideal) examples of bases (functional building blocks), and they can be used to synthesize complicated patterns of fields and waves. Their complex conjugate counterparts can be employed to analyze fields and waves. The Fourier transform and inverse transform (FTIT) and their discrete (D-FTIT) and fast realizations (FD-FTIT) are, respectively, manifestations of the analysis and synthesis processes via planewaves.

The Taylor series expansion also plays significant roles in the entire mathematical physics and computational engineering, very often directly, and more often indirectly and in subtle ways. The Taylor series expansion too makes explicit use of the monomials and derivative operators , built from the ingredients and . It requires high-order derivatives of a function, which are evaluated at a single point, say, and uses the set of monomials to synthesize the function on an interval which contains as an interior or a boundary point. The Taylor series expansion is a fascinating and enabling tool. It reveals that while the high-order derivatives are required to be calculated at one point only (here, ), the very existence of the high-order derivatives (before being calculated at one point) encodes the information necessary and sufficient for synthesizing (expressing) the function on the entire interval. The information gathering and encoding are of peculiar types though: when applying to the test function and evaluating the result at , the result, denoted by , is a measure for the relevance of the building block in reconstructing the entire on the entire interval.

Seen from this perspective, it is in essence what the FTIT, or, for this matter, every transform and its associated inverse transform accomplish. The Fourier transform of a function analyses the function and determines the amount of the ingredient basis function that is required to synthesize the function. The Taylor transform of a function analyses the function and determines the amount of the basis function that is required to synthesize the function.

Engineers and scientists appreciate that transforming functions is a big deal. The functions can be scrutinized and manipulated much easier and more pointedly in the transformed domain, which are the main reasons for transforming functions in the first place. Every bona fide transform, however, must admit a well-behaved associated inverse transform. Since the transform of a function (analysis) followed by the corresponding back transform (synthesis) must recover the originating function, it is required that with being the identity operator (the unity matrix in discrete cases). All known transform and inverse transform techniques in the mathematical physics follow this scheme. It should emphatically be pointed out that the relationship does not involve any test function . The specifics of a particular transform technique is incorporated into or by design. Consecutively, or , respectively, must be determined by resorting to the back-transform transform regulating relationship .

In the 80s of the previous century, mathematician, physicists, engineers, and signal processing professionals jointly came up with original ideas to weaken centuries-old conditions imposed upon the basis functions, e.g., locality, orthonormality, and, completeness requirements. These efforts led to the development of the theory of wavelets and dual wavelets, and, in particular, the powerful concept of frames and dual frames. In the following decade, many researchers, including this author, expected utterly new and impact-rich opportunities in computational engineering. However, the anticipated pervasive spread of wavelets and frames in computational engineering did not materialize, despite the fact that wavelets and frames seemed to enable the design of problem-specific analysis and synthesis tools. The concept of developing problem-tailored computational tools is not only fascinating for its own merit but also guides and conditions the intuition in a variety of ways. This author (together with Gilbert Walter) has successfully engaged in constructing wavelets based on Green's functions. This author has also demonstrated that Dirac delta functions can be constructed from specific dyadic Green's functions.

The computational electromagnetic professionals are well familiar with the fact that the utilization of the basis functions and weighting functions in the method of moments (MoM) applications aims to synthesizing a desired function with a priori unknown coefficients and analyzing (determining) the coefficients. The involved basis and weighting functions are in general biorthogonal.

The author's fascination with '''' and '','' which ultimately led to the development of the Discrete Taylor Transform and Inverse Transform (D-TTIT), has its genesis not only in quantum physics but also all the way back to the works of Newton, Leibniz, and Cauchy. The lim-process, built in the conceptualization of , has inspired the author to search for alternative differential and integral formulations, and various summability and integrability methods, and in particular generalized functions, a prominent advocate of which is the Dirac's delta function. The set of monomials , being the archetypical basis, is the foundation of a myriad of orthonormal basis (ONB) functions in the mathematical physics. Each classical orthonormal basis has the potential to give rise to a transform and the corresponding inverse transform. The theories of wavelets and frames extend this concept to biorthogonal general systems of functions. Since between the archetypical basis and the canonical ONB in mathematical physics, a correspondence can be established, extensions of D-TTIT can be envisioned.

This book has been in making since 2019. It introduces D-TTIT and authentically shares its historical development with the reader. The ensuing occasional inconsistencies in notations are hoped to be overlooked. With each passing year new aspects of the theory unraveled themselves. With each proofreading session of a chapter, new connections with group theory, number theory and other branches in mathematics were recognized, which had to be investigated, examined, interpreted, refined, or discarded. The work is by no means complete. However, it has reached a fair level of maturity that can be used in engineering applications and hopefully inspires other researchers to examine any of the salient features of the theory which deserve further investigation. The story telling in the book is based on simplest toy models and variations thereof. The toy models involving univariate functions (e.g., ) are comparatively simple. At times the calculations are lengthy and laborious. All relevant details have been provided, without any reservation. The motivation behind this has been to enable the reader to compare their calculations against the solutions, in case they wish to do so. The interested reader may also benefit from the detailed ideation processes.

The contents of the book can be divided into three parts, without explicitly having introduced the notion of parts. Part I deals with univariate functions, Part II with bivariate functions, and Part III with trivariate functions. The introduction briefly and casually introduces the notions of orthonormal bases, nonorthonormal bases, frames, frame operator, inverse frame operator, and dual frames. The notion of the identity operator has been introduced as well. Dirac's notion of bras (row vectors) and kets (column vectors), along with the inner product and the exterior product, have been discussed. The Fourier transform and inverse transform have been reviewed within the concept of the resolution of identity.

Chapters 1-11 concern toy models involving univariate functions. Thereby, the early chapters invite the reader to search for patterns, identify them, and utilize them. The latter chapters provide recipes for calculating difference operators of arbitrary complexity.

Chapter 12 analyses bivariate functions (e.g., ). It deals with the simplest possible toy model in this category.

Chapter 13, discussing trivariate functions (e.g., ) is by far the longest chapter comprising nearly one third of the book, despite the fact it deals with the simplest possible toy model. Every conceivable detail has been provided to encourage further research. The book introduces D-TTIT and authentically shares its historical development with the reader. The ensuing occasional inconsistencies in...