Lattice Point Identities and Shannon-Type Sampling

Willi Freeden, M. Zuhair Nashed

• Publisher: Chapman and Hall/CRC
• Year: 2019
• ISBN:
  9781000757743
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Signals & Signal Processing
• Index: Technology & Engineering

Lattice Point Identities and Shannon-Type Sampling demonstrates that significant roots of many recent facets of Shannon's sampling theorem for multivariate signals rest on basic number-theoretic results.

This book leads the reader through a research excursion, beginning from the Gaussian circle problem of the early nineteenth century, via the classical Hardy-Landau lattice point identity and the Hardy conjecture of the first half of the twentieth century, and the Shannon sampling theorem (its variants, generalizations and the fascinating stories about the cardinal series) of the second half of the twentieth century. The authors demonstrate how all these facets have resulted in new multivariate extensions of lattice point identities and Shannon-type sampling procedures of high practical applicability, thereby also providing a general reproducing kernel Hilbert space structure of an associated Paley-Wiener theory over (potato-like) bounded regions (cf. the cover illustration of the geoid), as well as the whole Euclidean space.

All in all, the context of this book represents the fruits of cross-fertilization of various subjects, namely elliptic partial differential equations, Fourier inversion theory, constructive approximation involving Euler and Poisson summation formulas, inverse problems reflecting the multivariate antenna problem, and aspects of analytic and geometric number theory.

Features:

• 
  
  

• New convergence criteria for alternating series in multi-dimensional analysis

• 
  
  

• Self-contained development of lattice point identities of analytic number theory

• 
  
  

• Innovative lattice point approach to Shannon sampling theory

• 
  
  

• Useful for students of multivariate constructive approximation, and indeed anyone interested in the applicability of signal processing to inverse problems.

Part I

Central Theme

Chapter 1

From Lattice Point to Shannon-Type Sampling Identities

In the year 1765, J.L. Lagrange proposed a sampling theorem for bandlimited periodic functions by stating that the knowledge of functional values at 2n + 1 equidistant points within a period is sufficient to represent uniquely a periodic function by sine and cosine terms. A.L. Cauchy [1841] found an interpolation formula for a bandlimited function, i.e., a function representable by a complex Fourier expansion showing no more than a certain number of coefficients. These essential results may be seen as the prehistory of sampling.

1.1Classical Framework of Shannon Sampling

Following P.L. Butzer and R.L. Stens [1992], the history of today’s understanding of the sampling theory can be traced back to the interpolation theory using equidistant nodes published by the Belgian mathematician Charles–Jean Baron de la Vallée Poussin (1866-1962). He probably was the first person to consider sampling for not necessarily bandlimited functions already in 1908. In fact, Ch.–J. de la Vallée Poussin [1908] treated the particular case of non-bandlimited (more concretely, spacelimited) functions, a class of functions that actually represent signals occurring often in engineering practice (note that these signals cannot be bandlimited simultaneously due to the uncertainty principle). The German mathematician M. Theis [1919] continued de la Vallée Poussin’s work in a considerable way.

Early in one-dimensional theory, variants of bandlimited lattice point sampling were given by E.T. Whittaker [1915]; E.T. Whittaker [1923]; E.T. Whittaker [1929], and K. Ogura [1920]. In later years, it became known that the classical sampling theorem had been presented before C.E. Shannon [1949a,b] to the Russian communication community by V.A. Kotel’nikov [1933]. In a more implicit verbal form, i...