Abstract
Sampling theory has been studied in a variety of function spaces and our purpose here is to develop the theory in a Fourier algebra setting. This aspect is not as well-known as it might be, possibly because the approximate sampling theorem, central to our discussion, had what could be called a mysterious birth and a confused adolescence. We also discuss functions of the familiar bandlimited and bandpass types, showing that they too have a place in this Fourier algebra setting. This paper combines an expository and historical treatment of the origins of exact and approximate sampling, including bandpass sampling, all in the Fourier algebra setting. It has two objectives. The first is to provide an accessible and rigorous account of this sampling theory. The various cases mentioned above are each discussed in order to show that the Fourier algebra, a Banach space, is a broad and natural setting for the theory. The second objective is to clarify the early development of approximate sampling in the Fourier algebra and to unravel its origins.
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Notes
- 1.
The name classical sampling theorem already occurs in [8, p. 75].
- 2.
The adjective ‘exact’ is in contrast with ‘approximate’ sampling; the two forms are defined in Sect. 2.4.
- 3.
This terminology was not used by these authors but at the time it was relatively recent [27].
- 4.
The time and frequency domains are both represented by the real line because it is self-dual [47].
- 5.
In the interval ψ has only finitely many discontinuities and its real and imaginary parts have only finitely many maxima and minima [55, p. 407].
- 6.
The translates of the period are essentially cosets of the quotient group \(\mathbb {R}/\mathbb {S}^1\cong \mathbb {Z}\).
- 7.
- 8.
The extensions to \(\mathbb {C}\) are entire [33, Theorem 7.2].
- 9.
- 10.
The formula is
$$\displaystyle \begin{aligned} C(t) = \frac{1}{2} + \frac{1}{\uppi\, t} \sin \frac{\uppi}{2} t^2 + O\left(\frac{1}{t^2} \right); \end{aligned}$$this also follows from [42, p. 356].
- 11.
Most of this note concerns summation formulae and establishes Theorem 6 (the approximate sampling theorem for A). Brown [8] and Weiss [56] are cited but not Standish [52] or Stickler [54]. In a short concluding section, Theorem 3 (the exact sampling theorem for A w) is proved under rather general hypotheses, namely that the signal f is in A w.
- 12.
Their paper discusses continuous analogues of the binomial series and does not mention sampling.
- 13.
There is one simple discontinuity at each of the end points ±π w of the interval and the number of maxima and minima of \(\cos \, t\uptau \) and \(\sin \,t\uptau \) is O(tw).
- 14.
The symmetry of the intervals is consistent with the signals being real but this restriction is not imposed on f.
- 15.
Known as the ‘quadrature’ function of g in signal processing [7, p. 269]. We are grateful to Professor Alister Burr of the Dept. of Electronics, University of York for explaining its significance in signal processing.
- 16.
A striking case of multiple discovery occurs for exact sampling in the L 2 setting [29].
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Acknowledgements
We are grateful to the editors for the opportunity to contribute to this centennial ANHA volume for Claude Shannon and particularly to Stephen Casey for his patience and long-distance support. Our thanks for their long standing interest and helpful comments also go to Paul Butzer, Paulo Ferreira and Simon Eveson, who also gave us invaluable assistance with LaTeX, and to the referee for suggesting an improvement to the presentation. We are also grateful to our families for their wonderful support which enabled us to complete this paper.
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Maurice Dodson, M., Higgins, J.R. (2020). Sampling Theory in a Fourier Algebra Setting. In: Casey, S., Okoudjou, K., Robinson, M., Sadler, B. (eds) Sampling: Theory and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36291-1_3
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