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Uniform bounds for sampling expansions. (English) Zbl 1033.42007

Summary: Let \(f\in B^2_\sigma\), i.e., \(f\in L^2(\mathbb{R})\) and its Fourier transform \(F(s)= \int_{\mathbb{R}} f(t)\,e^{-2ist}dt\) vanishes outside of \([-\sigma,\sigma]\), then the Shannon sampling theorem says that \(f\) can be reconstructed by its infinitely many sampling points at \(\{k/(2)\}\), \(k\in\mathbb{Z}\), i.e., \[ f(t)= \sum^\infty_{k=-\infty} f\Biggl({k\over 2\sigma}\Biggr) {\sin\pi(2\sigma t-k)\over \pi(2\sigma t-k)},\quad \forall t\in\mathbb{R}. \] But, in practice, only finitely many samples are available, so one would like to study the truncation error \[ T_N(t)= f(t)- \sum^N_{k=-N} f\Biggl({k\over 2\sigma}\Biggr) \text{sinc}(2\sigma t-k),\quad\forall f\in B^2_\sigma. \] The error bounds commonly seen in literature are not uniform. In this paper, the author gives uniform bounds for the truncation error for \(f\in B^2\), when its Fourier transform satisfies some smooth conditions.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
94A20 Sampling theory in information and communication theory
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References:

[1] Beutler, F. J., On the truncation error of the cardinal sampling expansion, IEEE Trans. Inform. Theory, IT-22, 568-573 (1976) · Zbl 0336.94002
[2] Brown, J. L., On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem, J. Math. Anal. Appl., 18, 75-84 (1967) · Zbl 0167.47804
[3] Butzer, P. L.; Engels, W.; Scheben, U., Magnitude of the truncation error in sampling expansions of bandlimited signals, IEEE Trans. Acoust. Speech Signal process, ASSP-30, 906-912 (1982) · Zbl 0564.94004
[4] Butzer, P. L.; Stens, R. L., Sampling theory for not necessarily band-limited functions: A historical overview, SIAM Rev., 34, 40-53 (1992) · Zbl 0746.94002
[5] Butzer, P. L.; Engels, W., On the implementation of the sampling series for bandlimited signals, IEEE Trans. Inform. Theory, IT-32, 314-318 (1986) · Zbl 0513.65100
[6] Cambanis, S., Truncation error bounds for the cardinal sampling expansion of band-limited signals, IEEE Trans. Inform. Theory, IT-28, 605-612 (1982) · Zbl 0491.94005
[7] Eoff, C., The discrete nature of the Paley-Wiener space, Proc. Amer. Math. Soc., 123, 505-512 (1995) · Zbl 0820.30017
[8] Marks, R. J., Advanced Topics on Shannon Sampling and Interpolation Theory (1992), Springer-Verlag: Springer-Verlag New York
[9] Piper, H. S., Bounds for truncation error in sampling expansions of finite energy band-limited signals, IEEE Trans. Inform. Theory, IT-21, 482-485 (1975) · Zbl 0304.94007
[10] Shannon, C. E., A mathematical theory of communication, Bell System Techn. J., 27, 379-423 (1948) · Zbl 1154.94303
[11] Stens, R. L., Error estimates for sampling sums based on convolution integrals, Information and Control, 45, 37-47 (1980) · Zbl 0456.94003
[12] Yao, K.; Thomas, J. B., On truncation error bounds for sampling representations of band-limited signals, IEEE Trans. Aerospace Electronic Syst., AES-2, 640-647 (1966)
[13] Zayed, A. I., Advances in Shannon’s Sampling Theorem (1993), CRC Press: CRC Press Boca Raton · Zbl 0868.94011
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