An Inter-Extrapolation Formula in Sampling Theorem for Band-Limited Distributions
Summary :
The sampling theorem can be extended to cover all distributions in L. Schwartz's sense. As any band-limited distribution reduces to an entire function of exponential type, the sampling theorem for band-limited signals is nothing but an interpolation or extrapolation of such a function. Here, an explicit inter-extrapolation formula with equally-spaced sampling points on the negative time axis is presented. The formula can be applied to all band-limited signals appearing in practice because on additional requirement is imposed on signals such as being square-integrable, being bounded or being tempered. And moreover, since the sampling points are entirely contained in the negative time axis, a signal at any time can be reconstructed from the signal values up to the present without delay, and in principle a deterministic prediction is possible for arbitrarily far future. The formula is obtained combining the extension of Newton interpolation formula for polynomials with Mittag-Leffler summation for divergent series.
- Publication
- IEICE TRANSACTIONS on transactions Vol.E67-E No.3 pp.147-152
- Publication Date
- 1984/03/25
- Publicized
- Online ISSN
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- Type of Manuscript
- PAPER
- Category
- Mathematics
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Hirosi SUGIYAMA, "An Inter-Extrapolation Formula in Sampling Theorem for Band-Limited Distributions" in IEICE TRANSACTIONS on transactions,
vol. E67-E, no. 3, pp. 147-152, March 1984, doi: .
Abstract: The sampling theorem can be extended to cover all distributions in L. Schwartz's sense. As any band-limited distribution reduces to an entire function of exponential type, the sampling theorem for band-limited signals is nothing but an interpolation or extrapolation of such a function. Here, an explicit inter-extrapolation formula with equally-spaced sampling points on the negative time axis is presented. The formula can be applied to all band-limited signals appearing in practice because on additional requirement is imposed on signals such as being square-integrable, being bounded or being tempered. And moreover, since the sampling points are entirely contained in the negative time axis, a signal at any time can be reconstructed from the signal values up to the present without delay, and in principle a deterministic prediction is possible for arbitrarily far future. The formula is obtained combining the extension of Newton interpolation formula for polynomials with Mittag-Leffler summation for divergent series.
URL: https://global.ieice.org/en_transactions/transactions/10.1587/e67-e_3_147/_p
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@ARTICLE{e67-e_3_147,
author={Hirosi SUGIYAMA, },
journal={IEICE TRANSACTIONS on transactions},
title={An Inter-Extrapolation Formula in Sampling Theorem for Band-Limited Distributions},
year={1984},
volume={E67-E},
number={3},
pages={147-152},
abstract={The sampling theorem can be extended to cover all distributions in L. Schwartz's sense. As any band-limited distribution reduces to an entire function of exponential type, the sampling theorem for band-limited signals is nothing but an interpolation or extrapolation of such a function. Here, an explicit inter-extrapolation formula with equally-spaced sampling points on the negative time axis is presented. The formula can be applied to all band-limited signals appearing in practice because on additional requirement is imposed on signals such as being square-integrable, being bounded or being tempered. And moreover, since the sampling points are entirely contained in the negative time axis, a signal at any time can be reconstructed from the signal values up to the present without delay, and in principle a deterministic prediction is possible for arbitrarily far future. The formula is obtained combining the extension of Newton interpolation formula for polynomials with Mittag-Leffler summation for divergent series.},
keywords={},
doi={},
ISSN={},
month={March},}
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TY - JOUR
TI - An Inter-Extrapolation Formula in Sampling Theorem for Band-Limited Distributions
T2 - IEICE TRANSACTIONS on transactions
SP - 147
EP - 152
AU - Hirosi SUGIYAMA
PY - 1984
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E67-E
IS - 3
JA - IEICE TRANSACTIONS on transactions
Y1 - March 1984
AB - The sampling theorem can be extended to cover all distributions in L. Schwartz's sense. As any band-limited distribution reduces to an entire function of exponential type, the sampling theorem for band-limited signals is nothing but an interpolation or extrapolation of such a function. Here, an explicit inter-extrapolation formula with equally-spaced sampling points on the negative time axis is presented. The formula can be applied to all band-limited signals appearing in practice because on additional requirement is imposed on signals such as being square-integrable, being bounded or being tempered. And moreover, since the sampling points are entirely contained in the negative time axis, a signal at any time can be reconstructed from the signal values up to the present without delay, and in principle a deterministic prediction is possible for arbitrarily far future. The formula is obtained combining the extension of Newton interpolation formula for polynomials with Mittag-Leffler summation for divergent series.
ER -
English
