# SAMPLING EXPANSION OF BANDLIMITED FUNCTIONS OF POLYNOMIAL GROWTH ON THE REAL LINE

Shin, Chang Eon

• Published : 2014.04.30
• 34 9

#### Abstract

For a bandlimited function with polynomial growth on the real line, we derive a nonuniform sampling expansion using a special bandlimited function which has polynomial decay on the real line. The series converges uniformly on any compact subsets of the real line.

#### Keywords

Shannon sampling;nonuniform sampling;bandlimited function;polynomial growth

#### References

1. M. A. Al-Gwaiz, Theory of Distributions, Marcel Dekker Inc., 1992.
2. L. L. Campbell, Sampling theorem for the Fourier transform of a distribution with bounded support, SIAM J. Appl. Math. 16 (1968), 626-636. https://doi.org/10.1137/0116051
3. J. R. Higgins, A sampling theorem for irregularly spaced sample points, IEEE Trans. Inform. Theory IT-22 (1976), no. 5, 621-622.
4. J. R. Higgins, Sampling Theory in Fourier and Function Analysis: Foundations, Oxford Science, 1996.
5. G. Hinsen, Irregular sampling of bandlimited $L^p$-functions, J. Approx. Theory 72 (1993), no. 3, 346-364. https://doi.org/10.1006/jath.1993.1027
6. M. I. Kadec, The exact value of the Paley-Wiener constant, Soviet Math. Dokl. 5 (1964) 559-561.
7. A. J. Lee, Characterization of bandlimited functions and processes, Inform. Control 31 (1976), no. 3, 258-271. https://doi.org/10.1016/S0019-9958(76)90409-5
8. N. Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, v. 26. American Mathematical Society, New York, 1940.
9. R. Paley and N. Wiener, Fourier Transforms in the Complex Domain, (Amer. Math. Soc. Colloq. Pub. Ser.), Vol. 19, Privindence, RI: Amer. Math. Soc. 1934.
10. E. Pfaffelhuber, Sampling series for band-limited generalized functions, IEEE Trans. Inform. Theory IT-17 (1971), 650-654.
11. C. E. Shin, M. B. Lee, and K. S. Rim, Nonuniform sampling of bandlimited functions, IEEE Trans. Inform. Theory 54 (2008), no. 8, 3814-3819. https://doi.org/10.1109/TIT.2008.926458
12. C. E. Shin, K. S. Rim, and Y. Kim, A weighted OFDM signal scheme for peak-to-average ratio reduction of OFDM signals, IEEE Trans. Veh. Technol. 62 (2013), 1406-1409. https://doi.org/10.1109/TVT.2012.2232949
13. G. G. Walter, Sampling bandlimited functions of polynomial growth, SIAM J. Math. Anal. 19 (1988), no. 5, 1198-1201. https://doi.org/10.1137/0519084
14. G. G. Walter, Nonuniform sampling of bandlimited functions of polynomial growth, SIAM J. Math. Anal. 23 (1992), no. 4, 995-1003. https://doi.org/10.1137/0523056
15. M. Zakai, Band-limited functions and the sampling theorem, Inform. Control 8 (1965), 143-158. https://doi.org/10.1016/S0019-9958(65)90038-0
16. A. I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, FL, 1993.
17. A. I. Zayed and A. G. Garcia, Nonuniform sampling of bandlimited signals with polynomial growth on the real axis, IEEE Trans. Inform. Theory 43 (1997), no. 5, 1717-1721. https://doi.org/10.1109/18.623180

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)