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Average sampling in spline subspaces. (English) Zbl 0998.94518

Summary: We show that every function in a spline subspace is uniquely determined and can be reconstructed by its local averages near certain points. Regular and irregular average sampling theorems for spline subspaces are obtained.

MSC:

94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
41A15 Spline approximation
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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[1] Aldroubi, A.; Gröchenig, K., Beuling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces, J. Fourier Anal. Appl., 6, 93-103 (2000) · Zbl 0964.42020
[2] Aldroubi, A.; Unser, M., Families of wavelet transforms in connection with Shannon’s sampling theory and the Gabor transform, (Chui, C. K., Wavelets—A Tutorial in Theory and Applications (1992), Academic Press: Academic Press Boston, MA), 509-528 · Zbl 0769.42014
[3] Aldroubi, A.; Unser, M.; Eden, M., Cardinal spline filters: Stability and convergence to the ideal sinc interpolator, Signal Processing, 28, 127-138 (1992) · Zbl 0755.41007
[4] Chen, W.; Itoh, S., A sampling theorem for shift-invariant subspaces, IEEE Trans. Signal Processing, 46, 2822-2824 (1998) · Zbl 0978.94033
[5] Janssen, A. J.E. M., The Zak transform and sampling theorem for wavelet subspaces, IEEE Trans. Signal Processing, 41, 3360-3364 (1993) · Zbl 0841.94011
[6] Liu, Y., Irregular sampling for spline wavelet subspaces, IEEE Trans. Inform. Theory, 42, 623-627 (1996) · Zbl 0852.94003
[7] Sun, W.; Zhou, X., Frames and sampling theorem, Science in China, Series A, 41, 606-612 (1998) · Zbl 0959.42021
[8] Sun, W.; Zhou, X., Sampling theorem for multiwavelet subspaces, Chinese Science Bulletin, 44, 1283-1286 (1999) · Zbl 1039.42037
[9] Sun, W.; Zhou, X., Sampling theorem for wavelet subspaces: Error estimate and irregular sampling, IEEE Trans. Signal Processing, 48, 223-226 (2000) · Zbl 1011.94008
[10] Walter, G., A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory, 38, 881-884 (1992) · Zbl 0744.42018
[11] Xia, X. G.; Zhang, Z., On sampling theorem, wavelets, and wavelet transforms, IEEE Transactions on Signal Processing, 41, 2535-3524 (1993) · Zbl 0841.94022
[12] Zhou, X.; Sun, W., On the sampling theorem for wavelet subspaces, Journal of Fourier Analysis and Applications, 5, 347-354 (1999) · Zbl 0931.42022
[13] Wiley, R. G., Recovery of band-limited signals from unequally spaced samples, IEEE Trans. Comm., 26, 135-137 (1978) · Zbl 0372.94013
[14] Butzer, P. L.; Lei, J., Approximation of signals using measured sampled values and error analysis, Communications in Applied Analysis, 4, 245-255 (2000) · Zbl 1089.94503
[15] Gröchenig, K., Reconstruction algorithms in irregular sampling, Math. Comput., 59, 181-194 (1992) · Zbl 0756.65159
[16] Feichtinger, H.; Gröchenig, K., Theory and practice of irregular sampling, (Benedetto, J.; Frazier, M., Wavelets: Mathematics and Applications (1994), CRC Press), 305-363 · Zbl 1090.94524
[17] Chui, C. K., An Introduction to Wavelets (1992), Academic Press: Academic Press New York · Zbl 0925.42016
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