Abstract
As an extension of the average sampling, we analyse generalized average sampling and reconstruction problem over some wavelet subspaces of \(L^{2}({{\mathbb {R}}})\). For closed subspaces \(V_{0}\) of \(L^{2}({{\mathbb {R}}})\), we present a necessary and sufficient condition under which there is the generalized average sampling expansion for every \(f \in V_{0}\).
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References
Aldroubi, A., and K. Grochenig. 2001. Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Review 43 (4): 585–620.
Aldroubi, A., and K. Grochenig. 2000. Beurling–Landau type theorems for non-uniform sampling in shift invariant spline spaces. The Journal of Fourier Analysis and Applications 6 (1): 93–103.
Chen, Wen, Shuichi Itoh, and Junji Shiki. 1998. Irregular sampling theorems for wavelet subspaces. IEEE Transactions on Information Theory 44 (3): 1131–1142.
Chui, Charles K. 1992. An Introduction to Wavelets. Cambridge: Academic Press, Inc.
Janseen, A.J.E.M. 1993. The Zak transform and sampling theorems for wavelet subspaces. IEEE Transactions on Signal Processing 41 (12): 3360–3364.
Liu, Y.M., and G.G. Walter. 1995. Irregular sampling in wavelet subspaces. The Journal of Fourier Analysis and Applications 2 (2): 181–189.
Sun, W., and X. Zhou. 1998. Frames and sampling theorem. Science in China (Series A) 41 (6): 606–612.
Sun, W., and X. Zhou. 2000. Sampling theorem for wavelet subspaces: error estimate and irregular sampling. IEEE Transactions on Signal Processing 48 (1): 223–226.
Sun, W., and X. Zhou. 2002. Average sampling in spline subspaces. Applied Mathematics Letters 15: 233–237.
Walter, Gilbert G. 1992. A sampling theorem for wavelet subspaces. IEEE Transactions on Information Theory 38 (2): 881–884.
Zhou, X., and Wenchang Sun. 1999. On the sampling theorem for wavelet subspaces. The Journal of Fourier Analysis and Applications 5 (4): 347–354.
Kang, S., and K.H. Kwon. 2011. Generalized average sampling in shift invariant spaces. Journal of Mathematical Analysis and Applications 377: 70–78.
Devaraj, P., and S. Yugesh. 2017. A local weighted average sampling and reconstruction theorem over shift invariant subspaces. Results in Mathematics 71: 319–332.
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First author would like to thank the Management of SSN College of Engineering, Kalavakkam-603 110, Tamil Nadu, India.
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Yugesh, S., Devaraj, P. Generalized average sampling and reconstruction for wavelet subspaces. J Anal 26, 333–342 (2018). https://doi.org/10.1007/s41478-018-0148-8
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DOI: https://doi.org/10.1007/s41478-018-0148-8