The Ill-Posedness of Derivative Interpolation and Regularized Derivative Interpolation for Non-Bandlimited Functions
Abstract: In this paper, the ill-posedness of derivative interpolation is discussed, and a regularized derivative interpolation for non-bandlimited signals is presented. The convergence of the regularized derivative interpolation is studied. The numerical results are given and compared with derivative interpolation using the Tikhonov regularization method. The regularized derivative interpolation in this paper is more accurate in computation.
Cite this paper: Chen, W. (2022) The Ill-Posedness of Derivative Interpolation and Regularized Derivative Interpolation for Non-Bandlimited Functions. Applied Mathematics, 13, 87-100. doi: 10.4236/am.2022.131008.
References

[1]   Zhao, X.R., Sheng, Z., Li, J.W., Yu, H. and Wei, K.J. (2019) Determination of the “Wave Turbopause” Using a Numerical Differentiation Method. Journal of Geophysical Research: Atmospheres, 124, 10592-10607.
https://doi.org/10.1029/2019JD030754

[2]   Shannon, C.E. (1948) A Mathematical Theory of Communication. The Bell System Technical Journal, 27, 623-656.
https://doi.org/10.1002/j.1538-7305.1948.tb00917.x

[3]   Steiner, A. (1980) Plancherel’s Theorem and the Shannon Series Derived Simultaneously. The American Mathematical Monthly, 87, 193-197.
https://doi.org/10.1080/00029890.1980.11994990

[4]   Marks, R. (1983) Noise Sensitivity of Band-Limited Signal Derivative Interpolation. IEEE Transactions on Acoustics, Speech, and Signal Processing, 31, 1028-1032.
https://doi.org/10.1109/TASSP.1983.1164168

[5]   Bruno, O. and Hoch, D. (2012) Numerical Differentiation of Approximated Functions with Limited Order-of-Accuracy Deterioration. SIAM Journal on Numerical Analysis, 50, 1581-1603.
https://doi.org/10.1137/100805807

[6]   Chen, W.D. (2020) The Ill-Posedness of Derivative Interpolation and Regularized Derivative Interpolation for Band-Limited Functions. EURASIP Journal on Advances in Signal Processing, 2020, Article No. 32.
https://doi.org/10.1186/s13634-020-00668-5

[7]   Chen, W.D. (2006) An Efficient Method for An Ill-Posed Problem—Band-Limited Extrapolation by Regularization. IEEE Transactions on Signal Processing, 54, 4611-4618.
https://doi.org/10.1109/TSP.2006.881255

[8]   Chen, W.D. (2011) Computation of Fourier Transforms for Noisy Bandlimited Signals. SIAM Journal on Numerical Analysis, 49, 1-14.
https://doi.org/10.1137/100784758

[9]   Chen, W.D. (2016) A Regularized Sampling Algorithm in Reconstructing Non-Bandlimited Functions. Journal of Computational and Applied Mathematics, 301, 259-270.
https://doi.org/10.1016/j.cam.2015.11.032

[10]   Chen, W.D. (2011) The Ill-Posedness of the Sampling Problem and Regularized Sampling Algorithm. Digital Signal Processing, 21, 375-390.
https://doi.org/10.1016/j.dsp.2010.06.003

[11]   Chen, W.D. (2017) A Regularized Two-Dimensional Sampling Algorithm. Journal of Inverse and Ill-Posed Problems, 26, 67-84.
https://doi.org/10.1515/jiip-2015-0049

[12]   Brown, J.L. (1967) On the Error in Reconstructing a Non-Bandlimited Function by Means of the Bandpass Sampling Theorem. Journal of Mathematical Analysis and Applications, 18, 75-84.
https://doi.org/10.1016/0022-247X(67)90183-7

[13]   Griesbaum, A., Barbara, B. and Vexler, B. (2008) Efficient Computation of the Tikhonov Regularization Parameter by Goal-Oriented Adaptive Discretization. Inverse Problems, 24, Article ID: 025025.
https://doi.org/10.1088/0266-5611/24/2/025025

[14]   Belge, M., Kilmer, M.E. and Miller, E.L. (2002) Efficient Determination of Multiple Regularization Parameters in a Generalized L-Curve Framework. Inverse Problems, 18, 1161-1183.
https://doi.org/10.1088/0266-5611/18/4/314

[15]   Kilmer, M.E. and O’leary, D.P. (2001) Choosing Regularization Parameters in Iterative Methods for Ill-Posed Problems. SIAM Journal on Matrix Analysis and Applications, 22, 1204-1221.
https://doi.org/10.1137/S0895479899345960

[16]   Tikhonov, A.N. and Arsenin, V.Y. (1977) Solution of Ill-Posed Problems. Winston/Wiley, Washington DC.

[17]   Wang, Y.B., Hon, Y.C. and Cheng, J. (2006) Reconstruction of High Order Derivatives from Input Data. Journal of Inverse and Ill-Posed Problems, 14, 205-218.
https://doi.org/10.1515/156939406777571085

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