Authors:
(1) Michael Sorochan Armstrong, Computational Data Science (CoDaS) Lab in the Department of Signal Theory, Telematics, and Communications at the University of Granada;
(2) Jose Carlos P´erez-Gir´on, part of the Interuniversity Institute for Research on the Earth System in Andalucia through the University of Granada;
(3) Jos´e Camacho, Computational Data Science (CoDaS) Lab in the Department of Signal Theory, Telematics, and Communications at the University of Granada;
(4) Regino Zamora, part of the Interuniversity Institute for Research on the Earth System in Andalucia through the University of Granada. Table of Links Abstract & Introduction Optimization of the Optical Interpolation Materials and Methods Results and Discussion Conclusion Appendix A: Proof of Hermitian Self-Adjoint product identity for Equidistant Time-Domain Measurements Appendix B: AAH ̸= MIN I in the Non-Equidistant Case Acknowledgments & References ACKNOWLEDGMENTS The proof for the equidistant case was adapted from a proof by David J. Fleet and Allan D. Jepson from the University of Toronto [18]. REFERENCES [1] C. Haley, “Missing-data nonparametric coherency estimation,” IEEE Signal Processing Letters, vol. 28, pp. 1704–1708, 2021. [2] S. Dilmaghani, I. C. Henry, P. Soonthornnonda, E. R. Christensen, and R. C. Henry, “Harmonic analysis of environmental time series with missing data or irregular sample spacing,” Environmental science & technology, vol. 41, no. 20, pp. 7030–7038, 2007. [3] J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Mathematics of computation, vol. 19, no. 90, pp. 297–301, 1965. [4] F. Rasheed, P. Peng, R. Alhajj, and J. Rokne, “Fourier transform based spatial outlier mining,” in Intelligent Data Engineering and Automated Learning-IDEAL 2009: 10th International Conference, Burgos, Spain, September 23-26, 2009. Proceedings 10. Springer, 2009, pp. 317–324. [5] C. Van Loan, Computational frameworks for the fast Fourier transform. SIAM, 1992. [6] H. Carslaw, “A historical note on gibbs’ phenomenon in fourier’s series and integrals,” Bulletin of the American Mathematical Society, 1925. [7] J. Markel, “Fft pruning,” IEEE transactions on Audio and Electroacoustics, vol. 19, no. 4, pp. 305–311, 1971. [8] J. C. Bowman and Z. Ghoggali, “The partial fast fourier transform,” Journal of Scientific Computing, vol. 76, pp. 1578–1593, 2018. [9] M. Kircheis and D. Potts, “Direct inversion of the nonequispaced fast fourier transform,” Linear Algebra and its Applications, vol. 575, pp. 106–140, 2019. [10] S. Kunis and D. Potts, “Stability results for scattered data interpolation by trigonometric polynomials,” SIAM Journal on Scientific Computing, vol. 29, no. 4, pp. 1403–1419, 2007. [11] K. B. Petersen, M. S. Pedersen et al., “The matrix cookbook,” Technical University of Denmark, vol. 7, no. 15, p. 510, 2008. [12] C. M. Bishop, Neural networks for pattern recognition. Oxford university press, 1995. [13] H. Wickham, M. Averick, J. Bryan, W. Chang, L. D. McGowan, R. Franc¸ois, G. Grolemund, A. Hayes, L. Henry, J. Hester, M. Kuhn, T. L. Pedersen, E. Miller, S. M. Bache, K. Muller, J. Ooms, D. Robin- ¨ son, D. P. Seidel, V. Spinu, K. Takahashi, D. Vaughan, C. Wilke, K. Woo, and H. Yutani, “Welcome to the tidyverse,” Journal of Open Source Software, vol. 4, no. 43, p. 1686, 2019. [14] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2023. [Online]. Available: https://www.R-project.org/ [15] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del R´ıo, M. Wiebe, P. Peterson, P. Gerard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, ´ H. Abbasi, C. Gohlke, and T. E. Oliphant, “Array programming with NumPy,” Nature, vol. 585, no. 7825, pp. 357–362, Sep. 2020. [Online]. Available: https://doi.org/10.1038/s41586-020-2649-2 [16] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,” Nature Methods, vol. 17, pp. 261– 272, 2020. [17] J. D. Hunter, “Matplotlib: A 2d graphics environment,” Computing in Science & Engineering, vol. 9, no. 3, pp. 90–95, 2007. [18] “Notes on fourier analysis,” http://www.cs.toronto.edu/ ∼jepson/csc320/notes/fourier.pdf, accessed: 2023-09-01. This paper is available on arxiv under CC 4.0 license. Authors: (1) Michael Sorochan Armstrong, Computational Data Science (CoDaS) Lab in the Department of Signal Theory, Telematics, and Communications at the University of Granada; (2) Jose Carlos P´erez-Gir´on, part of the Interuniversity Institute for Research on the Earth System in Andalucia through the University of Granada; (3) Jos´e Camacho, Computational Data Science (CoDaS) Lab in the Department of Signal Theory, Telematics, and Communications at the University of Granada; (4) Regino Zamora, part of the Interuniversity Institute for Research on the Earth System in Andalucia through the University of Granada. Authors: Authors: (1) Michael Sorochan Armstrong, Computational Data Science (CoDaS) Lab in the Department of Signal Theory, Telematics, and Communications at the University of Granada; (2) Jose Carlos P´erez-Gir´on, part of the Interuniversity Institute for Research on the Earth System in Andalucia through the University of Granada; (3) Jos´e Camacho, Computational Data Science (CoDaS) Lab in the Department of Signal Theory, Telematics, and Communications at the University of Granada; (4) Regino Zamora, part of the Interuniversity Institute for Research on the Earth System in Andalucia through the University of Granada. Table of Links Abstract & Introduction Abstract & Introduction Optimization of the Optical Interpolation Optimization of the Optical Interpolation Materials and Methods Materials and Methods Results and Discussion Results and Discussion Conclusion Conclusion Appendix A: Proof of Hermitian Self-Adjoint product identity for Equidistant Time-Domain Measurements Appendix A: Proof of Hermitian Self-Adjoint product identity for Equidistant Time-Domain Measurements Appendix B: AAH ̸= MIN I in the Non-Equidistant Case Appendix B: AAH ̸= MIN I in the Non-Equidistant Case Acknowledgments & References Acknowledgments & References ACKNOWLEDGMENTS The proof for the equidistant case was adapted from a proof by David J. Fleet and Allan D. Jepson from the University of Toronto [18]. REFERENCES [1] C. Haley, “Missing-data nonparametric coherency estimation,” IEEE Signal Processing Letters, vol. 28, pp. 1704–1708, 2021. [2] S. Dilmaghani, I. C. Henry, P. Soonthornnonda, E. R. Christensen, and R. C. Henry, “Harmonic analysis of environmental time series with missing data or irregular sample spacing,” Environmental science & technology, vol. 41, no. 20, pp. 7030–7038, 2007. [3] J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Mathematics of computation, vol. 19, no. 90, pp. 297–301, 1965. [4] F. Rasheed, P. Peng, R. Alhajj, and J. Rokne, “Fourier transform based spatial outlier mining,” in Intelligent Data Engineering and Automated Learning-IDEAL 2009: 10th International Conference, Burgos, Spain, September 23-26, 2009. Proceedings 10. Springer, 2009, pp. 317–324. [5] C. Van Loan, Computational frameworks for the fast Fourier transform. SIAM, 1992. [6] H. Carslaw, “A historical note on gibbs’ phenomenon in fourier’s series and integrals,” Bulletin of the American Mathematical Society, 1925. [7] J. Markel, “Fft pruning,” IEEE transactions on Audio and Electroacoustics, vol. 19, no. 4, pp. 305–311, 1971. [8] J. C. Bowman and Z. Ghoggali, “The partial fast fourier transform,” Journal of Scientific Computing, vol. 76, pp. 1578–1593, 2018. [9] M. Kircheis and D. Potts, “Direct inversion of the nonequispaced fast fourier transform,” Linear Algebra and its Applications, vol. 575, pp. 106–140, 2019. [10] S. Kunis and D. Potts, “Stability results for scattered data interpolation by trigonometric polynomials,” SIAM Journal on Scientific Computing, vol. 29, no. 4, pp. 1403–1419, 2007. [11] K. B. Petersen, M. S. Pedersen et al., “The matrix cookbook,” Technical University of Denmark, vol. 7, no. 15, p. 510, 2008. [12] C. M. Bishop, Neural networks for pattern recognition. Oxford university press, 1995. [13] H. Wickham, M. Averick, J. Bryan, W. Chang, L. D. McGowan, R. Franc¸ois, G. Grolemund, A. Hayes, L. Henry, J. Hester, M. Kuhn, T. L. Pedersen, E. Miller, S. M. Bache, K. Muller, J. Ooms, D. Robin- ¨ son, D. P. Seidel, V. Spinu, K. Takahashi, D. Vaughan, C. Wilke, K. Woo, and H. Yutani, “Welcome to the tidyverse,” Journal of Open Source Software, vol. 4, no. 43, p. 1686, 2019. [14] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2023. [Online]. Available: https://www.R-project.org/ [15] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del R´ıo, M. Wiebe, P. Peterson, P. Gerard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, ´ H. Abbasi, C. Gohlke, and T. E. Oliphant, “Array programming with NumPy,” Nature, vol. 585, no. 7825, pp. 357–362, Sep. 2020. [Online]. Available: https://doi.org/10.1038/s41586-020-2649-2 [16] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,” Nature Methods, vol. 17, pp. 261– 272, 2020. [17] J. D. Hunter, “Matplotlib: A 2d graphics environment,” Computing in Science & Engineering, vol. 9, no. 3, pp. 90–95, 2007. [18] “Notes on fourier analysis,” http://www.cs.toronto.edu/ ∼jepson/csc320/notes/fourier.pdf, accessed: 2023-09-01. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Proof of Hermitian Self-Adjoint Product Identity for Equidistant Time-Domain Measurements

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