Optimization of the Optimal Interpolation Problem

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 2 OPTIMIZATION OF THE OPTIMAL INTERPOLATION PROBLEM 2.1 With respect to the predicted Fourier coefficients, ˆf which through their respective norms is equivalent to: which unpacked as a polynomial expression: 2.1.1 Optimal interpolation problem for equispaced nodes Factoring out the constant M: 2.1.2 Optimal interpolation problem for non-equispaced nodes substituting Equation 34 into the denominator of Equation 28, the function that minimises the cost function for the optimal interpolation problem as a function of ˆf in Equation 19: where it is clear that in Equation 36 we avoid the problem of directly inverting the kernel function along the diagonal of Wˆ , as a substitute for Equation 28. 2.2 Second derivative test - equispaced case 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 2 OPTIMIZATION OF THE OPTIMAL INTERPOLATION PROBLEM 2.1 With respect to the predicted Fourier coefficients, ˆf which through their respective norms is equivalent to: which unpacked as a polynomial expression: 2.1.1 Optimal interpolation problem for equispaced nodes 2.1.1 Optimal interpolation problem for equispaced nodes Factoring out the constant M: 2.1.2 Optimal interpolation problem for non-equispaced nodes 2.1.2 Optimal interpolation problem for non-equispaced nodes substituting Equation 34 into the denominator of Equation 28, the function that minimises the cost function for the optimal interpolation problem as a function of ˆf in Equation 19: where it is clear that in Equation 36 we avoid the problem of directly inverting the kernel function along the diagonal of Wˆ , as a substitute for Equation 28. 2.2 Second derivative test - equispaced case 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