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Optimization of the Optimal Interpolation Problem

by The Interpolation PublicationMarch 10th, 2024

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This paper delves into optimizing Fourier coefficients prediction and solving the optimal interpolation problem, considering both equispaced and non-equispaced nodes. It offers insights into polynomial expressions and the second derivative test, providing valuable strategies for improving interpolation accuracy. Available on arXiv under CC 4.0 license.

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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.