Abstract
The goal of this chapter is to solve a concentration of energy problem associated with the Special Affine Fourier Transformation (SAFT). Since an explicit, closed form solution seems to be elusive, we will solve the problem numerically. The problem can be reduced to finding the largest eigenvalues and their associated eigenfunctions of two-dimensional integral equations. The numerical solutions are obtained by using the Gaussian quadrature method in two dimensions. We use also the Gaussian quadrature method in two dimensions to solve concentration of energy problems in other cases, including a problem for kernels of convolution type, and to compute the so-called generalized prolate spheroidal wave functions (GPSWF).
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Notes
- 1.
With a little abuse of notation, we use \(\underline{\lambda }^{-1}\) which should be understood as a parameter vector corresponding to the inverse–SAFT.
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Ammari, A., Moumni, T., Zayed, A. (2017). Numerical Solution to an Energy Concentration Problem Associated with the Special Affine Fourier Transformation. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Frames and Other Bases in Abstract and Function Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55550-8_8
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DOI: https://doi.org/10.1007/978-3-319-55550-8_8
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