Resolvent Based Hilbert Transform

Seiichi Kuwata

doi:10.4028/p-0dHWan

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HomeAdvances in Science and TechnologyAdvances in Science and Technology Vol. 144Resolvent Based Hilbert Transform

Resolvent Based Hilbert Transform

Abstract:

To perform the Hilbert transform H_il of a non-integrable function φ, such as φ(x) = 1, x, in a numerical calculation-friendly way, we propose a method of rewriting H_il in terms of the resolvent for a differential operator R whose eigenfunctions satisfy the orthogonality and the completeness, so that the resolvent kernel 〈x|R^-1y〉can be given by the eigenfunction expansion. We deal with two cases for the choice of R: one is the harmonic oscillator Hamiltonian, which is commutative with the Fourier transform F; and the other is such that is commutative with H_il itself. We show how the calculation of H_ilφ is made in a numerical calculation-friendly way, to find that Π_k=0,1 H_ilf_k (f_k (x) = x^k) satisfies quite a simple relation.

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Periodical:

Advances in Science and Technology (Volume 144)

Pages:

29-36

DOI:

https://doi.org/10.4028/p-0dHWan

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Online since:

March 2024

Authors:

Seiichi Kuwata*

Keywords:

Confluent Hypergeometric Function, Conformal Symmetry, Fourier Transform, Hilbert Transform, Resolvent

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