NDLI: Weighted Finite Fourier Transform Operator: Uniform Approximations of the Eigenfunctions, Eigenvalues Decay and Behaviour

Content Provider	Springer Nature Link
Author	Karoui, Abderrazek Souabni, Ahmed
Copyright Year	2016
Abstract	In this paper, we investigate two uniform asymptotic approximations as well as some spectral properties of the eigenfunctions of the weighted finite Fourier transform operator, defined by $${\displaystyle {\mathcal {F}}_c^{(\alpha )} f(x)=\int _{-1}^1 e^{icxy} f(y)\,(1-y^2)^{\alpha }\, dy.}$$ Here, $$ c >0, \alpha \ge -1/2$$ are two fixed real numbers. These eigenfunctions are called generalized prolate spheroidal wave functions (GPSWFs) and they are firstly introduced and studied in Wang and Zhang (Appl Comput Harmon Anal 29(3):303–329, 2010). The present study is motivated by the promising concrete applications of the GPSWFs in various scientific area such as numerical analysis, mathematical physics and signal processing. We should mention that these two uniform approximation results of the GPSWFs can be considered as generalizations of the results given in the joint work of one of us (Bonami and Karoui in Constr Approx 43(1):15–45, 2016). As it will be seen, these generalizations require some involved extra work, especially in the case where $$\alpha > 1/2.$$ By using the uniform asymptotic approximations of the GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the operator $${\mathcal {F}}_c^{(\alpha )}$$ in the case where $$0<\alpha < 3/2.$$ Moreover, by computing the trace and an estimate of the norm of the operator $${\displaystyle {\mathcal {Q}}_c^{(\alpha )}=\frac{c}{2\pi } {\mathcal {F}}_c^{{(\alpha )}^*} \circ {\mathcal {F}}_c^{(\alpha )},}$$ we give a lower bound for the counting number of the eigenvalues of $$Q_c^{(\alpha )},$$ when $$c>>1.$$ Finally, we provide the reader with some numerical examples that illustrate the different results of this work.
Starting Page	547
Ending Page	570
Page Count	24
File Format	PDF
ISSN	08857474
Journal	Journal of Scientific Computing
Volume Number	71
Issue Number	2
e-ISSN	15737691
Language	English
Publisher	Springer US
Publisher Date	2016-10-21
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Sturm–Liouville operators Weighted finite Fourier transform Asymptotic approximations of eigenvalues and eigenfunctions Generalized prolate spheroidal wave functions Fourier series in special orthogonal functions Computation of special functions, construction of tables Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions Approximation by other special function classes Algorithms Computational Mathematics and Numerical Analysis ApplicationMathematics/Computational Methods of Engineering Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Engineering Computational Theory and Mathematics Computational Mathematics Numerical Analysis Software

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