72 research outputs found
An inverse Sturm-Liouville problem with a fractional derivative
In this paper, we numerically investigate an inverse problem of recovering
the potential term in a fractional Sturm-Liouville problem from one spectrum.
The qualitative behaviors of the eigenvalues and eigenfunctions are discussed,
and numerical reconstructions of the potential with a Newton method from finite
spectral data are presented. Surprisingly, it allows very satisfactory
reconstructions for both smooth and discontinuous potentials, provided that the
order of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Computational Physic
Integral representations for some weighted classes of functions holomorphic in matrix domains
In 1945 the first author introduced the classes , 1 ≤ p -1, of holomorphic functions in the unit disk with finite integral
(1)
and established the following integral formula for :
(2) f(z) = (α+1)/π ∬_\mathbb{D} f(ζ) ((1-|ζ|²)^α)//((1-zζ̅)^{2+α}) dξdη, z∈ \mathbb{D} .
We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes , where:
1) , ;
2) Ω is the matrix domain consisting of those complex m × n matrices W for which is positive-definite, and ;
3) Ω is the matrix domain consisting of those complex n × n matrices W for which is positive-definite, and K(W) = det[Im W].
Here dm is Lebesgue measure in the corresponding domain, denotes the unit m × m matrix and W* is the Hermitian conjugate of the matrix W
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