Direct approach for the characteristic function of a dissipative operator with distributional potentials

Abstract In this paper, maximal dissipative fourth order operators with equal deficiency indices are investigated. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which based on the method of Pavlov and define its characteristic function. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative fourth order operators.

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Spectral theory of dissipative q-Sturm-Liouville problems

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.3.1289 ◽

2014 ◽

Vol 51 (3) ◽

pp. 366-383

Author(s):

Aytekin Eryilmaz ◽

Hüseyin Tuna

Keyword(s):

Hilbert Space ◽

Characteristic Function ◽

Spectral Theory ◽

Difference Operator ◽

Functional Model ◽

Dissipative Operator ◽

Eigenvalues And Eigenvectors ◽

Maximal Dissipative Operator ◽

Sturm Liouville ◽

Liouville Operators

This paper is devoted to studying a q-analogue of Sturm-Liouville operators. We formulate a dissipative q-difference operator in a Hilbert space. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which is based on the method of Pavlov and define its characteristic function. Finally, we prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative q-Sturm-Liouville difference operator.

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A direct approach to the stable distributions

Advances in Applied Probability ◽

10.1017/apr.2016.55 ◽

2016 ◽

Vol 48 (A) ◽

pp. 261-282 ◽

Cited By ~ 4

Author(s):

E. J. G. Pitman ◽

Jim Pitman

Keyword(s):

Functional Equation ◽

Characteristic Function ◽

Integral Representation ◽

Explicit Form ◽

Regular Variation ◽

Stable Distribution ◽

Stable Distributions ◽

Direct Approach ◽

Infinitely Divisible Distributions ◽

Infinitely Divisible

AbstractThe explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.

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Spectral analysis of dissipative Schrödinger operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026962 ◽

1997 ◽

Vol 127 (6) ◽

pp. 1113-1121 ◽

Cited By ~ 15

Author(s):

B. P. Allahverdiev ◽

Ahmet Canoǧlu

Keyword(s):

Characteristic Function ◽

Schrödinger Operators ◽

Functional Model ◽

Dissipative Operator ◽

Schrodinger Operators ◽

Associated Functions ◽

Index 2 ◽

Eigenfunctions And Associated Functions ◽

Spectral Representations ◽

Symmetric Operators

Dissipative Schrodinger operators are studied in L2(0, ∞) which are extensions of symmetric operators with defect index (2, 2). We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix according to the scheme of Lax and Phillips. With the help of the incoming spectral representation, we construct a functional model of the dissipative operator and construct its characteristic function in terms of solutions of the corresponding differential equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operator.

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Invariant subspaces of a dissipative operator and divisors of its characteristic function

Functional Analysis and Its Applications ◽

10.1007/bf01075983 ◽

1971 ◽

Vol 4 (4) ◽

pp. 342-343

Author(s):

Ya. S. Shvartsman

Keyword(s):

Characteristic Function ◽

Invariant Subspaces ◽

Dissipative Operator

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Coordinate-Free Approach for the Characteristic Function of a Fourth-Order Dissipative Operator

Numerical Functional Analysis and Optimization ◽

10.1080/01630563.2019.1639730 ◽

2019 ◽

Vol 40 (16) ◽

pp. 1877-1897

Author(s):

Ekin Uğurlu

Keyword(s):

Characteristic Function ◽

Fourth Order ◽

Dissipative Operator

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Information and estimation in image processing

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100125142 ◽

1987 ◽

Vol 45 ◽

pp. 14-17

Author(s):

B. Roy Frieden

Keyword(s):

Image Processing ◽

Optical System ◽

Large Amplitude ◽

Communication Theory ◽

Image Data ◽

Direct Approach ◽

Ill Posed

Despite the skill and determination of electro-optical system designers, the images acquired using their best designs often suffer from blur and noise. The aim of an “image enhancer” such as myself is to improve these poor images, usually by digital means, such that they better resemble the true, “optical object,” input to the system. This problem is notoriously “ill-posed,” i.e. any direct approach at inversion of the image data suffers strongly from the presence of even a small amount of noise in the data. In fact, the fluctuations engendered in neighboring output values tend to be strongly negative-correlated, so that the output spatially oscillates up and down, with large amplitude, about the true object. What can be done about this situation? As we shall see, various concepts taken from statistical communication theory have proven to be of real use in attacking this problem. We offer below a brief summary of these concepts.

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