On the surjectivity of Hankel convolution operators on Beurling-type distribution spaces

2005 ◽  
Vol 135 (3) ◽  
pp. 479-512
Author(s):  
M. Belhadj ◽  
J. J. Betancor
Keyword(s):  
Convolution Operator ◽  
Differential Operators ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Hankel Transform ◽  
Convolution Operators ◽  
Type Distribution ◽  
Hankel Convolution ◽  
Distribution Spaces ◽  
Necessary And Sufficient

In this paper we consider Beurling-type distributions in the Hankel setting. The Hankel transform and Hankel convolution are studied on Beurling-type distributions. We also introduce a class of ultra-differential operators that allows us to show a Hankel version of the second structure theorem of Komatsu and Braun. Necessary and sufficient conditions are established in order that a Beurling distribution generates a surjective Hankel convolution operator.

On the surjectivity of Hankel convolution operators on Beurling-type distribution spaces

2005 ◽  
Vol 135 (3) ◽  
pp. 479-512
Author(s):  
M. Belhadj ◽  
J. J. Betancor
Keyword(s):  
Convolution Operator ◽  
Differential Operators ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Hankel Transform ◽  
Convolution Operators ◽  
Type Distribution ◽  
Hankel Convolution ◽  
Distribution Spaces ◽  
Necessary And Sufficient

In this paper we consider Beurling-type distributions in the Hankel setting. The Hankel transform and Hankel convolution are studied on Beurling-type distributions. We also introduce a class of ultra-differential operators that allows us to show a Hankel version of the second structure theorem of Komatsu and Braun. Necessary and sufficient conditions are established in order that a Beurling distribution generates a surjective Hankel convolution operator.

Some Properties of Hankel Convolution Operators

1993 ◽  
Vol 36 (4) ◽  
pp. 398-406 ◽  
Author(s):  
J. J. Betancor ◽  
I. Marrero
Keyword(s):  
Sufficient Conditions ◽  
Hankel Transform ◽  
Generalized Functions ◽  
Necessary And Sufficient Conditions ◽  
Convolution Operators ◽  
Zemanian Space ◽  
Hankel Convolution ◽  
Necessary And Sufficient

AbstractLet be the Zemanian space of Hankel transformable generalized functions and let be the space of Hankel convolution operators for . This is the dual of a subspace of for which is also the space of Hankel convolutors. In this paper the elements of are characterized as those in and in that commute with Hankel translations. Moreover, necessary and sufficient conditions on the generalized Hankel transform are established in order that every such that in .

Necessary and Sufficient Conditions for Hypoellipticity for a Class of Convolution Operators

1994 ◽  
Vol 46 (1) ◽  
pp. 212-224
Author(s):  
Luo Xuebo
Keyword(s):  
Convolution Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Convolution Operators ◽  
Necessary And Sufficient ◽  
Complex Zeros

AbstractIn this paper the Corwin's conjecture is proved, which says that if d is a function analytic near ∞, then the hypoellipticity of the convolution operator Ad, defined by for every u ∊ S'(ℝn), implies that P(x)/ logx → ∞ as x → ∞, where P(x) is the distance from x ∊ ℝn to the set of complex zeros of d.

scholarly journals The Zak Transform on Gelfand–Shilov and Modulation Spaces with Applications to Operator Theory

2020 ◽  
Vol 15 (1) ◽  
Author(s):  
Joachim Toft
Keyword(s):  
Operator Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Modulation Spaces ◽  
Zak Transform ◽  
Periodic Functions ◽  
Distribution Spaces ◽  
Necessary And Sufficient

AbstractWe characterize Gelfand–Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.

Embedding theorems and the spectra of certain differential operators

1991 ◽  
Vol 434 (1892) ◽  
pp. 643-656 ◽  
Keyword(s):  
Differential Operators ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Independent Interest ◽  
Embedding Theorems ◽  
Necessary And Sufficient

This paper is concerned with problems of the form n Ʃ k =0 (─1) k ( ρ 2 k y ( k ))( k ) = λ r 2 y on R , ry ∈ L 2 ( R ) and gives conditions on the coefficients sufficient to ensure that the spectrum is discrete; necessary conditions are also given. In certain circumstances, necessary and sufficient conditions for discreteness are given, thus extending the celebrated Molcanov criterion. These results follow from embedding theorems which have independent interest.

scholarly journals A NEW CONSTRUCTION FOR REGULAR SEMIGROUPS WITH QUASI-IDEAL ORTHODOX TRANSVERSALS

2009 ◽  
Vol 86 (2) ◽  
pp. 177-187 ◽  
Author(s):  
XIANGJUN KONG ◽  
XIANZHONG ZHAO
Keyword(s):  
Regular Semigroup ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
New Construction ◽  
Necessary And Sufficient ◽  
Equivalent Conditions ◽  
Orthodox Semigroups

AbstractIn any regular semigroup with an orthodox transversal, we define two sets R and L using Green’s relations and give necessary and sufficient conditions for them to be subsemigroups. By using R and L, some equivalent conditions for an orthodox transversal to be a quasi-ideal are obtained. Finally, we give a structure theorem for regular semigroups with quasi-ideal orthodox transversals by two orthodox semigroups R and L.

scholarly journals An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions

2019 ◽  
Vol 50 (3) ◽  
pp. 207-221 ◽  
Author(s):  
Sergey Buterin
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
A Priori ◽  
Sufficient Conditions ◽  
Robin Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Robin Boundary

The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

scholarly journals Hypoelliptic differential operators with generalized constant coefficients

1998 ◽  
Vol 41 (1) ◽  
pp. 47-60 ◽  
Author(s):  
M. Nedeljkov ◽  
S. Pilipović
Keyword(s):  
Small Parameter ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Necessary And Sufficient Conditions ◽  
Order Estimate ◽  
Colombeau Generalized Functions ◽  
Constant Coefficients ◽  
Necessary And Sufficient

The space of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter ε.There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on ε and behave like powers of ε as ε→0. The solutions of such family of equations should also satisfy the power order estimate with respect to ε.

Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators

1981 ◽  
Vol 33 (1) ◽  
pp. 229-246 ◽  
Author(s):  
Calvin D. Ahlbrandt ◽  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Differential Operators ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Dimensional Vector ◽  
Selfadjoint Extension ◽  
Image Position ◽  
Vector Valued Function ◽  
Necessary And Sufficient ◽  
Vector Valued

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l bywhere y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfyingwith inner productwhere . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

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