scholarly journals A study on a class of modified Bessel-type integrals in a Fréchet space of Boehmians

2019 ◽  
Vol 38 (4) ◽  
pp. 145-156 ◽  
Author(s):  
Shrideh Khalaf Al-Omari
Keyword(s):  
Generalized Functions ◽  
Convolution Theorem ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Basic Properties ◽  
Type Integral ◽  
Classical Integral ◽  
Convolution Products

In this paper, an attempt is being made to discuss a class of modified Bessel- type integrals on a set of generalized functions known as Boehmians. We show that the modified Bessel-type integral, with appropriately defined convolution products, obeys a fundamental convolution theorem which consequently justifis pursuing analysis in the Boehmian spaces. We describe two Fréchet spaces of Boehmians and extend the modifid Bessel-type integral between the diferent spaces. Furthermore, a convolution theorem and a class of basic properties of the extended integral such as linearity, continuity and compatibility with the classical integral, which provide a convenient extention to the classical results, have been derived

scholarly journals A New Version of the Generalized Krätzel-Fox Integral Operators

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 222 ◽  
Author(s):  
Shrideh Al-Omari ◽  
Ghalib Jumah ◽  
Jafar Al-Omari ◽  
Deepali Saxena
Keyword(s):  
Integral Operator ◽  
Integral Operators ◽  
Convolution Theorem ◽  
Fréchet Space ◽  
Frechet Space ◽  
Generalized Convolution ◽  
Real Numbers ◽  
Convolution Products ◽  
Fox’S H Function ◽  
Krätzel Integral

This article deals with some variants of Krätzel integral operators involving Fox’s H-function and their extension to classes of distributions and spaces of Boehmians. For real numbers a and b > 0 , the Fréchet space H a , b of testing functions has been identified as a subspace of certain Boehmian spaces. To establish the Boehmian spaces, two convolution products and some related axioms are established. The generalized variant of the cited Krätzel-Fox integral operator is well defined and is the operator between the Boehmian spaces. A generalized convolution theorem has also been given.

Interpolation in Separable Frechet Spaces with Applications to Spaces of Analytic Functions

1975 ◽  
Vol 27 (5) ◽  
pp. 1110-1113 ◽  
Author(s):  
Paul M. Gauthier ◽  
Lee A. Rubel
Keyword(s):  
Sufficient Conditions ◽  
Fréchet Space ◽  
Complex Numbers ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear Functional ◽  
Interpolating Sequence ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Spaces Of Analytic Functions

Let E be a separable Fréchet space, and let E* be its topological dual space. We recall that a Fréchet space is, by definition, a complete metrizable locally convex topological vector space. A sequence {Ln} of continuous linear functional is said to be interpolating if for every sequence {An} of complex numbers, there exists an ƒ ∈ E such that Ln(ƒ) = An for n = 1, 2, 3, … . In this paper, we give necessary and sufficient conditions that {Ln} be an interpolating sequence. They are different from the conditions in [2] and don't seem to be easily interderivable with them.

scholarly journals Some results on asymptotically normable Frechet spaces

BIBECHANA ◽  
1970 ◽  
Vol 7 ◽  
pp. 39-43
Author(s):  
GK Palei ◽  
NP Sah
Keyword(s):  
Banach Spaces ◽  
Tensor Products ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Norm ◽  
Köthe Spaces ◽  
Köthe Space

In this paper, it is shown that the asymptotically normable spaces are the smallest class of Frechet spaces which contains the nuclear Kothe spaces with continuous norm, the Banach spaces and is closed under e-tensor products and sub-spaces. Again our main aim will be to construct an example of a Kothe space which is Montel, admits a continuous norm, but still is not asymptotically normable. Keywords: Asymptotically normable; Frechet space; Kothe space DOI: 10.3126/bibechana.v7i0.4043BIBECHANA 7 (2011) 39-43

A Question of Valdivia on Quasinormable Fréchet Spaces

1991 ◽  
Vol 34 (3) ◽  
pp. 301-304 ◽  
Author(s):  
José Bonet
Keyword(s):  
Positive Answer ◽  
Null Sequence ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Strong Dual

AbstractIt is proved that a Fréchet space is quasinormable if and only if every null sequence in the strong dual converges equicontinuously to the origin. This answers positively a question raised by Valdivia. As a consequence a positive answer to a problem of Jarchow on Fréchet Schwartz spaces is obtained.

scholarly journals Selfadjoint Operators and Generalized Central Algorithms in Frechet Spaces

2006 ◽  
Vol 13 (2) ◽  
pp. 363-382
Author(s):  
Soso Tsotniashvili ◽  
David Zarnadze
Keyword(s):  
Differential Operators ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Von Neumann ◽  
Selfadjoint Operators ◽  
Elliptic Differential Operators ◽  
Dense Set

Abstract The paper gives an extension of the fundamental principles of selfadjoint operators in Fréchet–Hilbert spaces, countable-Hilbert and nuclear Fréchet spaces. Generalizations of the well known theorems of von Neumann, Hellinger–Toeplitz, Friedrichs and Ritz are obtained. Definitions of generalized central and generalized spline algorithms are given. The restriction 𝐴∞ of a selfadjoint operator 𝐴 defined on a dense set 𝐷(𝐴) of the Hilbert space 𝐻 to the Frechet space 𝐷(𝐴∞) is substantiated. The extended Ritz method is used for obtaining an approximate solution of the equation 𝐴∞𝑢 = 𝑓 in the Frechet space 𝐷(𝐴∞). It is proved that approximate solutions of this equation constructed by the extended Ritz method do not depend on the number of norms that generate the topology of the space 𝐷(𝐴∞). Hence this approximate method is both a generalized central and generalized spline algorithm. Examples of selfadjoint and positive definite elliptic differential operators satisfying the above conditions are given. The validity of theoretical results in the case of a harmonic oscillator operator is confirmed by numerical calculations.

scholarly journals Distinguishedness of weighted Fréchet spaces of continuous functions

1992 ◽  
Vol 35 (2) ◽  
pp. 271-283 ◽  
Author(s):  
Françoise Bastin
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Density Condition ◽  
Locally Compact ◽  
Spaces Of Continuous Functions ◽  
Locally Compact Hausdorff Space

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

scholarly journals The q-Sumudu transform and its certain properties in a generalized q-calculus theory

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari
Keyword(s):  
Inversion Formula ◽  
Generalized Functions ◽  
Convolution Theorem ◽  
General Properties ◽  
Integrable Functions ◽  
Sumudu Transform ◽  
Convolution Products ◽  
Set Up

AbstractIn this paper we consider a generalization to the q-calculus theory in the space of q-integrable functions. We introduce q-delta sequences and develop q-convolution products to derive certain q-convolution theorem. By using the concept of q-delta sequences, we establish various axioms and set up q-spaces of generalized functions named q-Boehmian spaces. The new assigned spaces of q-generalized functions are acceptable and compatible with the classical spaces of the ordinary functions. Consequently, we extend the generalized q-Sumudu transform to the sets of q-Boehmian spaces. On top of that, we nominate the canonical q-embeddings between the q-integrable sets of functions and the q-integrable sets of q-Boehmians. Furthermore, we address the general properties of the generalized q-Sumudu transform and its inversion formula in some detail.

An integro-differential equation from population genetics and perturbations of differentiable semigroups in Fréchet spaces

1991 ◽  
Vol 118 (1-2) ◽  
pp. 63-73 ◽  
Author(s):  
Reinhard Bürger
Keyword(s):  
Differential Equation ◽  
Population Genetics ◽  
Quantitative Trait ◽  
Existence And Uniqueness ◽  
Directional Selection ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Uniqueness Of Solutions ◽  
Integro Differential Equation

SynopsisExistence and uniqueness of solutions of an integro-differential equation that arises in population genetics are proved. This equation describes the evolution of type densities in a population that is subject to mutation and directional selection on a quantitative trait. It turns out that a certain Fréchet space is the natural framework to show existence and uniqueness. One of the main steps in the proof is the investigation of perturbations of generators of differentiable semigroups in Fréchet spaces.

On the Weak Basis Theorem in F-spaces

1974 ◽  
Vol 26 (6) ◽  
pp. 1294-1300 ◽  
Author(s):  
Joel H. Shapiro
Keyword(s):  
Banach Spaces ◽  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Weak Basis ◽  
Basis Theorem ◽  
Locally Convex ◽  
Convex Spaces

It is well-known that every weak basis in a Fréchet space is actually a basis. This result, called the weak basis theorem was first given for Banach spaces in 1932 by Banach [1, p. 238], and extended to Fréchet spaces by Bessaga and Petczynski [3]. McArthur [12] proved an analogue for bases of subspaces in Fréchet spaces, and recently W. J. Stiles [18, Corollary 4.5, p. 413] showed that the theorem fails in the non-locally convex spaces lp (0 < p < 1). The purpose of this paper is to prove the following generalization of Stiles' result.

scholarly journals On some variant of a whittaker integral operator and its representative in a class of square integrable Boehmians

2018 ◽  
Vol 38 (1) ◽  
pp. 173 ◽  
Author(s):  
Shrideh Khalaf Al-Omari
Keyword(s):  
Integral Operator ◽  
Integral Operators ◽  
Convolution Theorem ◽  
Classical Integral ◽  
Convolution Products

This paper investigates some variant of Whittaker integral operators on a class of square integrable Boehmians. We define convolution products and derive the convolution theorem which substantially satisfy the axioms necessary for generating the Whittaker spaces of Boehmians. Relied on this analysis, we give a definition and properties of the Whittaker integral operator in the class of square integrable Boehmians. The extended Whittaker integral operator, is well-defined, linear and coincides with the classical integral in certain properties.

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