Integral Representations and Continuous Projectors in Some Spaces of Analytic and Pluriharmonic Functions

2008 ◽  
Vol 15 (4) ◽  
pp. 739-752
Author(s):  
Gigla Oniani ◽  
Lamara Tsibadze
Keyword(s):  
Integral Representations ◽  
Fractional Integrals ◽  
Boundary Values ◽  
Pluriharmonic Functions

Abstract We consider analytic and pluriharmonic functions belonging to the classes 𝐵𝑝(Ω) and 𝑏𝑝(Ω) and defined in the ball . The theorems established in the paper make it possible to obtain some integral representations of functions of the above-mentioned classes. The existence of bounded projectors from the space 𝐿(ρ, Ω) into the space 𝐵𝑝(Ω) and from the space 𝐿(ρ, Ω) into the space 𝑏𝑝(Ω) is proved. Also, consideration is given to the existence of boundary values of fractional integrals of functions of the spaces 𝐵𝑝(Ω) and 𝑏𝑝(Ω).

Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values

1981 ◽  
Vol 91 (1-2) ◽  
pp. 49-62 ◽  
Author(s):  
R. S. Pathak
Keyword(s):  
Analytic Function ◽  
Growth Condition ◽  
Compact Support ◽  
Analytic Functions ◽  
Convex Cone ◽  
Integral Representations ◽  
Boundary Values ◽  
Cauchy Integral ◽  
Boundedness Condition ◽  
Poisson Integrals

SynopsisUltradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.

scholarly journals The J-Generalized P - K Mittag-Leffler Function

2020 ◽  
Author(s):  
Kuldeep Singh Gehlot ◽  
Anjana Bhandari
Keyword(s):  
Mellin Transform ◽  
Integral Representations ◽  
Fractional Integrals ◽  
Linear Differential ◽  
Euler Transform ◽  
Relationship Of ◽  
The Relationship ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation ◽  
Differential And Integral Equations

We know that the classical Mittag-Leffler function play an important role as solution of fractional order differential and integral equations. We introduce the j-generalized p - k Mittag-Leffler function. We evaluate the second order differential recurrence relation and four different integral representations and introduce a homogeneous linear differential equation whose one of the solution is the j-generalized p-k Mittag-Leffler function. Also we evaluate the certain relations that exist between j-generalized p - k Mittag-Leffler function and Riemann-Liouville fractional integrals and derivatives. We evaluate Mellin-Barnes integral representation of j-generalized p-k Mittag-Le er Function. The relationship of j-generalized p-k Mittag-Leffler Function with Fox H-Function and Wright hypergeometric function is also establish. we obtained its Euler transform, Laplace Transform and Mellin transform. Finally we derive some particular cases.

scholarly journals Pluriharmonic functions on symmetric tube domains with BMO boundary values

2002 ◽  
Vol 94 (1) ◽  
pp. 67-86
Author(s):  
Ewa Damek ◽  
Jacek Dziubański ◽  
Andrzej Hulanicki ◽  
Jose L. Torrea
Keyword(s):  
Boundary Values ◽  
Tube Domains ◽  
Pluriharmonic Functions

scholarly journals A new transform method II: the global relation and boundary-value problems in polar coordinates

2010 ◽  
Vol 466 (2120) ◽  
pp. 2283-2307 ◽  
Author(s):  
E. A. Spence ◽  
A. S. Fokas
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Integral Representations ◽  
New Method ◽  
Polar Coordinates ◽  
Nonlinear Pdes ◽  
Boundary Values ◽  
Transform Method ◽  
Global Relation ◽  
The Fourier Transform

A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In a previous paper, a simplified way of obtaining these representations was presented. In the current paper, first, the second ingredient of the new method, namely the derivation of the so-called ‘global relation’ (GR)—an equation involving transforms of the boundary values—is presented. Then, using the GR as well as the IR derived in the previous paper, certain BVPs in polar coordinates are solved. These BVPs elucidate the fact that this method has substantial advantages over the classical transform method.

scholarly journals On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 232 ◽  
Author(s):  
Ahmed Bakhet ◽  
Fuli He
Keyword(s):  
Integral Representations ◽  
Fractional Integrals ◽  
Matrix Polynomials ◽  
Generating Matrix ◽  
Finite Sum

In this paper, we first introduce the 2-variables Konhauser matrix polynomials; then, we investigate some properties of these matrix polynomials such as generating matrix relations, integral representations, and finite sum formulae. Finally, we obtain the fractional integrals of the 2-variables Konhauser matrix polynomials.

Corrigenda: Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values

1984 ◽  
Vol 96 (1-2) ◽  
pp. 179-179 ◽  
Author(s):  
R. S. Pathak
Keyword(s):  
Compact Support ◽  
Integral Representations ◽  
Poisson Integral ◽  
Boundary Values

Page 56: In the estimate , C1 = CSn/(2πσ)n, h = (2πσ)−1 where σ depends on y. Therefore, {fy(t)} may not be bounded in as y → 0, y ∈ C′.

scholarly journals Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan ◽  
Salah Mahmoud Boulaaras ◽  
Bahri-Belkacem Cherif
Keyword(s):  
Probability Theory ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integrals ◽  
Convergence Properties ◽  
Fractional Integral Operators ◽  
Derivative Formulas

Hypergeometric functions have many applications in various areas of mathematical analysis, probability theory, physics, and engineering. Very recently, Hidan et al. (Math. Probl. Eng., ID 5535962, 2021) introduced the (p, k)-extended hypergeometric functions and their various applications. In this line of research, we present an expansion of the k-Gauss hypergeometric functions and investigate its several properties, including, its convergence properties, derivative formulas, integral representations, contiguous function relations, differential equations, and fractional integral operators. Furthermore, the current results contain several of the familiar special functions as particular cases, and this extension may enrich the theory of special functions.

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