scholarly journals Weighted -Integral Representations of -Functions in

2012 ◽  
Vol 2012 ◽  
pp. 1-27
Author(s):  
Arman H. Karapetyan
Keyword(s):  
Integral Operator ◽  
Complex Space ◽  
Entire Functions ◽  
Integral Representations ◽  
Orthogonal Projector ◽  
Weighted Integral

For -functions , given in the complex space , integral representations of the form are obtained. Here, is the orthogonal projector of the space onto its subspace of entire functions and the integral operator appears by means of explicitly constructed kernel Φ which is investigated in detail.

Universal oscillations of high derivatives

2005 ◽  
Vol 461 (2058) ◽  
pp. 1735-1751 ◽  
Author(s):  
M.V Berry
Keyword(s):  
Real Axis ◽  
Wide Class ◽  
Entire Functions ◽  
Saddle Points ◽  
Integral Representations ◽  
Simple Pole ◽  
Branch Points ◽  
The Real

Differentiation generates oscillations. For the n th derivative f ( n ,  t ) of a function f ( t ) that is analytic in a strip, including the real t -axis, the oscillations occupy a t interval that gets larger as n increases. The oscillations are studied in detail using integral representations and large- n asymptotics. For functions with singularities (poles or branch-points) in the complex t -plane, the oscillations of high derivatives are determined by the singularities; for entire functions, the oscillations originate in complex saddle-points. In a wide class of cases, the oscillations are contained in a Gaussian envelope in the t interval where f ( n ,  t ) is largest, with the envelope including about oscillations. Examples of the universal oscillations are given for f ( t ) with a simple pole, competing branch-points, a single saddle, competing pole and saddle and where the zeros are confined to half the real axis.

scholarly journals Weighted integral representations of harmonic functions in the unit disc by means of Mittag-Leffler type kernels

2021 ◽  
pp. 1-11
Author(s):  
Feliks Hayrapetyan
Keyword(s):  
Weight Function ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Integral Representations ◽  
Weighted Integral

For weighted $L^p$-classes of functions harmonic in the unit disc, we obtain a family of weighted integral representations with weight function of the type $|w|^{2\varphi}\cdot(1-|w|^{2\rho})^{\beta}$.

scholarly journals The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functions

2021 ◽  
Vol 13 (3) ◽  
pp. 805-817
Author(s):  
D.M. Bushev ◽  
F.G. Abdullayev ◽  
I.V. Kal'chuk ◽  
M. Imashkyzy
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Integral Representations ◽  
Periodic Functions ◽  
Approximation Of Functions ◽  
Functions Of Exponential Type

In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.

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