scholarly journals Note on the Heaviside Expansion Formula

1931 ◽  
Vol 17 (12) ◽  
pp. 678-684 ◽  
Author(s):  
J. M. D. Valle
Keyword(s):  
Expansion Formula

scholarly journals Befriending Askey–Wilson polynomials

2014 ◽  
Vol 17 (03) ◽  
pp. 1450015 ◽  
Author(s):  
Paweł J. Szabłowski
Keyword(s):  
Characteristic Function ◽  
Hermite Polynomials ◽  
Probabilistic Interpretation ◽  
Conjugate Pair ◽  
Linear Combinations ◽  
Expansion Formula ◽  
Wilson Polynomials ◽  
The Way

We recall five families of polynomials constituting a part of the so-called Askey–Wilson scheme. We do this to expose properties of the Askey–Wilson (AW) polynomials that constitute the last, most complicated element of this scheme. In doing so we express AW density as a product of the density that makes q-Hermite polynomials orthogonal times a product of four characteristic function of q-Hermite polynomials (2.9) just pawing the way to a generalization of AW integral. Our main results concentrate mostly on the complex parameters case forming conjugate pairs. We present new fascinating symmetries between the variables and some newly defined (by the appropriate conjugate pair) parameters. In particular in (3.12) we generalize substantially famous Poisson–Mehler expansion formula (3.16) in which q-Hermite polynomials are replaced by Al-Salam–Chihara polynomials. Further we express Askey–Wilson polynomials as linear combinations of Al-Salam–Chihara (ASC) polynomials. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we reach AW polynomials and AW densities that have clear probabilistic interpretation.

scholarly journals Comultiplication Rules for the Double Schur Functions and Cauchy Identities

10.37236/102 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
A. I. Molev
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Primitive Elements ◽  
Expansion Formula ◽  
New Family ◽  
Dual Object ◽  
Multiplication Rule ◽  
Distinguished Basis ◽  
Natural Way

The double Schur functions form a distinguished basis of the ring $\Lambda(x\!\parallel\!a)$ which is a multiparameter generalization of the ring of symmetric functions $\Lambda(x)$. The canonical comultiplication on $\Lambda(x)$ is extended to $\Lambda(x\!\parallel\!a)$ in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood–Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood–Richardson coefficients provide a multiplication rule for the dual Schur functions.

The Parseval Equality and Expansion Formula for Singular Hahn-Dirac System

2020 ◽  
pp. 209-235
Author(s):  
Bilender P. Allahverdiev ◽  
Hüseyin Tuna
Keyword(s):  
Spectral Function ◽  
Spectral Parameter ◽  
Continuous Functions ◽  
Dirac System ◽  
Parseval Equality ◽  
Expansion Formula

This work studies the singular Hahn-Dirac system given by Here 𝜇 is a complex spectral parameter, p(.) and r(.) are real-valued continuous functions at 𝜔0, defined on [𝜔0,∞) and q∈(0,1), , 𝜔>0, x∈[𝜔0,∞). The existence of a spectral function for this system is proved. Further, a Parseval equality and an expansion formula in eigenfunctions are proved in terms of the spectral function.

scholarly journals Fast Imaging of Thin, Curve-Like Electromagnetic Inhomogeneities without a Priori Information

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 799 ◽  
Author(s):  
Won-Kwang Park
Keyword(s):  
Imaging Technique ◽  
A Priori ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Mathematical Structure ◽  
Fast Imaging ◽  
A Priori Information ◽  
Expansion Formula ◽  
Priori Information ◽  
Thin Curve

It is well-known that subspace migration is a stable and effective non-iterative imaging technique in inverse scattering problem. However, for a proper application, a priori information of the shape of target must be estimated. Without this consideration, one cannot retrieve good results via subspace migration. In this paper, we identify the mathematical structure of single- and multi-frequency subspace migration without any a priori of unknown targets and explore its certain properties. This is based on the fact that elements of so-called multi-static response (MSR) matrix can be represented as an asymptotic expansion formula. Furthermore, based on the examined structure, we improve subspace migration and consider the multi-frequency subspace migration. Various results of numerical simulation with noisy data support our investigation.

Gradient approach for recursive estimation and control in finite Markov chains

1981 ◽  
Vol 13 (04) ◽  
pp. 778-803 ◽  
Author(s):  
Yousri M. El-Fattah
Keyword(s):  
Likelihood Function ◽  
Transition Probabilities ◽  
Control Policy ◽  
Recursive Estimation ◽  
Gradient Algorithm ◽  
Expansion Formula ◽  
Estimation Algorithms ◽  
Stochastic Gradient Algorithm ◽  
Long Run ◽  
Estimation And Control

The problem studied is that of controlling a finite Markov chain so as to maximize the long-run expected reward per unit time. The chain's transition probabilities depend upon an unknown parameter taking values in a subset [ a, b ] of Rn . A control policy is defined as the probability of selecting a control action for each state of the chain. Derived is a Taylor-like expansion formula for the expected reward in terms of policy variations. Based on that result, a recursive stochastic gradient algorithm is presented for the adaptation of the control policy at consecutive times. The gradient depends on the estimated transition parameter which is also recursively updated using the gradient of the likelihood function. Convergence with probability 1 is proved for the control and estimation algorithms.

BIRGET–RHODES EXPANSION OF GROUPS AND INVERSE MONOIDS OF MÖBIUS TYPE

2002 ◽  
Vol 12 (04) ◽  
pp. 525-533 ◽  
Author(s):  
KEUNBAE CHOI ◽  
YONGDO LIM
Keyword(s):  
Hausdorff Space ◽  
Inverse Monoid ◽  
Expansion Formula ◽  
Inverse Monoids ◽  
Isolated Point

In this paper we prove that if a group G acts faithfully on a Hausdorff space X and acts freely at a non-isolated point, then the Birget–Rhodes expansion [Formula: see text] of the group G is isomorphic to an inverse monoid of Möbius type obtained from the action.

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