A difference equation approach to Plancherel‐Rotach asymptotics for ‐orthogonal polynomials

2020 ◽  
Vol 145 (4) ◽  
pp. 696-718
Author(s):  
Mourad Ismail ◽  
Chun‐Kong Law
Keyword(s):  
Difference Equation ◽  
Orthogonal Polynomials ◽  
Equation Approach

Ladder operators and a differential equation for varying generalized Freud-type orthogonal polynomials

2018 ◽  
Vol 07 (04) ◽  
pp. 1840005 ◽  
Author(s):  
Galina Filipuk ◽  
Juan F. Mañas-Mañas ◽  
Juan J. Moreno-Balcázar
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Inner Product ◽  
Ladder Operators ◽  
First Order ◽  
Standard Family ◽  
Differential Difference Equation ◽  
First Order Differential Equations

In this paper, we introduce varying generalized Freud-type polynomials which are orthogonal with respect to a varying discrete Freud-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as find a second-order differential–difference equation that these polynomials satisfy. To reach this objective, it is necessary to consider the standard Freud orthogonal polynomials and, in the meanwhile, we find new difference relations for the coefficients in the first-order differential equations that this standard family satisfies.

scholarly journals H∞Channel Estimation for DS-CDMA Systems: A Partial Difference Equation Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Wei Wang ◽  
Chunyan Han
Keyword(s):  
Channel Estimation ◽  
Difference Equation ◽  
Multiple Access ◽  
Multipath Fading ◽  
Channel Noise ◽  
Equation Approach ◽  
Partial Difference Equation ◽  
Partial Difference ◽  
Channel Estimator ◽  
Code Division Multiple

In the communications literature, a number of different algorithms have been proposed for channel estimation problems with the statistics of the channel noise and observation noise exactly known. In practical systems, however, the channel parameters are often estimated using training sequences which lead to the statistics of the channel noise difficult to obtain. Moreover, the received signals are corrupted not only by the ambient noises but also by multiple-access interferences, so the statistics of observation noises is also difficult to obtain. In this paper, we will investigate theH∞channel estimation problem for direct-sequence code-division multiple-access (DS-CDMA) communication systems with time-varying multipath fading channels. The channel estimator is designed by applying a partial difference equation approach together with the innovation analysis theory. This method can give a sufficient and necessary condition for the existence of anH∞channel estimator.

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