Orthogonal polynomials with respect to the form u = λ(x – a)–1 ν + δb

2010 ◽  
Vol 17 (3) ◽  
pp. 581-596
Author(s):  
Mabrouk Sghaier
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Linear Functional ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Regular Form ◽  
Necessary And Sufficient ◽  
Three Term Recurrence Relation

Abstract We study properties of the form (linear functional) u = λ(x – a)–1 ν + δb , where ν is a regular form. We give a necessary and sufficient condition for the regularity of the form u. The coefficients of a three-term recurrence relation, satisfied by the corresponding sequence of orthogonal polynomials, are given explicitly. The semi-classical character of the founded families is studied. We conclude by giving some examples.

The Associated Ultraspherical Polynomials and their q-Analogues

1982 ◽  
Vol 34 (3) ◽  
pp. 718-736 ◽  
Author(s):  
Joaquin Bustoz ◽  
Mourad E. H. Ismail
Keyword(s):  
Orthogonal Polynomials ◽  
Positive Measure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Ultraspherical Polynomials ◽  
Associated Polynomials ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Precise Degree

A sequence of polynomials {Pn(x)} is orthogonal if Pn(x) is of precise degree n and there is a finite positive measure dμ such that1.1A necessary and sufficient condition for orthogonality [9] is that {Pn(x)} satisfies a three term recurrence1.2with1.3Given a sequence of orthogonal polynomials {Pn(x)} satisfying (1.2), the associated polynomials {Pn(γ)(x)}, γ > 0, are defined by1.4with P(γ)-1(x) = 0, P0(γ)(x) = 1, when An+γ and Bn+γ are well-defined.

Mean Convergence of Hermite-Fejér Interpolation Based on the Zeros of Lascenov Polynomials

1996 ◽  
Vol 39 (1) ◽  
pp. 117-128 ◽  
Author(s):  
Ying Guang Shi
Keyword(s):  
Orthogonal Polynomials ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mean Convergence ◽  
Zeros Of Orthogonal Polynomials ◽  
Necessary And Sufficient

AbstractWeighted LP mean convergence of Hermite-Fejér interpolation based on the zeros of orthogonal polynomials with respect to the weight |x|2α+1(l — x2)β(α, β > — 1) is investigated. A necessary and sufficient condition for such convergence for all continuous functions is given. Meanwhile divergence of Hermite-Fejér interpolation in LP with p > 2 is obtained. This gives a possible answer to Problem 17 of P. Turân [J. Approx. Theory, 29(1980), p. 40].

Generating Some Symmetric Semi-classical Orthogonal Polynomials

2015 ◽  
Vol 58 (4) ◽  
pp. 877-890
Author(s):  
Mohamed Zaatra
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Linear Functional ◽  
Symmetric Form ◽  
Classical Orthogonal Polynomials ◽  
Classical Form ◽  
Knowing That ◽  
Structure Relation ◽  
Three Term Recurrence Relation ◽  
Structure Relation Coefficients

AbstractWe show that if v is a regular semi-classical form (linear functional), then the symmetric form u defined by the relation x2σu = -λv, where (σƒ )(x) = f (x2) and the odd moments of u are 0, is also regular and semi-classical form for every complex λ except for a discrete set of numbers depending on v. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with u and the class of the form u knowing that of v. We conclude with an illustrative example.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

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