Sobolev-Gegenbauer-type orthogonality and a hydrodynamical interpretation. (English) Zbl 1231.42024
The paper is devoted to study orthogonality with respect to a discrete-continuous Sobolev inner product defined in the space of algebraic polynomials as follows:
\[
\langle f, g \rangle_{\lambda,\zeta}= \eta f(\zeta) g(\zeta)+ \int_{-1}^1 f'(x)g'(x) d \mu_{\lambda}(x),
\]
where \(\eta >0, \zeta\) is a fixed complex point and \(d \mu_{\lambda}(x)=(1-x^2)^{\lambda-\frac{1}{2}}dx\), with \(\lambda>- \frac{1}{2}\), is the classical Gegenbauer or ultraspherical measure.
Several properties of the corresponding monic orthogonal polynomials are obtained such as the recurrence relation that they satisfy and the relative asymptotic with respect to the Gegenbauer polynomials and their derivatives.
Besides, the location of the zeros of the orthogonal discrete-continuous Sobolev polynomials is obtained and a hydrodynamical interpretation like the location of source points and its corresponding strength is also given.
Several properties of the corresponding monic orthogonal polynomials are obtained such as the recurrence relation that they satisfy and the relative asymptotic with respect to the Gegenbauer polynomials and their derivatives.
Besides, the location of the zeros of the orthogonal discrete-continuous Sobolev polynomials is obtained and a hydrodynamical interpretation like the location of source points and its corresponding strength is also given.
Reviewer: Alicia Cachafeiro López (Vigo)
MSC:
42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
33C47 | Other special orthogonal polynomials and functions |
76B99 | Incompressible inviscid fluids |
Keywords:
asymptotic behavior; hydrodynamical model; orthogonal polynomials; recurrence relation; zero locationReferences:
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