Sobolev-Type Orthogonal Polynomials: The Nondiagonal Case

Alfaro, M.; Marcellán, Francisco; Rezola, M. L.; Ronveaux, A.

doi:10.1006/jath.1995.1121

Cited by 25 publications

(34 citation statements)

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“…This problem has been considered before when E is the interval [−1, 1] (see [1], [14], [16]), or when E is the unit circle (see [4], [13]), although in all these works a wider class of measures has been considered.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc

2002

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Abstract. Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner productwhere E is a rectifiabl Jordan curve or arc in the complex planeA is an M × M Hermitian matrix, Ml 1 + · · · + l m + m, |dξ | denotes the arc length measure, ρ is a nonnegative function on E, and z i ∈ , i = 1, 2, . . . , m, where is the exterior region to E.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…The (r, s) block is the (l r + 1) × (l s + 1) matrix whose (i, j) (u) , and F(i, k) is given by (31). Notice that the elements of the matrix C n are o (1), and R n is a block matrix. The r, s block is the (l r + 1) × (l s + 1) matrix …”

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confidence: 99%

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc

2002

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“…Let {q (α,β;M) n (t)} n 0 be the orthogonal polynomials with respect to this inner product, normalized with leading coefficient k (α,β) n given in (2.3). Some properties for the monic orthogonal polynomials with respect to this inner product can be found in [2,4].…”

Section: The Univariate Non-diagonal Sobolev Inner Productmentioning

confidence: 99%

Sobolev orthogonal polynomials on the unit ball via outward normal derivatives

Delgado

Fernández

Lubinsky

et al. 2016

Journal of Mathematical Analysis and Applications

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Abstract. We analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which involves the outward normal derivatives on the sphere. Using their representation in terms of spherical harmonics, algebraic and analytic properties will be deduced. First, we deduce explicit connection formulas relating classical multivariate ball polynomials and our family of Sobolev orthogonal polynomials. Then explicit representations for the norms and the kernels will be obtained. Finally, the asymptotic behaviour of the corresponding Christoffel functions is studied.

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“…non-diagonal) Sobolev inner products with respect to measures supported on the complex plane can be found in [1,4,6,13]. Results related to asymptotics for extremal polynomials associated to non-diagonal Sobolev norms may be seen in [12,[18][19][20].…”

Section: Introductionmentioning

confidence: 99%

“…in [−1, 1], then μ ∈ M(0, 1). In [11], using an approach based on the theory of Padé approximants, the authors obtain the outer relative asymptotics for orthogonal polynomials with respect to the Sobolev-type inner product (1) provided that μ belongs to Nevai class M(0, 1) and the mass points b k belong to C\suppμ. The same problem with the mass points in supp μ = [−1, 1] was solved in [24], provided that μ (x) > 0 a.e.…”

Section: Introductionmentioning

confidence: 99%

Recurrence relations and outer relative asymptotics of orthogonal polynomials with respect to a discrete Sobolev type inner product

et al. 2013

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We investigate algebraic and analytic properties of sequences of polynomials orthogonal with respect to the Sobolev type inner product polynomials. When the values M k,i are nonnegative real numbers, we can deduce the coefficients of the recurrence relation in terms of the connection coefficients for the sequences of polynomials orthogonal with respect to the Sobolev type inner product and those orthogonal with respect to the measure μ. The matrix of a symmetric multiplication operator in terms of the above sequence of Sobolev type orthogonal polynomials is obtained from the Jacobi matrix associated with the measure μ. Finally, we focus our attention on some outer relative asymptotics of such polynomials, which are deduced by using the above connection formulas.

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Sobolev-Type Orthogonal Polynomials: The Nondiagonal Case

Cited by 25 publications

References 0 publications

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc

Sobolev orthogonal polynomials on the unit ball via outward normal derivatives

Recurrence relations and outer relative asymptotics of orthogonal polynomials with respect to a discrete Sobolev type inner product

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