scholarly journals Linear combinations of orthogonal polynomials generating positive quadrature formulas

1990 ◽  
Vol 55 (191) ◽  
pp. 231-231 ◽  
Author(s):  
Franz Peherstorfer
Keyword(s):  
Orthogonal Polynomials ◽  
Quadrature Formulas ◽  
Linear Combinations ◽  
Positive Quadrature Formulas

Padé approximation and orthogonal polynomials

1974 ◽  
Vol 10 (2) ◽  
pp. 263-270 ◽  
Author(s):  
G.D. Allen ◽  
C.K. Chui ◽  
W.R. Madych ◽  
F.J. Narcowich ◽  
P.W. Smith
Keyword(s):  
Power Series ◽  
Variational Method ◽  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Quadrature Formulas ◽  
The Past ◽  
Gaussian Quadrature Formulas ◽  
Poles And Zeros ◽  
Formal Power ◽  
Determinant Theory

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

scholarly journals A characterization of the four Chebyshev orthogonal families

2005 ◽  
Vol 2005 (13) ◽  
pp. 2071-2079 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. M. Garcia-Amor
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Linear Combinations ◽  
Classical Orthogonal Polynomial ◽  
Polynomial Sequences ◽  
Fixed Length ◽  
Constant Coefficients ◽  
Fourth Kind

We obtain a property which characterizes the Chebyshev orthogonal polynomials of first, second, third, and fourth kind. Indeed, we prove that the four Chebyshev sequences are the unique classical orthogonal polynomial families such that their linear combinations, with fixed length and constant coefficients, can be orthogonal polynomial sequences.

scholarly journals When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

2010 ◽  
Vol 233 (6) ◽  
pp. 1446-1452 ◽  
Author(s):  
Manuel Alfaro ◽  
Francisco Marcellán ◽  
Ana Peña ◽  
M. Luisa Rezola
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Combinations

scholarly journals Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 617 ◽  
Author(s):  
Dmitry Dolgy ◽  
Dae Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Connection Problem ◽  
Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 .

Zeros of linear combinations of orthogonal polynomials

1995 ◽  
Vol 117 (3) ◽  
pp. 533-544 ◽  
Author(s):  
Franz Peherstorfer
Keyword(s):  
Distribution Function ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Weak Assumption ◽  
Linear Combinations ◽  
Simple Zeros ◽  
Jacobi Weights

AbstractLet ψ be a distribution function on [−1,1] from the Szegö-class, which contains in particular all Jacobi weights, and let (pn) be the monic polynomials orthogonal with respect to dψ. Let m(n)∈ℕ, n∈ℕ, be non-decreasing with limn → ∞ (n − m(n)) = ∞, l(n)∈ℕ with 0 ≤ l(n) ≤ m(n), and μj, n ∈ℝ for j = 0, …, m(n), n∈ℕ. It is shown that for each sufficiently large n, has n−l(n) simple zeros in (−1, 1)and l(n) zeros in ℂ\[−1,1] if for n ≥ n0, has m(n) − l(n) zeros in the disc |z| ≤ r < 1, l(n) zeros outside of the disc |z| ≥ R > 1 and where q > 2 max {r, 1/R}. If m(n) is constant for n ≥ n0 then the statement holds even for such polynomials (pn) orthogonal with respect to a distribution dψ satisfying the weak assumption ψ′ > 0 a.e. on [−1, 1]. For linear combinations of polynomials orthogonal on the unit circle corresponding results are derived.

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