Necessary and Sufficient Conditions Relating the Coefficients in the Recurrence Formula to the Spectral Function for Orthogonal Polynomials

1983 ◽  
Vol 14 (3) ◽  
pp. 622-637 ◽  
Author(s):  
Jeffrey S. Geronimo ◽  
Paul G. Nevai
Keyword(s):  
Orthogonal Polynomials ◽  
Spectral Function ◽  
Recurrence Formula ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Mean Convergence of Lagrange Interpolation for Exponential weights on [-1, 1]

1998 ◽  
Vol 50 (6) ◽  
pp. 1273-1297 ◽  
Author(s):  
D. S. Lubinsky
Keyword(s):  
Orthogonal Polynomials ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exponential Weights ◽  
Mean Convergence ◽  
Zeros Of Orthogonal Polynomials ◽  
Necessary And Sufficient

AbstractWe obtain necessary and sufficient conditions for mean convergence of Lagrange interpolation at zeros of orthogonal polynomials for weights on [-1, 1], such asw(x) = exp(-(1 - x2)-α), α > 0orw(x) = exp(-expk(1 - x2)-α), k≥1, α > 0,where expk = exp(exp(. . . exp( ) . . .)) denotes the k-th iterated exponential.

Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights

1996 ◽  
Vol 48 (4) ◽  
pp. 710-736 ◽  
Author(s):  
S. B. Damelin ◽  
D. S. Lubinsky
Keyword(s):  
Continuous Function ◽  
Orthogonal Polynomials ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Interpolation Polynomial ◽  
Necessary And Sufficient Conditions ◽  
Lagrange Interpolation Polynomial ◽  
Mean Convergence ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdös weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), whereα > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < ∞, Δ ∊ ℝ, k > 0. Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then forto hold for every continuous function ƒ: ℝ —> ℝ satisfyingit is necessary and sufficient that

scholarly journals Generating new classes of orthogonal polynomials

1996 ◽  
Vol 19 (4) ◽  
pp. 643-656 ◽  
Author(s):  
Amílcar Branquinho ◽  
Francisco Marcellán
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Functional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Given a sequence of monic orthogonal polynomials (MOPS),{Pn}, with respect to a quasi-definite linear functionalu, we find necessary and sufficient conditions on the parametersanandbnfor the sequencePn(x)+anPn−1(x)+bnPn−2(x),   n≥1P0(x)=1,P−1(x)=0to be orthogonal. In particular, we can find explicitly the linear functionalvsuch that the new sequence is the corresponding family of orthogonal polynomials. Some applications for Hermite and Tchebychev orthogonal polynomials of second kind are obtained.We also solve a problem of this type for orthogonal polynomials with respect to a Hermitian linear functional.

A generalisation of the inverse spectral theorem of Levitan and Gasymov

1979 ◽  
Vol 84 (3-4) ◽  
pp. 185-196 ◽  
Author(s):  
Christine Thurlow
Keyword(s):  
Spectral Function ◽  
Sufficient Conditions ◽  
Liouville Problem ◽  
Necessary And Sufficient Conditions ◽  
Spectral Theorem ◽  
Sturm Liouville ◽  
Inverse Spectral ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisIn the Inverse Spectral Theorem in the form given by Levitan and Gasymov, necessary and sufficient conditions are given for a non-decreasing function, p(ℷ), to be the spectral function of a Sturm–Liouville problem. In these conditions, p(ℷ) is compared with the spectral function for the particular Strum–Liouville problemIf the method of Levitan and Gasymov's proof is slightly adapted, the necessary and sufficient conditions can be stated in a more general form in which p(ℷ) is compared with the spectral function for any problem of the formwherehOis real and qo(x) locally integrable.

scholarly journals Mean convergence of Grünwald interpolation operators

2003 ◽  
Vol 2003 (33) ◽  
pp. 2083-2095
Author(s):  
Zhixiong Chen
Keyword(s):  
Orthogonal Polynomials ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Mean Convergence ◽  
Zeros Of Orthogonal Polynomials ◽  
General Weight ◽  
Interpolation Operators ◽  
Necessary And Sufficient

We investigate weightedLpmean convergence of Grünwald interpolation operators based on the zeros of orthogonal polynomials with respect to a general weight and generalizedJacobiweights. We give necessary and sufficient conditions for such convergence for all continuous functions.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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