2 edition of **On mechanical quadratures formulae involving the classical orthogonal polynomials ...** found in the catalog.

On mechanical quadratures formulae involving the classical orthogonal polynomials ...

Clement Winston

- 2 Want to read
- 20 Currently reading

Published
**1933**
in Philadelphia
.

Written in English

- Polynomials.,
- Interpolation.,
- Processes, Infinite.

**Edition Notes**

Other titles | Mechanical quadratures formulae., Orthogonal polynomials. |

Statement | [by] Clement Winston. |

Classifications | |
---|---|

LC Classifications | QA311 .W53 1933 |

The Physical Object | |

Pagination | 1 p. l., 658-677 p. |

Number of Pages | 677 |

ID Numbers | |

Open Library | OL6319757M |

LC Control Number | 35007777 |

OCLC/WorldCa | 6526766 |

They generalize quadrature formulae involving zeros of Bessel functions, which were first designed by Frappier and Olivier. Bessel quadratures correspond to the Fourier-Hankel integral transform. Some other examples, connected with the Jacobi integral transform, Fourier series in Jacobi orthogonal polynomials and the general Sturm-Liouville. Some of the basic properties on classical orthogonal polynomials, used in Sec-tions 3 through 5, are summarized in Appendix A. For the general theory, we refer the readers to Szego˝’s book Orthogonal Polynomials [33, Chapter IV and § and § ]. A quasi-orthogonal polynomial of degree n and order r is a polynomial of type Φn,r(x.

[26] G. V. MILOVANOVIC and A. S. Cvetkovic, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balkanica, 26 (), pp. [27] G. V. MILOVANOVIC and M. M. SPALEVIC, Kronrod extensions with multiple nodes of quadrature formulas for Fourier coefficients, Math. Anti-Gauss quadrature formulae associated with four classical Chebyshev weight functions are considered. Golub, G.H., Opfer, G. (Eds.), Applications and Computation of Orthogonal Polynomials, Birkhäuser, Basel. pp. Google Scholar [3]. Ehrich, S., On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas.

Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm-Liouville problem, which are exact for entire functions of exponential type, are established. This volume contains the Proceedings of the NATO Advanced Study Institute on "Orthogonal Polynomials and Their Applications" held at The Ohio State University in Columbus, Ohio, U.S.A. between and June 3,

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In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: probability, such as the Edgeworth series;; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus;; in numerical analysis as Gaussian quadrature;; in physics, where they give rise to the eigenstates of the quantum harmonic oscillator.

ON MECHANICAL QUADRATURES FORMULAE INVOLVING THE CLASSICAL ORTHOGONAL POLYNOMIALS' BY C. WINSTON (Received December 8, ) Introduction. Let {(x) be a function bounded and non-decreasing, with infinitely many points of increase, over an interval (a, b), finite or infinite, and rb such that all "moments" an = fXnd,(x) exist (n = 0, 1.

by Shohat (see [22]) concerning about mechanical quadratures. The problem was revisited by Maroni in [19], into a more general algebraic frame, who gives an expression for the MOPS associated with (1) in terms of the so called co-recursive polynomials of the classical orthogonal polynomials.

InCited by: 3. The main part of the topic is orthogonal trigonometric systems on [0, 2 π) (or on [− π, π)) with respect to some weight functions w (x). We prove that the so-called orthogonal trigonometric polynomials of semi-integer degree satisfy a five-term recurrence relation.

In particular, we study some cases with symmetric weight by: On mechanical quadratures formulae involving the classical orthogonal polynomials can be found in my draft book Approximation Theory and Approximation Practice, available at Gaussian quadratures and the classical orthogonal polynomials hav e formulas.

F or orthogonal polynomials in general, the contin uous cases, the existence o f mechanical quadratures. Quantum mechanical matrix elements often involve orthogonal polynomials, whose properties can be exploited for evaluation. Nullity conditions are considered for hydrogenic and harmonic oscillator radial integrals, and Gaunt's triangular condition for integrals over triple products of associated Legendre functions is generalized so that the superscripts form a triangle of even perimeter.

New York, WINSTON, C. On mechanical quadratures formulae involving the classical orthogonal polynomials. Ann. Math. 35, (). Recommended articles Citing articles (0) References 1. C WinstonOn mechanical quadratures formulae involving the classical orthogonal polynomials.

Ann. Math., 35 (), pp. Google Scholar. () Symbolic–numeric computation of orthogonal polynomials and Gaussian quadratures with respect to the cardinal B-spline. Numerical Algorithms() A fractional spectral method with applications to some singular problems.

Advancing research. Creating connections. Spectral and pseudospectral methods in chemistry and physics are based on classical and nonclassical orthogonal polynomials defined in terms of a three term recurrence relation.

The coefficients in the three term recurrence relations for the nonclassical polynomials can be calculated with the Gautschi-Stieltjes procedure. as well as for constructing the so-called half range quadratures (with the same weight function on (0,1)), was recently presented by Shizgal [28], including a construc-tion of orthogonal polynomials by using discretizing Stieltjes-Gautschi procedure (cf.

[12, –]). The classical Chebyshev method of moments is ill-conditioned. Laden, Fundamental polynomials of Lagrange interpolation and coefficients of mechanical quadrature, Duke Math.

10 (), – MR [27] Giovanni Monegato, Some new inequalities related to certain ultra-spherical polynomials. Perhaps the most intuitive procedure to use for building the orthogonal polynomials required by the mechanical quadrature to be used in this work is to proceed using the Gram–Schmidt orthogonalization method.

The algorithm to be used relies on two theorems which according to [18 (pp 30, 31)] state the following: Theorem We derive explicit formulas for the discriminants of classical quasi-orthogonal polynomials, as a full generalization of the result of Dilcher and Stolarsky ().

We consider a certain system of Diophantine equations, originally designed by Hausdorff () as a simplification of Hilbert's solution () of Waring's problem, and then create. This is the first book that provides graphs and references to online datasets that enable the generation of a large number of orthogonal polynomials with classical, quasi-classical, and nonclassical weight functions.

Useful numerical tables are also included. The book will be of interest to scientists, engineers, applied mathematicians, and. It would be interesting to determine this value for other classical weights.

In Bernstein showed that the Chebyshev polynomials of ﬁrst kind are s-orthogonal on [¡1;1] for each s ‚ 0. Ossicini (with Rosati) [14] in found three other measures for which the s-orthogonal polynomials. Orthogonal polynomials We begin with some basic facts about orthogonal polynomials on the real line and introduce appropriate notation as we go along.

Suppose d is a positive measure supported on an interval (or a set of disjoint intervals) on the real line such that all. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting.

Then, we categorize recent advances into those that propose a new sampling approach, and those centered on. Orthogonal Polynomials on the Real Line. Orthogonal Polynomials on the Semicircle.

Chebyshev Quadrature. Kronrod and Other Quadratures. Gauss-type Quadrature. Selected Works with Commentaries, Vol. Linear Difference Equations.

Ordinary Differential Equations. Software. History and Biography. Miscellanea. Works of Werner Gautschi. strong influence and interest: orthogonal polynomials (his book on Classical and Quantum Orthogonal Polynomials in One Variable should be on the desk of anyone reading this Newsletter), integrable systems and their applications.

Special functions were not mentioned in .4 Where the P n(z) are the polynomials of Eqn. (3), and the P n(z,1) are the related numerator OPs, see Ismail Ref.

(2), sections and Eqn. (6) implies that the zeros of the P n(x) are poles of the nth approximant, a result famous for its role in the early developments of the theory of orthogonal polynomials, see, again, Refs.given in terms of entropic integrals of the involved classical orthogonal polynomials.

Section 3 explores the connection of the entropy with the logarithmic potential and the Lp-norms of the polynomials, which leads to the asymptotic results stated in Section 4. The problem of the explicit.