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Positive interpolatory quadrature rules generated by some biorthogonal polynomials

2009
*
Mathematics of Computation
*

As a consequence, we establish

doi:10.1090/s0025-5718-09-02299-6
fatcat:ywre4buxnja5pd6apyd3ctpjkq
*positivity*of the weights in a*quadrature**rule*introduced*by*the second author in 1980,*generated**by*a*polynomial*that is*biorthogonal*to (log x) j , 0 ≤ j ≤ n − 1. ...*Interpolatory**quadrature**rules*whose abscissas are zeros of a*biorthogonal**polynomial*have proved to be useful, especially in numerical integration of singular integrands. ... Let p n be the monic*polynomial*of degree n determined*by*the*biorthogonality*conditions (1.5) . Then the weights {λ j } n j=1 in the*interpolatory**rule*I n*generated**by*p n and w are all*positive*. ...##
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Page 313 of Mathematical Reviews Vol. , Issue 83a
[page]

1983
*
Mathematical Reviews
*

This formula was derived from the trapezoidal

*rule*formula*by*F. Stenger. “An explicit form of the error is given for the case where the integrand has a factor of the form (1—x)*%(1+x)*, a,B>-1. ... Authors’ summary: “We formulate a*quadrature**rule*for numerical evaluation of Cauchy principal value integrals of the type {'\(x—a)"f(x)dx where -1<a<1 and f(x) possesses complex singularities near the ...##
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The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization
[article]

2020
*
arXiv
*
pre-print

It is well known that the (classical) Gauss

arXiv:1903.11395v3
fatcat:qelm6z6lknhtbepnlao7fubjfm
*quadrature*for*positive*definite linear functionals is connected with orthogonal*polynomials*, and with the (Hermitian) Lanczos algorithm. ... Original proofs of the Mismatch Theorem and of the Matching Moment Property are given*by*using the properties of formal orthogonal*polynomials*and the Gauss*quadrature*for linear functionals. ... This work has been supported*by*Charles University Research program No. UNCE/SCI/023. ...##
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Page 2390 of Mathematical Reviews Vol. , Issue 84f
[page]

1984
*
Mathematical Reviews
*

in

*general*normed spaces with*some*additional properties (for example, C or L,). ...*rule*evaluation*quadrature*routine could easily have solved the problem. ...##
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Quadrature integration for orthogonal wavelet systems

1999
*
Journal of Chemical Physics
*

*General*

*quadrature*formulae are developed for accurate evaluation of integrals involving compact support wavelet families, and their use is demonstrated in examples of spectral analysis and integrals over ... Adaptive wavelet

*quadrature*schemes are also presented for the purpose of meeting specific accuracy criteria without excessive oversampling. ... Acknowledgments This work was supported

*by*the National Science Foundation and the Robert A. Welch Foundation. The authors also wish to thank R. O. Wells, C. S. Burrus, J. Tian and W. ...

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The Lanczos algorithm and complex Gauss quadrature

2018
*
Electronic Transactions on Numerical Analysis
*

In particular, the existence of the n-weight (complex) Gauss

doi:10.1553/etna_vol50s1
fatcat:il5nyk6smffwbixadaf7u54t7u
*quadrature*corresponds to successfully performing the first n steps of the Lanczos algorithm for*generating**biorthogonal*bases of the two associated ... Gauss*quadrature*can be naturally*generalized*in order to approximate quasi-definite linear functionals, where the interconnections with (formal) orthogonal*polynomials*, (complex) Jacobi matrices, and ... In this case we say that the*quadrature*(3.2) is*interpolatory*since it can be obtained*by*applying the linear functional L to the*generalized*(Hermite) interpolating*polynomial*for the function f at the ...##
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Trigonometric multiple orthogonal polynomials of semi-integer degree and the corresponding quadrature formulas

2014
*
Publications de l'Institut Mathématique (Beograd)
*

Also, an optimal set of

doi:10.2298/pim1410211m
fatcat:lyxchksqgbb6nhjecfoau456jy
*quadrature**rules*is characterized*by*trigonometric multiple orthogonal*polynomials*of semiinteger degree. ... Finally, theoretical results are illustrated*by**some*numerical examples. ... Type II multiple orthogonal*polynomials*are also connected with the*generalized*Birkhoff-Young*quadrature**rules*[8, 11, 13] . ...##
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Composite quadrature formulae for the approximation of wavelet coefficients of piecewise smooth and singular functions

2005
*
Journal of Computational and Applied Mathematics
*

We develop

doi:10.1016/j.cam.2004.10.005
fatcat:ebq7hfipwndmtncntz4fzxjmqy
*quadrature**rules*for those integrals that converge fast for piecewise smooth and singular functions. ... The results of [3] were*generalized*in [13] to higher order*interpolatory**quadrature**rules*, and extended with a stable method to construct those*quadrature**rules*using Chebyshev*polynomials*. ... Since for Gaussian*rules*the weight function has to be*positive*, for*some*scaling functions g(x) := (x) + c is used instead. ...##
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Gradimir Milovanovic - a master in approximation and computation: Part II

2020
*
Applicable Analysis and Discrete Mathematics
*

higher-order iterative methods with local convergence; In their paper Branquinho, García-Ardila and Marcellán study matrix

doi:10.2298/aadm201210037r
fatcat:i2xllzoi55bfteqyez5f57nymq
*biorthogonal**polynomials*sequences that satisfy a nonsymmetric three term recurrence ... relation with unbounded matrix coefficients; The paper*by*Jovanović and Voß describes a matheuristic approach for solving the 2-connected dominating set problem; In their paper Landon, Carley and Mohapatra ... In the first section on*quadrature*formulae they present*some*special Newton-Cotes*rules*, the Gauss-Christoffel, Gauss-Radau and Gauss-Lobatto*quadratures*, the socalled product integration*rules*, as well ...##
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Reviews and Descriptions of Tables and Books

1993
*
Mathematics of Computation
*

., the differences between the ATARI ST version and other systems) to deep mathematical issues (at least briefly, the algorithms for

doi:10.1090/s0025-5718-93-99737-2
fatcat:xwxys2wfgbfcxop6hwxv4db5mm
*polynomial*factoring and indefinite integration are discussed). ... Thus,*biorthogonal**polynomials*have been defined in [3] as follows: a nonzero nth-degree*polynomial*p"(-; px, p2, ... , pn) is*biorthogonal*with respect to the parametrized Borel measure dtp(x, p) if ... It was in 1973 that Frank Stenger surprised us with his paper [4] in which he demonstrated that one can get exponential convergence of*quadrature**rules*based on the trapezoidal*rule*even when the integrand ...##
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Borislav D. Bojanov: 18 November 1944–8 April 2009

2010
*
Journal of Approximation Theory
*

We hope that the following few pages of reminiscences, written

doi:10.1016/j.jat.2009.09.003
fatcat:bxmndj2iibbznmsr6tvakaou7m
*by*a few of his good friends and colleagues, and appearing in the order in which we received them, give something of the spirit of the man ... Studying*quadrature*formulae and interpolation processes with multiple nodes, Bojanov arrived at the necessity of studying*generalized*Chebyshev*polynomials*. ... The*generalized*Chebyshev*polynomials*T m (t; k) = (t − t * 1 ) k 1 · · · (t − t * n ) k n , (m = k 1 + · · · + k n ) preserve*some*properties of the classical Chebyshev*polynomials*(see [54] ). ...##
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BIORTHOGONAL WAVELETS WITH SIX-FOLD AXIAL SYMMETRY FOR HEXAGONAL DATA AND TRIANGLE SURFACE MULTIRESOLUTION PROCESSING

2011
*
International Journal of Wavelets, Multiresolution and Information Processing
*

Under

doi:10.1142/s0219691311004316
fatcat:izjjg7cisnf3tkpd3tih5gfitu
*some*condition, sum*rule*order is equivalent to the approximation order of ϕ, see [14]. ... In this paper, we consider L2 (IR2 ) inner product and the wavelets constructed*generate**biorthogonal*(Riesz) bases. ...##
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Page 1806 of Mathematical Reviews Vol. , Issue Subject Index
[page]

*
Mathematical Reviews
*

(English summary) 2001a:65032
Novak, Erich Intractability results for

*positive**quadrature*formulas and extremal problems for trigonometric*polynomials*. ...*Interpolatory**quadrature*formulae with Bernstein-Szeg6 abscissae. ...##
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Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

2016
*
Applied Numerical Mathematics
*

also explored

doi:10.1016/j.apnum.2016.05.008
fatcat:rdfov6gpdfh2pmg7ih46wkrjlq
*some*further consequences of these sequences of*polynomials*such as their connections to sequences of orthogonal*polynomials*on the interval [-1,1]. ... In this case, Delsarte and Genin, in 1986, have shown that the two sequences of para-orthogonal*polynomials*{zS_n(z) + S_n^∗(z)} and {zS_n(z) - S_n^∗(z)} satisfy three term recurrence formulas and have ... Moreover, the*interpolatory**quadrature**rules*based on the zeros of these*polynomials*are those*quadrature**rules*on the unit circle which are analogous to the Gaussian*quadrature**rules*on the real line. ...##
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Multivariate orthonormal interpolating scaling vectors

2007
*
Applied and Computational Harmonic Analysis
*

Our method is substantiated

doi:10.1016/j.acha.2006.06.002
fatcat:aq44v4zd4bac7jllogz6wzmkte
*by*constructing several examples of bivariate scaling vectors for quincunx and box-spline dilation matrices. ... As the main ingredients of our recipe we derive*some*implementable conditions for accuracy and orthonormality of an interpolating scaling vector in terms of its mask. ... These schemes are also*interpolatory*in the more*general*sense of [6] , cf. Proposition 1 in [6] . ...
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