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### Positive interpolatory quadrature rules generated by some biorthogonal polynomials

D. S. Lubinsky, A. Sidi
2009 Mathematics of Computation
As a consequence, we establish 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.  ...

### 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  ...

### The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization [article]

Stefano Pozza, Miroslav S. Pranić
2020 arXiv   pre-print
It is well known that the (classical) Gauss 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.  ...

### 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.  ...

### Quadrature integration for orthogonal wavelet systems

Bruce R. Johnson, Jason P. Modisette, Peter J. Nordlander, James L. Kinsey
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.  ...

### The Lanczos algorithm and complex Gauss quadrature

Stefano Pozza, Miroslav S. Pranić, Zdeněk Strakoš
2018 Electronic Transactions on Numerical Analysis
In particular, the existence of the n-weight (complex) Gauss 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  ...

### Trigonometric multiple orthogonal polynomials of semi-integer degree and the corresponding quadrature formulas

Gradimir Milovanovic, Marija Stanic, Tatjana Tomovic
2014 Publications de l'Institut Mathématique (Beograd)
Also, an optimal set of 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] .  ...

### Composite quadrature formulae for the approximation of wavelet coefficients of piecewise smooth and singular functions

Daan Huybrechs, Stefan Vandewalle
2005 Journal of Computational and Applied Mathematics
We develop 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.  ...

### Gradimir Milovanovic - a master in approximation and computation: Part II

Themistocles Rassias, Miodrag Spalevic
2020 Applicable Analysis and Discrete Mathematics
higher-order iterative methods with local convergence; In their paper Branquinho, García-Ardila and Marcellán study matrix 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  ...

### 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 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  ...

### Borislav D. Bojanov: 18 November 1944–8 April 2009

András Kroó, Allan Pinkus, Yuan Xu
2010 Journal of Approximation Theory
We hope that the following few pages of reminiscences, written 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] ).  ...

### BIORTHOGONAL WAVELETS WITH SIX-FOLD AXIAL SYMMETRY FOR HEXAGONAL DATA AND TRIANGLE SURFACE MULTIRESOLUTION PROCESSING

QINGTANG JIANG
2011 International Journal of Wavelets, Multiresolution and Information Processing
Under 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.  ...

### 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.  ...

### Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

C.F. Bracciali, A. Sri Ranga, A. Swaminathan
2016 Applied Numerical Mathematics
also explored 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.  ...

### Multivariate orthonormal interpolating scaling vectors

Karsten Koch
2007 Applied and Computational Harmonic Analysis
Our method is substantiated 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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