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1
October 2017
Numerical Algorithms: Volume 76 Issue 2, October 2017
Publisher: SpringerVerlag New York, Inc.
The first 60 coefficients in the threeterm recurrence relation for monic polynomials orthogonal with respect to cardinal Bsplines źm as the weight functions on [0, m] (m ź ź) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to ...
Keywords:
Moment, 42C05, Cardinal Bspline, Gaussian quadrature formula, Recurrence relation, 65D30, 65D32, 41A15, 41A55, Orthogonal polynomial, Symbolic computation
2
June 2017
Calcolo: a quarterly on numerical analysis and theory of computation: Volume 54 Issue 2, June 2017
Publisher: SpringerVerlag New York, Inc.
A class of Fredholm integral equations of the second kind, with respect to the exponential weight function $$w(x)=\exp ((x^{\alpha }+x^\beta ))$$w(x)=exp((x�+x�)), $$\alpha >0$$�>0, $$\beta >1$$�>1, on $$(0,+\infty )$$(0,+�), is considered. The kernel k(x,�y) and the function g(x) in such kind of equations, $$\begin{aligned} f(x)\mu \int _0^{+\infty }k(x,y)f(y)w(y)\mathrm {d}y =g(x),\quad x\in ...
Keywords:
41A55, Orthogonal polynomials, 65R10, Fredholm integral equation, Gaussian quadrature formula, Nyström method, Truncation, Weighted polynomial approximation, 65D30, 65D32, Error estimate
3
December 2016
Journal of Computational and Applied Mathematics: Volume 308 Issue C, December 2016
Publisher: Elsevier Science Publishers B. V.
In this paper, we study the numerical methods for the evaluation of two kinds of highly oscillatory Bessel transforms. Firstly, we rewrite both integrals as the sum of two integrals. By rewriting the Bessel function as a linear combination of Whittaker W function, we then transform one of integrals into ...
Keywords:
65D30, 65D32, Oscillatory Bessel Hilbert transform, Gaussian quadrature rules, Oscillatory Bessel transform, Whittaker W function
4
August 2016
Computational Statistics & Data Analysis: Volume 100 Issue C, August 2016
Publisher: Elsevier Science Publishers B. V.
A modification of one of the most popular stochastic model in describing financial indexes dynamics is introduced. For describing a nonlinear behavior of volatility, a threshold noise indicator in the autoregressive time series of stochastic volatility is used. Toward this end, the model named the SplitSV model is introduced and ...
Keywords:
SplitSV process, Convolutions, Empirical characteristic function estimation, Noiseindicator, Contaminated Gaussian distribution
5
June 2015
Applied Mathematics and Computation: Volume 261 Issue C, June 2015
Publisher: Elsevier Science Inc.
In this paper, we consider the evaluation of two kinds of oscillatory integrals with a Hankel function as kernel. We first rewrite these integrals as the integrals of Fouriertype. By analytic continuation, these Fouriertype integrals can be transformed into the integrals on 0, +∞), the integrands of which are not ...
Keywords:
65D30, 65D32, Hankel function, Error analysis, GaussLaguerre quadrature rule, Oscillatory integral
6
May 2015
Applied Mathematics and Computation: Volume 259 Issue C, May 2015
Publisher: Elsevier Science Inc.
An optimal quadrature formula in the sense of Sard in the Hilbert space K 2 ( P m ) is constructed. New optimal quadrature formula of such a type and explicit expressions for the corresponding optimal coefficients are obtained using S.L. Sobolev's method. The obtained optimal quadrature formula is exact ...
Keywords:
Optimal coefficients, Optimal quadrature formulas, Error functional, Hilbert space, Extremal function
7
April 2015
Journal of Computational and Applied Mathematics: Volume 278 Issue C, April 2015
Publisher: Elsevier Science Publishers B. V.
An efficient method for constructing a class of generalized quadrature formulae of Gaussian type on (  1 , 1 ) for integrands having logarithmic singularities is developed. That kind of singular integrals are very common in the boundary element method. Several special cases for n point quadratures, which are ...
Keywords:
33C45, Orthogonalization, 65D30, 65D32, Gaussian quadrature, Singular integrals, 41A55, Boundary element method, Numerical integration
8
December 2014
Applied Mathematics and Computation: Volume 248 Issue C, December 2014
Publisher: Elsevier Science Inc.
The purpose of this paper is to present a systematic study of some extended special functions like B b ; � , λ α , β x , y , 2 F 1 α , β ; � , λ z ; b and p F q α , β ...
Keywords:
Extended beta function, Extended generalized hypergeometric function, Fractional integral, Extended Gauss hypergeometric function, Hfunction, Laguerre polynomial
9
June 2014
Calcolo: a quarterly on numerical analysis and theory of computation: Volume 51 Issue 2, June 2014
Publisher: SpringerVerlag New York, Inc.
Using S.L. Sobolev's method, we construct the interpolation splines minimizing the seminorm in $$K_2(P_2)$$ , where $$K_2(P_2)$$ is the space of functions $$\phi $$ such that $$\phi ^{\prime } $$ is absolutely continuous, $$\phi ^{\prime \prime } $$ belongs to $$L_2(0,1)$$ and $$\int _0^1(\varphi ^{\prime \prime }(x)+\varphi (x))^2dx<\infty $$ . ...
Keywords:
Hilbert space, S.L. Sobolev's method, 41A15, A seminorm minimizing property, Discrete argument function, Coefficients of interpolation splines, Discrete analogue of a differential operator, Interpolation splines, MSC 41A05
10
December 2013
Journal of Global Optimization: Volume 57 Issue 4, December 2013
Publisher: Kluwer Academic Publishers
In this paper we consider properties and power expressions of the functions $$f:(1,1)\rightarrow \mathbb{R }$$ and $$f_L:(1,1)\rightarrow \mathbb{R }$$ , defined by $$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{1}^1 \frac{(1+xt)^\gamma }{\sqrt{1t^2}}\,\mathrm{d}t \quad \text{ and}\quad f_L(x;\gamma )=\frac{1}{\pi }\int \limits _{1}^1 \frac{(1+xt)^\gamma \log (1+x t)}{\sqrt{1t^2}}\,\mathrm{d}t, \end{aligned}$$ respectively, where $$\gamma $$ is a real ...
Keywords:
26D07, 41A10, Approximation, Minimum, 26D15, 49K05, Expansion, Gamma function, Hypergeometric function, Maximum, 33C05, Digamma function, 41A17, Turán type inequality
11
November 2013
Applied Mathematics and Computation: Volume 224, November, 2013
Publisher: Elsevier Science Inc.
The paper deals with threeterm recurrence relations for Boubaker and related polynomials, as well as some properties including zero distribution of such kinds of polynomials. Also, an application of these polynomials for obtaining approximate analytical solution of Love's integral equation is presented. This Fredholm integral equation of the second kind ...
Keywords:
Boubaker polynomials, Chebyshev polynomials, Fredholm integral equation, Love's integral equation, Recurrence relation, Zeros
12
March 2013
Applied Mathematics and Computation: Volume 219 Issue 14, March, 2013
Publisher: Elsevier Science Inc.
The paper deals with the approximation of the solution of the following bivariate Fredholm integral equationf(y) [email protected]@!"DK(x,y)f(x)@w~(x)dx=g(y), [email protected]?D,where the domain D is a triangle. The proposed procedure, by a suitable transformation, is essentially the Nystrom method based on the zeros of univariate Jacobi orthogonal polynomials. Convergence, stability and well conditioning of the ...
Keywords:
Projection methods, Fredholm integral equations, Nyström method
13
August 2011
Numerical Algorithms: Volume 57 Issue 4, August 2011
Publisher: SpringerVerlag New York, Inc.
In this paper we construct an optimal quadrature formula in the sense of Sard in the Hilbert space K 2 ( P 2 ). Using S.L. Sobolev's method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order ...
Keywords:
Extremal function, Optimal coefficients, Optimal quadrature formulas, Error functional, 65D32, Hilbert space
14
March 2011
Mathematical and Computer Modelling: An International Journal: Volume 53 Issue 56, March, 2011
Publisher: Elsevier Science Publishers B. V.
A weighted quadrature rule of interpolatory type is represented as @!"a^bf(x)w(x) [email protected]?k=0nw"kf(x"k)+R"n"+"1[f], where w(x) is a weight function, {x"k}"k"="0^n are integration nodes, {w"k}"k"="0^n are the corresponding weight coefficients, and R"n"+"1[f] denotes the error term. During the past decades, various kinds of formulae of the above type have been developed. In this ...
Keywords:
Newton interpolation, qbinomial theorem, Weighted quadrature rules, Geometric nodes, Node polynomials
15
January 2010
Applied Numerical Mathematics: Volume 60 Issue 12, January, 2010
Publisher: Elsevier Science Publishers B. V.
Keywords:
secondary, Error bound, GaussTurán quadrature formula, Remainder term for analytic functions, primary, Contour integral representation
16
December 2009
Journal of Computational and Applied Mathematics: Volume 233 Issue 3, December, 2009
Publisher: Elsevier Science Publishers B. V.
For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points +/1 and the sum of semiaxes @r>1 for the Chebyshev weight functions of ...
Keywords:
Remainder term for analytic functions, primary, Contour integral representation, Chebyshev weight function, Gaussian quadrature formula, secondary, Error bound
17
April 2009
Numerische Mathematik: Volume 112 Issue 3, April 2009
Publisher: SpringerVerlag New York, Inc.
In this paper we consider interpolatory quadrature formulae with multiple nodes, which have the maximal trigonometric degree of exactness. Our approach is based on a procedure given by Ghizzeti and Ossicini (Quadrature formulae, AcademieVerlag, Berlin, 1970). We introduce and consider the socalled σorthogonal trigonometric polynomials of semiinteger degree and give ...
Keywords:
42A05, 65D32
18
February 2009
Numerical Analysis and Its Applications: 4th International Conference, NAA 2008, Lozenetz, Bulgaria, June 1620, 2008. Revised Selected Papers
Publisher: SpringerVerlag
In this paper we study the numerical integration on ( − 1,1) with respect to the Jacobi weight function (1 − x) α (1 + x) β , where α and β are complex parameters. The problem arises in some applications of computational models in quantum mechanics. We discuss two methods for integration. One is suitable ...
19
December 2008
Computers & Mathematics with Applications: Volume 56 Issue 11, December, 2008
Publisher: Pergamon Press, Inc.
Quadrature rules with maximal even trigonometric degree of exactness are considered. We give a brief historical survey on such quadrature rules. Special attention is given on an approach given by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (3) (2005) 337359. Translation ...
Keywords:
Quadrature rules of Gaussian type, Trigonometric degree of exactness, Nodes, Orthogonality, Recurrence relation, Trigonometric interpolation, Weights
20
October 2008
The classical books on interpolation address numerous negative results, i.e., results on divergent interpolation processes, usually constructed over some equidistant systems of nodes. The authors present, with complete proofs, recent results on convergent interpolation processes, for trigonometric and algebraic polynomials of one real variable, not yet published in other textbooks ...

