Computation of the Riesz-Herglotz transform and its application to quadrature formulas over the unit circle

Summary A method is proposed for the computation of the Riesz-Herglotz transform. Numerical experiments show the effectiveness of this method. We study its application to the computation of integrals over the unit circle in the complex plane of analytic functions. This approach leads us to the integration by Taylor polynomials. On the other hand, with the goal of minimizing the quadrature error bound for analytic functions, in the set of quadrature formulas of Hermite interpolatory type, we found that this minimum is attained by the quadrature formula based on the integration of the Taylor polynomial. These two different approaches suggest the effectiveness of this formula. Numerical experiments comparing with other quadrature methods with the same domain of validity, or even greater such as Szeg? formulas, (traditionally considered as the counterpart of the Gauss formulas for integrals on the unit circle) confirm the superiority of the numerical estimations. This work was supported by the ministry of education and culture of Spain under contract PB96-1029.     

A survey of known and new cubature formulas for the unit disk

In addition to some new cubature formulas for the approximation of integrals over the unit disk, we present a survey of all known cubature formulas of algebraic degree for this region.     

An application of Szegő polynomials to the computation of certain weighted integrals on the real line

In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong Stieltjes weight functions. Several numerical experiments are finally carried out.     

Orthogonal polynomials and some q-beta integrals of Ramanujan

Two integrals of Ramanujan are used to define a q-analogue of the Euler beta integral on the real line and of the Cauchy beta-integral on the complex unit circle. Such integrals are connected to orthogonal, biorthogonal and Laurent polynomials. Explicit examples of Laurent orthogonal polynomials are given on the real line and on the circle.     

Representations of solutions of Lindblad equations by randomized Feynman formulas

Representations of solutions of Lindblad equations by randomized Feynman integrals over trajectories are obtained by averaging similar representations for solutions of stochastic Schrödinger equations (Schrödinger–Belavkin equations). An approach based on the application of Chernoff’s theorem is applied. First, (randomized) Feynman formulas approximating Feynman path integrals are obtained; these formulas contain integrals over finite Cartesian powers of the space of values of the functions over which the Feynman integrals are taken.     

Some unit square integrals

In this article, we prove some identities which allow us to evaluate some multiple unit square integrals. In our examples, we will give the value of some double and triple integrals. We then prove several classical integral formulas with the help of these identities and present others that seem to be new. Finally, we get double integrals for classical constants and different expressions for two Ramanujan’s integral formulas.     

Product Gauss quadrature rules vs. cubature rules in the meshless local Petrov–Galerkin method

A crucial point in the implementation of meshless methods such as the meshless local Petrov–Galerkin (MLPG) method is the evaluation of the domain integrals arising over circles in the discrete local weak form of the governing partial differential equation. In this paper we make a comparison between the product Gauss numerical quadrature rules, which are very popular in the MLPG literature, with cubature formulas specifically constructed for the approximation of an integral over the unit disk, but not yet applied in the MLPG method, namely the spherical, the circularly symmetrical and the symmetric cubature formulas. The same accuracy obtained with 64×64 points in the product Gauss rules may be obtained with symmetric quadrature formulas with very few points.     

Rotation invariant cubature formulas over the n-dimensional unit cube

We consider cubature formulas to approximate multivariate integrals that remain unchanged under the orientation preserving transformations of a cube. The use of invariant theory (Molien series, Reynold operator) for the construction of such cubature formulas is investigated. Some new cubature formulas for the unit cube that are obtained using this approach, are presented.     

Sums over Graphs and Integration over Discrete Groupoids

We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pull-back or push-forward formulas for integrals over suitable groupoids.     

Sequences of orthogonal Laurent polynomials,bi-orthogonality and quadrature formulas on the unit circle

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.     

Sequences of orthogonal Laurent polynomials,bi-orthogonality and quadrature formulas on the unit circle

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.     

On Taylor coefficients of Cauchy-type integrals of finite complex measures

Boundary values of Cauchy-type integrals of finite complex measures given on a unit circle, generally speaking, are not Lebesgue integrable, and therefore at expansion of Cauchy-type integrals in Taylor series, the expansion coefficients cannot be expressed by boundary values using the Lebesgue integral. In this paper, using the notion of A-integration and N-integration, we get a formula for calculating the Taylor expansion coefficients of Cauchy-type integrals of finite complex measures.     

Derivatives of probability functions and some applications

Probability functions depending upon parameters are represented as integrals over sets given by inequalities. New derivative formulas for the intergrals over a volume are considered. Derivatives are presented as sums of integrals over a volume and over a surface. Two examples are discussed: probability functions with linear constraints (random right-hand sides), and a dynamical shut-down problem with sensors.     

A matricial computation of rational quadrature formulas on the unit circle

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.     

逻辑度量空间

取赋值格为［０,１］,用积分工具引入了公式的真度和相似度的概念,并从而在公式集Ｆ上建立了伪距离,为近似推理提供了一种可能的框架．     

Quadrature formulas on the unit circle based on rational functions

Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szeg polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.     

On an algorithm for calculating diffraction integrals

An algorithm for calculating integrals of rapidly oscillating functions given on a smooth two-dimensional surface is proposed. The surface is approximated by a collection of flat triangles with the values of the integrand known at their vertices. These values are used as reference ones to extend the function to other points of a triangle. The integral of the extended function over the surface of a triangle is calculated exactly. The desired value of the full diffraction integral is determined as the sum of the integrals calculated over the surfaces of all triangles. The resulting formulas for integral calculation involve singularities (indeterminate forms). Much attention is given to representations of these formulas in such a way that the indeterminate forms are automatically evaluated. Numerical results are presented.     

Unit Circle Elliptic Beta Integrals

We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations. 2000 Mathematics Subject Classification: Primary—11F50, 11L05, 33D05, 33D67     

Integral geometry of tensor valuations

We prove a complete set of integral geometric formulas of Crofton type (involving integrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Minkowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger's general integral geometric theorem, the Crofton formulas yield also kinematic formulas for Minkowski tensors. The explicit calculations of integrals over affine Grassmannians require several integral geometric and combinatorial identities. The latter are derived with the help of Zeilberger's algorithm.     

Equivariant Tautological Integrals on Flag Varieties

The author apples the Atiyah-Bott-Berline-Vergne formula to the equivariant tautological integrals over flag varieties of types A, B, C, D, and recovers the formulas expressing the integrals as iterated residues at infinity, which were first obtained by Zielenkiewicz using symplectic reduction.