Convergence Of Product Integration Rules Research Articles

L∞ convergence of interpolation and associated product integration for exponential weights

We investigate convergence in a weighted L ∞ norm of Hermite–Fejér, Hermite, and Grünwald interpolations at zeros of orthogonal polynomials with respect to exponential weights such as Freud, Erdős, and exponential weight on (−1,1). Convergence of product integration rules induced by the various approximation processes is deduced. We also give more precise weight conditions for convergence of interpolations with respect to above three types of weights, respectively.

Convergence of product integration rules over (0,∞) for functions with weak singularities at the origin

In this paper we consider integrals of the form\[∫0∞e−xK(x,y)f(x)dx,\int _0^\infty {{e^{ - x}}K(x,y)f(x)dx,}\]withf∈Cp[0,∞)∩Cq(0,∞),q≥p≥0f \in {C^p}[0,\infty ) \cap {C^q}(0,\infty ),q \geq p \geq 0, andxif(p+i)(x)∈C[0,∞),i=1,…,q−p{x^i}{f^{(p + i)}}(x) \in C[0,\infty ),i = 1, \ldots ,q - p, whenq>pq > p. They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nyström method. To discretize the integral above, we propose to use a product rule of interpolatory type based on the zeros of Laguerre polynomials. For this rule we derive (weighted) uniform convergence estimates and present some numerical examples.

Convergence of Product Integration Rules Over (0, ∞) for Functions with Weak Singularities at the Origin

In this paper we consider integrals of the form ∫ 0 ∞ e −x K(x, y) f(x) dx, with f ∈ C p [0, ∞) ∩ C q (0, ∞), q ≥ p ≥ 0, and x i f (p+1) (x) ∈ C[0, ∞), i = 1,..., q − p, when q > p. They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nystrom method. To discretize the integral above, we propose to use a product rule of interpolatory type based on the zeros of Laguerre polynomials. For this rule we derive (weighted) uniform convergence estimates and present some numerical examples

Hermite-Fejér-Related Interpolation and Product Integration

Convergence of product integration rules based on Hermite-Fejér interpolation with end conditions is shown for all Riemann-integrable functions when the interpolation points are zeros of generalized Jacobi polynomials even in cases where the corresponding Hermite-Fejér operator is indefinite.

Hermite and Hermite-Fejér interpolation and associated product integration rules on the real line: The L∞ theory

We investigate convergence in a weighted L ∞-norm of Hermite-Fejér and Hermite interpolation and related approximation processes, when the interpolation points are zeros of orthogonal polynomials associated with weights W 2 = e −2 Q on the real line. For example, if H n(W 2,ƒ,x) denotes the nth Hermite-Fejér interpolation polynomial for W 2 = e −2 Q and the function ƒ, then we show that lim n→∞ xϵR { sup|H n(W 2, f, x)−f(x)| W 2(x)[1+|Q′(x)|] −k(1+|x|) −1}=0 under suitable conditions on ƒ, W 2, and κ. The weights to which the results are applicable include W 2(x) = exp(−¦x¦ α) , α > 1, or W 2(x) = exp(− exp k(¦x¦ α)) , α > 1, k ⩾ 1, where exp k denotes the kth iterated exponential. Convergence of product integration rules induced by the various approximation processes is then deduced. Essentially the conclusion of the paper is that by damping the error in approximation of ƒ Hermite-Fejér or Hermite interpolation by a factor [1 + ¦Q′(x)¦] −κ(1 + ¦x¦) −1 , which decays much more slowly than the weight W 2, we can ensure sup-norm convergence under quite general conditions.

Product integration of singular integrands based on cubic spline interpolation at equally spaced nodes

In this paper the convergence of product integration rules, based on cubic spline interpolation at equally spaced nodes, with "not-a-knot" end condition, is investigated for integrand functions with a interior or endpoint singularity in the integration interval.

Mean convergence of Lagrange interpolation for Freud's weights with application to product integration rules

The connection between convergence of product integration rules and mean convergence of Lagrange interpolation in L p (1 < p < ∞) has been thoroughly analysed by Sloan and Smith [37]. Motivated by this connection, we investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials associated with Freud weights on R . Our results apply to the weights exp(− x m /2), m = 2, 4, 6 …, and for the Hermite weight ( m = 2) extend results of Nevai [28] and Bonan [2] in at least one direction. The results are sharp in L p , 1 < p ⩽ 2. As a consequence, we can improve results of Smith, Sloan and Opie [38] on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the Hermite weight. In the process, we prove a new Markov-Stieltjes inequality for Gauss quadrature sums, and solve a problem of Nevai on how to estimate certain quadrature sums.

The convergence of interpolatory product integration rules

Conditions are given for the convergence of product integration rules based on the zeros of orthogonal polynomials associated with a generalized smooth Jacobi weight, possibly augmented by one or both endpoints.

Convergence of product integration rules for functions with interior and endpoint singularities over bounded and unbounded intervals

The convergence of product integration rules, based on Gaussian quadrature points, is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals. The investigation is based on a new convergence result for Lagrangian interpolation and Gaussian quadrature of singular integrands.

Convergence of Product Integration Rules for Functions With Interior and Endpoint Singularities Over Bounded and Unbounded Intervals

The convergence of product integration rules, based on Gaussian quadrature points, is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals. The investigation is based on a new convergence result for Lagrangian interpolation and Gaussian quadrature of singular integrands.

The convergence of product integration rules

A recent theorem due to Nevai on the mean convergence of Lagrange interpolation is used to obtain sufficient conditions for the convergence of quadrature rules for product integration.