On an interpolatory product rule for evaluating Cauchy principal value integrals

Convergence results for interpolatory product rules for evaluating Cauchy principal value integrals of the form f _?1 ¹ v(x)f(x)/x ? λ dx wherev is an admissible weight function have been extended to integrals of the form f _?1 ¹ k(x)f(x)/x ? λ dx wherek is an arbitrary integrable function subject to certain conditions. Further, whereas the above convergence results were shown when the interpolation points were the Gauss points with respect to some admissible weight functionw, they are now shown to hold when the interpolation points are Radau or Lobatto points with respect tow.     

Uniform approximations to Cauchy principal value integrals of oscillatory functions

This paper presents an interpolatory type integration rule for the numerical evaluation of Cauchy principal value integrals of oscillatory integrands , where -1<τ<1, for a given smooth function f(x). The proposed method is constructed by interpolating f(x) at practical Chebyshev points and subtracting out the singularity. A numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast Fourier transform. The validity of the method has been demonstrated by several numerical experiments.     

A Dini type test on the pointwise convergence of double Fourier integrals

We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued function f ∈ L ¹(?²) with bounded support at a given point (x ₀,y ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functions f(x,y ₀), x ∈ ?, and f(x ₀,y), y ∈ ?, at the points x:= x ₀ and y:= y ₀, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.     

A product quadrature algorithm by Hermite interpolation

This paper is concerned with numerical integration of ∫¹₋₁f(x)k(x)dx by product integration rules based on Hermite interpolation. Special attention is given to the kernel k(x) = e^iτx, with a view to providing high precision rules for oscillatory integrals. Convergence results and error estimates are obtained in the case where the points of integration are zeros of p_n(W; x) or of (1 − x²)p_n−2(W; x), where p_n(W; x), n = 0, 1, 2…, are the orthonormal polynomials associated with a generalized Jacobi weight W. Further, examples are given that test the performance of the algorithm for oscillatory weight functions.     

Pringsheim type tests on the pointwise convergence of integrals conjugate to double fourier integrals

We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued function f ∈ L ¹ (?²) with bounded support at a given point (x ₀, g ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier integral of the marginal functions f(x, y ₀), x ∈ ?, and f(x ₀, y), y ∈ ?, at x:= x ₀ and y:= y ₀, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.     

Rational interpolation to solutions of Riccati difference equations on elliptic lattices

It is shown how to define difference equations on particular lattices {x_n}, n∈Z, where the x_ns are values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special difference equations (elliptic Riccati equations) have remarkable simple (!) interpolatory continued fraction expansions.     

Whitney theorem of interpolatory type for k-monotone functions

One of the main results of this paper is the following Whitney theorem of interpolatory type for k-monotone functions (i.e., functions f such that divided differences f[x ₀,…, x _k ] are nonnegative for all choices of (k + 1) distinct points x ₀,…, x _k .     

On the numerical characterization of the reachability cone for an essentially nonnegative matrix

Given an n×n real matrix A with nonnegative off-diagonal entries, the solution to , x₀=x(0), t?0 is x(t)=e^tAx₀. The problem of identifying the initial points x₀ for which x(t) becomes and remains entrywise nonnegative is considered. It is known that such x₀ are exactly those vectors for which the iterates x^(k)=(I+hA)^kx₀ become and remain nonnegative, where h is a positive, not necessarily small parameter that depends on the diagonal entries of A. In this paper, this characterization of initial points is extended to a numerical test when A is irreducible: if x^(k) becomes and remains positive, then so does x(t); if x(t) fails to become and remain positive, then either x^(k) becomes and remains negative or it always has a negative and a positive entry. Due to round-off errors, the latter case manifests itself numerically by x^(k) converging with a relatively small convergence ratio to a positive or a negative vector. An algorithm implementing this test is provided, along with its numerical analysis and examples. The reducible case is also discussed and a similar test is described.     

Two-dimension periodic cardinal interpolatory wavelets

In this paper, we construct two-dimension periodic interpolatory scaling function and wavelets from a periodic function g(x₁, x₂), whose Fourier coefficients are positive, and obtain some properties of scaling functions and wavelets.     

Two-dimension periodic cardinal interpolatory wavelets

In this paper, we construct two-dimension periodic interpolatory scaling function and wavelets from a periodic function g(x₁, x₂), whose Fourier coefficients are positive, and obtain some properties of scaling functions and wavelets.     

Optimal cubature formulas for calculation of multidimensional integrals in weighted Sobolev spaces

Optimal cubature formulas are constructed for calculations of multidimensional integrals in weighted Sobolev spaces. We consider some classes of functions defined in the cube Ω = [-1, 1]^l, l = 1, 2,..., and having bounded partial derivatives up to the order r in Ω and the derivatives of jth order (r < j ≤ s) whose modulus tends to infinity as power functions of the form (d(x, Г))^-(j-r), where x ∈ Ω Г, x = (x₁,..., x_l), Г = ?Ω, and d(x, Г) is the distance from x to Г.     

A Family of Approximation Formulas and some Applications

The general scheme, suggested in [1] using a basis of an infinite-dimensional space and allowing to construct finite-dimensional orthogonal systems and interpolation formulas, is improved in the paper. This results particularly in a generalization of the well-known scheme by which periodic interpolatory wavelets are constructed. A number of systems which do not satisfy all the conditions for multiresolution analysis but have some useful properties are introduced and investigated.

Starting with general constructions in Hilbert spaces, we give a more careful consideration to the case connected with the classic Fourier basis.

Convergence of expansions which are similar to partial sums of the summation method of Fourier series, as well as convergence of interpolation formulas are considered.

Some applications to fast calculation of Fourier coefficients and to solution of integrodifferential equations are given. The corresponding numerical results have been obtained by means of MATHEMATICA 3.0 system.     

On the positivity of certain Cotes numbers

It is shown that the Cotes numbers are nonnegative for some interpolatory quadratic rules with nodes at the zeros of the ultraspherical polynomialsC _n , when integrating with respect to the weight functions w_µ(x)=(1–x²)^µ–1/2.     

Fractional integrals and weakly singular integral equations of the first kind in then-dimensional ball

The purpose of the paper is to introduce and to investigate a new class of fractional integrals connected with balls in ?ⁿ. A Riesz potentialI _Ω ^α ρ over a ball Ω is represented by a composition of such integrals. Using this representation we obtain necessary and sufficient solvability conditions for the equationI _Ω ^α ρ =f in the space L_p(Ωw) with a power weight w(x) and solve the equation in a closed form. The investigation is based on a special Fourier analysis adopted for operators commuting with rotations and dilations in ?ⁿ.     

Numerical analysis and orthogonal polynomials

Two topics of general interest are investigated. First an iterative improvement algorithm is given to reduce the accumulation of truncation errors which may occur when recursion formulae are utilized. Then some properties of orthogonal polynomials are derived that allow a successful construction of Gaussian type integration formulae employing the improvement algorithm. As special examples integrals of the ∫^b_aexp(-x²)f(x)dx and ∫^b_aexp(-|x|)f(x)dx are considered, where a and/o r b may be infinite.     

Fast integral equation solver for Maxwell’s equations in layered media with FMM for Bessel functions

The paper presents a new fast integral equation solver for Maxwell’s equations in 3-D layered media. First, the spectral domain dyadic Green’s function is derived, and the 0-th and the 1-st order Hankel transforms or Sommerfeld-type integrals are used to recover all components of the dyadic Green’s function in real space. The Hankel transforms are performed with the adaptive generalized Gaussian quadrature points and window functions to minimize the computational cost. Subsequently, a fast integral equation solver with O(N _z ² N _x N _y log(N _x N _y )) in layered media is developed by rewriting the layered media integral operator in terms of Hankel transforms and using the new fast multipole method for the n-th order Bessel function in 2-D. Computational cost and parallel efficiency of the new algorithm are presented.     

Product type quadrature formulas

This paper is concerned with the numerical approximation of integrals of the form _a ^b f(x)g(x)dx by means of a product type quadrature formula. In such a formula the functionf (x) is sampled at a set ofn+1 distinct points and the functiong(x) at a (possibly different) set ofm+1 distinct points. These formulas are a generalization of the classical (regular) numerical integration rules. A number of basic results for such formulas are stated and proved. The concept of a symmetric quadrature formula is defined and the connection between such rules and regular quadrature formulas is discussed. Expressions for the error term are developed. These are applied to a specific example.The work of the first author was supported in part by NIH Grant No. FRO 7129-01 and that of the second author in part by U.S. Army Ballistic Research Laboratories Contract DA-18-001-AMC-876 X.     

Error Estimates for Two Filters Based on Polynomial Interpolation for Recovering a Function from Its Fourier Coefficients

In this paper we derive error estimates for two filters based on piecewise polynomial interpolations of zeroth and first degrees. For a piecewise smooth function f(x) in [0,1], we show that, if all the discontinuity points of f(x) are nodes then, using these filters, we can reconstruct point values of f(x) accurately even near discontinuity points. If f(x) is a piecewise constant or a linear function, the reconstruction formulas are exact. We also propose reconstruction formulas such that we can compute the (approximate ) point values of f(x) using the fast Fourier transform, even when using non-uniform meshes. Several numerical experiments are also provided to illustrate the results.     

A new form of trigonometric orthogonality and Gaussian-type quadrature

Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(x_i), i = 1(1)2n+1, gives a trigonometric sum of the n^th order, S_2n+1(x = a₀ + ∑_jⁿ = 1^(a_jcos jx + b_jsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S_2r+1(x) is one that satisfies

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{2r+1}}\text{(x)S}{}_{\mathrm{2r\prime +1}}\text{(x)dx=0, r′<r}$

and two other arbitrarily imposable conditions needed to make S_2r1(x) unique. Two proofs are given of a fundamental factor theorem for any S_2n+1(x) (somewhat different from that for polynomials) from which we derive 2r-point Gaussian-type quadrature formulas, r = [n/2] + 1, which are exact for any S_4r?1(x). We have

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{4r?1(x)}}\text{dx=∑}{}_{\mathrm{j=1}}{}^{\mathrm{2r}}\text{A}{}_{\mathrm{j}}\text{S}{}_{\mathrm{4r?1}}\text{(x}{}_{\mathrm{j}}\text{)}$

where the nodes x_j, j = 1(1)2r, are the zeros of the orthogonal S_2r+1(x). It is proven that A_j > 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S_2r+1(x), x_jandA_j, j = 1(1)2r, were calculated for [a,b] = [0, π/4], 2r = 2 and 4, to single-precision accuracy.     

On Fourier Transforms of Radial Functions and Distributions

We find a formula that relates the Fourier transform of a radial function on R ⁿ with the Fourier transform of the same function defined on R ⁿ⁺². This formula enables one to explicitly calculate the Fourier transform of any radial function f(r) in any dimension, provided one knows the Fourier transform of the one-dimensional function t?f(|t|) and the two-dimensional function (x ₁,x ₂)?f(|(x ₁,x ₂)|). We prove analogous results for radial tempered distributions.