Showing papers on "Trigonometric interpolation published in 2013"

Interpolation and Approximation by Polynomials

Abstract: * Preface * Univariate Interpolation * Best Approximation * Numerical Integration * Peano's Theorem and Applications * Multivariate Interpolation * Splines * Bernstein Polynomials * Properties of the q-integers * References * Index *

Reconstructing Hyperbolic Cross Trigonometric Polynomials by Sampling along Rank-1 Lattices

Abstract: With given Fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a one-dimensional discrete Fourier transform. In many applications one is also interested in the reconstruction of the Fourier coefficients from samples in the spatial domain. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses. In addition, we suggest approaches for determining suitable rank-1 lattices using a component-by-component algorithm. We present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100.

Approximating optimal point configurations for multivariate polynomial interpolation

Abstract: Efficient and effective algorithms are designed to compute the coordinates of nearly optimal points for multivariate polynomial interpolation on a general geometry. “Nearly optimal” refers to the property that the set of points has a Lebesgue constant near to the minimal Lebesgue constant with respect to multivariate polynomial interpolation on a finite region. The proposed algorithms range from cheap ones that produce point configurations with a reasonably low Lebesgue constant, to more expensive ones that can find point configurations for several two-dimensional shapes which have the lowest Lebesgue constant in comparison to currently known results.

Interpolation in the Hardy space

Abstract: We introduce an interpolation formula for functions in the Hardy space on the right-half plane and prove its convergence in norm and pointwise under very general condition. We also obtain an inverse formula for the Laplace transform from data on a finite interval.

Padé---type rational and barycentric interpolation

Abstract: In this paper, we consider the particular case of the general rational Hermite interpolation problem where only the value of the function is interpolated at some points, and where the function and its first derivatives agree at the origin. Thus, the interpolants constructed in this way possess a Pade---type property at 0. Numerical examples show the interest of the procedure. The interpolation procedure can be easily modified to introduce a partial knowledge on the poles and the zeros of the function to approximated. A strategy for removing the spurious poles is explained. A formula for the error is proved in the real case. Applications are given.

A Global Interpolator With Low Sample Rate for Multilevel Fast Multipole Algorithm

Abstract: A new, improved version of a global interpolator utilizing trigonometric polynomials is presented for the high-frequency multilevel fast multipole algorithm. The number of required points to sample the outgoing and incoming field patterns is low, almost half in some levels, compared with the earlier published versions. Compared with local interpolators based on Lagrange interpolating polynomials, the proposed technique performs even more favorably and reduces the number of sample points by a factor of eight. The numerical examples demonstrate that the interpolator allows full numerical accuracy control during the aggregation and disaggregation phases, regardless of the number of the levels in the octree.

New limiting real interpolation methods and their connection with the methods associated to the unit square

Abstract: We study limit K-spaces for general Banach couples, not necessarily ordered. They correspond to the extreme choice θ = 0, 1 in the realization of the real method as a K-space. We also show the connection of these limit spaces with interpolation methods defined by the unit square.

A novel Interpolation technique to address the Edge-Reliability Problem

Abstract: In this paper we deal with the Edge-Reliability Problem: given a connected simple graph G = (V, E) with perfect nodes and links failing independently with equal probability 1-p, find the probability RG(p) that the network remains connected. The edge reliability RG(p) is a polynomial in p with degree m = |E|, and the exact computation of RG(p) is in the class of NP-Hard problems. However, the related literature is vast, and offers bounds and estimation techniques, as well as exponential algorithms to exactly find that polynomial. We propose a novel interpolation-based technique that exploits the structure of the Hilbert space L2[0; 1], and combines the efficiency of Newton interpolation with the simplicity of Monte Carlo reliability estimation in selected abscissas in [0; 1]. The aim is to guide the polynomial search respecting algebraic properties of the target polynomial coefficients. We illustrate the effectiveness of the algorithm in the lights of a naive graph with limited size, and discuss several hints for future work.

Adaptive trigonometric hermite wavelet finite element method for structural analysis

Abstract: Owing to its good approximation characteristics of trigonometric functions and the multi-resolution local characteristics of wavelet, the trigonometric Hermite wavelet function is used as the element interpolation function. The corresponding trigonometric wavelet beam element is formulated based on the principle of minimum potential energy. As the order of wavelet can be enhanced easily and the multi-resolution can be achieved by the multi-scale of wavelet, the hierarchical and multi-resolution trigonometric wavelet beam element methods are proposed for the adaptive analysis. Numerical examples have demonstrated that the aforementioned two methods are effective in improving the computational accuracy. The trigonometric wavelet finite element method (WFEM) proposed herein provides an alternative approach for improving the computational accuracy, which can be tailored for the problem considered.

Rate of Convergence of Hermite-Fejér Interpolation on the Unit Circle

Abstract: The paper deals with the order of convergence of the Laurent polynomials of Hermite-Fejer interpolation on the unit circle with nodal system, the roots of a complex number with modulus one. The supremum norm of the error of interpolation is obtained for analytic functions as well as the corresponding asymptotic constants.

Two-dimensional fast generalized Fourier interpolation of seismic records

Abstract: The fast generalized Fourier transform algorithm is extended to two-dimensional data cases. The algorithm provides a fast and non-redundant alternative for the simultaneous time-frequency and space-wavenumber analysis of data with time-space dependencies. The transform decomposes data based on local slope information and therefore making it possible to extract the weight function based on dominant dips from alias-free low frequencies. By projecting the extracted weight function to alias-contaminated high frequencies and utilizing a least-squares fitting algorithm, a beyond-alias interpolation method is accomplished. Synthetic and real data examples are provided to examine the performance of the proposed interpolation method.

On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation

Abstract: We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice.

Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data in Banach Spaces

Abstract: A class of cardinal basis functions is proposed in order to achieve a generalization to Banach spaces of Hermite-Birkhoff interpolation on arbitrarily distributed data. First, a constructive characterization of the class of cardinal basis functions is given. Then, the interpolation problem is solved by using a suitable combination of such functions and Taylor-Frechet expansions. The performance of the obtained interpolants is improved by applying a localizing scheme, and the corresponding approximation error is estimated. A noteworthy case in Hilbert spaces and a numerical test comparing the Hermite-Birkhoff and Lagrange interpolants complete the presentation.

Some new perspectives in best approximation and interpolation of random data

Abstract: A new notion of universally optimal experimental design is introduced, relevant from the perspective of adaptive nonparametric estimation. It is demonstrated that both discrete and continuous Chebyshev designs are universally optimal in the problem of fitting properly weighted algebraic polynomials to random data. The result is a direct consequence of the well-known relation between Chebyshev’s polynomials and the trigonometric functions. Optimal interpolating designs in rational regression proved particularly elusive in the past. The question can be effectively handled using its connection to elliptic interpolation, in which the ordinary circular sinus, appearing in the classical trigonometric interpolation, is replaced by the Abel-Jacobi elliptic sinus sn(x, k) of a modulus k. First, it is demonstrated that — in a natural setting of equidistant design — the elliptic interpolant is never optimal in the so-called normal case k ∈ (−1, 1), except for the trigonometric case k = 0. However, the equidistant elliptic interpolation is always optimal in the imaginary case k ∈ iℝ. Through a relation between elliptic and rational functions, the result leads to a long sought optimal design, for properly weighted rational interpolants. Both the poles and nodes of the interpolants can be conveniently expressed in terms of classical Jacobi’s theta functions.

Interpolation of uniformly absolutely continuous operators

Abstract: We develop a method suitable for interpolation of uniformly absolutely continuous operators. We then apply this method to establishing compactness of operators and embeddings especially in the limiting situations, where the classical interpolation methods fail. We study types of certain interpolation methods and their sharpness.

The Cell Method: quadratic interpolation with tetrahedra for 3D scalar fields

Abstract: The Cell Method (CM) is a numerical method to solve field equations starting from its direct algebraic formulation. For two-dimensional problems it has been demonstrated that using simplicial elements with an affine interpolation, the CM obtains the same fundamental equation of the Finite Element Method (FEM); using the quadratic interpolation functions, the fundamental equation differs depending on how the dual cell is defined. In spite of that, the CM can still provide the same convergence rate obtainable with the FEM. Particularly, adopting a uniform triangulation and basing the dual cells on the Gauss points of the primal edges, the CM is able to reach the 4th order of convergence. In this note the use of quadratic interpolation to solve the Laplace equation in threedimensional problems is presented, adopting as primal cell the 10-nodes tetrahedron. A convergence analysis demonstrates that, as in two-dimensional case, the CM with quadratic interpolation can obtain the fourth order of convergence in solving the Laplace equation.

Model-Based Calibration of Filter Imperfections in the Random Demodulator for Compressive Sensing

Abstract: The random demodulator is a recent compressive sensing architecture providing efficient sub-Nyquist sampling of sparse band-limited signals. The compressive sensing paradigm requires an accurate model of the analog front-end to enable correct signal reconstruction in the digital domain. In practice, hardware devices such as filters deviate from their desired design behavior due to component variations. Existing reconstruction algorithms are sensitive to such deviations, which fall into the more general category of measurement matrix perturbations. This paper proposes a model-based technique that aims to calibrate filter model mismatches to facilitate improved signal reconstruction quality. The mismatch is considered to be an additive error in the discretized impulse response. We identify the error by sampling a known calibrating signal, enabling least-squares estimation of the impulse response error. The error estimate and the known system model are used to calibrate the measurement matrix. Numerical analysis demonstrates the effectiveness of the calibration method even for highly deviating low-pass filter responses. The proposed method performance is also compared to a state of the art method based on discrete Fourier transform trigonometric interpolation.

Eigenvalue Methods for Interpolation Bases

Abstract: This thesis investigates eigenvalue techniques for the location of roots of polynomials expressed in the Lagrange basis. Polynomial approximations to functions arise in almost all areas of computational mathematics, since polynomial expressions can be manipulated in ways that the original function cannot. Polynomials are most often expressed in the monomial basis; however, in many applications polynomials are constructed by interpolating data at a series of points. The roots of such polynomial interpolants can be found by computing the eigenvalues of a generalized companion matrix pair constructed directly from the values of the interpolant. This affords the opportunity to work with polynomials expressed directly in the interpolation basis in which they were posed, avoiding the often ill-conditioned transformation between bases. Working within this framework, this thesis demonstrates that computing the roots of polynomials via these companion matrices is numerically stable, and the matrices involved can be reduced in such a way as to significantly lower the number of operations required to obtain the roots. Through examination of these various techniques, this thesis offers insight into the speed, stability, and accuracy of rootfinding algorithms for polynomials expressed in alternative bases.

Adaptive Trigonometric Wavelet Composite Beam Element Method

Abstract: It is always an issue to use as few finite elements as possible to reduce a computational load with the precision assured to solve engineering problems. In this paper, the trigonometric wavelet function and polynomial function are combined together to be the displacement interpolation function of beam element. Accordingly, the trigonometric wavelet composite beam element is formulated based on the principle of potential energy minimum to carry out the bending, free vibration and buckling analysis of beam structures. Such a trigonometric wavelet composite beam element utilizes the advantages of both conventional finite element and trigonometric wavelet finite element. As the order of wavelet function can be enhanced easily and the multi-resolution can be achieved by the multi-scales of wavelet function, the trigonometric wavelet composite finite element provides an alternate way to realize the adaptive analysis. Numerical examples have illustrated that the proposed trigonometric wavelet composite beam elem...

Polynomial and Rational Interpolation

Abstract: This chapter gives a detailed discussion of barycentric Lagrange and Hermite interpolation and extends this to rational interpolation with a specified denominator. We discuss the conditioning of these interpolants. A numerically stable method to find roots of polynomials expressed in barycentric form via a generalized eigenvalue problem is given. We conclude with a section on piecewise polynomial interpolants. ⊲

On some optimizations of trigonometric interpolation using Fourier discrete coefficients

Abstract: We investigate convergence of the rational-trigonometric-polynomial interpolations which perform convergence acceleration of the classical trigonometric interpolation by sequential application of polynomial and rational corrections. Rational corrections contain unknown parameters which determination outlines the behavior of the interpolations in different frameworks. We consider approach for determination of the unknown parameters by minimization of the constants of the asymptotic errors. We perform theoretical and numerical analysis of such optimal interpolations.

On a convergence of the Fourier-Pade interpolation

Abstract: We investigate convergence of the rational-trigonometric-polynomial interpolation that performs convergence acceleration of the classical trigonometric interpolation by sequential application of polynomial and rational correction functions. Unknown parameters of the rational corrections are determined along the ideas of the Fourier-Pade approximations. The resultant interpolation we call as Fourier-Pade interpolation and investigate its convergence in the regions away from singularities. Comparison with other rational-trigonometric-polynomial interpolations outlines the convergence properties of the Fourier-Pade interpolation.

Explicit interpolation bounds between hardy space and

Abstract: Abstract Every bounded linear operator that maps ${H}^{1} $ to ${L}^{1} $ and ${L}^{2} $ to ${L}^{2} $ is bounded from ${L}^{p} $ to ${L}^{p} $ for each $p\\in (1, 2)$, by a famous interpolation result of Fefferman and Stein. We prove ${L}^{p} $-norm bounds that grow like $O(1/ (p- 1))$ as $p\\downarrow 1$. This growth rate is optimal, and improves significantly on the previously known exponential bound $O({2}^{1/ (p- 1)} )$. For $p\\in (2, \\infty )$, we prove explicit ${L}^{p} $ estimates on each bounded linear operator mapping ${L}^{\\infty } $ to bounded mean oscillation ($\\mathit{BMO}$) and ${L}^{2} $ to ${L}^{2} $. This $\\mathit{BMO}$ interpolation result implies the ${H}^{1} $ result above, by duality. In addition, we obtain stronger results by working with dyadic ${H}^{1} $ and dyadic $\\mathit{BMO}$. The proofs proceed by complex interpolation, after we develop an optimal dyadic ‘good lambda’ inequality for the dyadic $\\sharp $-maximal operator.

Bivariate Symmetry Associated Continued Fractions Blending Rational Interpolation

Abstract: By introducing partial divided differences and partial inverse differences, bivariate symmetry associated continued fractions blending rational interpolation is constructed. We discuss the recursive algorithm, interpolation theorem and error estimation. We extend the conclusion to vector valued, matrix valued cases and the triangular net case.