Showing papers on "Trigonometric interpolation published in 1992"
TL;DR: In this paper, the authors give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling, and give a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice.
Abstract: We give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice, and similarly, a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von Neumann lattice
282 citations
TL;DR: In this article, it was shown that there is no robustly convergent linear algorithm for identifying exponentially stable systems in the presence of noise which is not tuned to prior information about the unknown system or noise.
117 citations
TL;DR: In this article, it was shown that for radial basis functions h associated with functions whose mth derivative (modulo a scalar multiple) is completely monotonic, there exists a function h(x) + am + am − 1r2 + … + a1r2m − 2 gives rise to an invertible interpolation matrix, and bounds on the norm of the inverse of this matrix were obtained.
111 citations
TL;DR: In this article, the authors consider the problem of finding a space P ⊂ Π which is correct for each continuous linear functional on a subspace Λ of the dual Π′ of the space Π of svariate polynomials, where each continuous functional can be interpolated by a unique p ∈ P.
Abstract: We consider the following problem: given a subspace Λ of the dual Π′ of the space Π of svariate polynomials, find a space P ⊂ Π which is correct for Λ in the sense that each continuous linear functional on Λ can be interpolated by a unique p ∈ P . We provide a map, Λ 7→ Λ↓ ⊂ Π, which we call the least map, that solves this interpolation problem and give a comprehensive discussion of its properties. This least solution, Λ↓, is a homogeneous space and is shown to have minimal degree among all possible solutions. It is the unique minimal degree solution which is dual (in a natural sense) to all minimal degree solutions. It also interacts nicely with various maps applied to Λ, such as convolution, translation, change of variables, and, particularly, differentiation. Our approach is illustrated by detailed examples, concerning finite-dimensional Λ’s spanned by point-evaluations or line integrals. Methods which facilitate the identification of the least solution are established. The paper is complemented by [BR3], in which an algorithmic approach for obtaining Λ↓ is presented whose computational aspects are detailed. AMS (MOS) Subject Classifications: primary 41A05, 41A63, 41A10; secondary 13F20, 13F25, 13A15
107 citations
TL;DR: In this article, the Lagrange polynomial interpolation on the nodes of Gauss type quadrature formulas is studied and asymptotic error estimates of the same order as the distance to the best fit are derived.
107 citations
01 Apr 1992
TL;DR: In this article, the distribution of lattice points on arcs of circles centered at the origin was studied and it was shown that on such a circle of radius R, an arc whose length is smaller than V2Rl12-l(4[m/2]+2) contains, at most, m lattices points.
Abstract: In this paper we study the distribution of lattice points on arcs of circles centered at the origin. We show that on such a circle of radius R, an arc whose length is smaller than V2Rl12-l(4[m/2]+2) contains, at most, m lattice points. We use the same method to obtain sharp L4-estimates for uncompleted, Gaussian sums
45 citations
TL;DR: The DCT algorithms are shown to be more accurate and outperform algorithms based on the Wang modified DCT-I which was proposed to replace the DCT for interpolation purposes.
Abstract: Zooming and zero padding interpolation algorithms based on the conventional discrete cosine transform (DCT) are presented. The DCT algorithms are shown to be more accurate and outperform algorithms based on the Wang modified DCT-I which was proposed to replace the DCT for interpolation purposes.
42 citations
TL;DR: In this paper, the convergence of derivatives of Lagrange interpolation at the union of zeros of Jacobi polynomials and some additional points is investigated, and it is shown that the convergence is uniform.
40 citations
TL;DR: This paper provides a method using Householder transformations to construct matrices of trigonometric series which leads to multivariate wavelet decompositions.
Abstract: In this paper we provide a method using Householder transformations to construct matrices of trigonometric series which leads to multivariate wavelet decompositions.
36 citations
TL;DR: In this paper, Erdős and Turan studied the mean convergence of Lagrange interpolation on an extended set of nodes that includes, in addition to the n zeros of the orthagonal (relative to some positive weight function w) polynomial πn of degree n, other n+1 nodes, which in turn are zeros corresponding to the weight functions w n =π 2 n w.
35 citations
TL;DR: In this article, known polynomial interpolation methods were generalized to obtain new algorithms for the computation of the inverse of such matrices using numerically stabilizable manipulations of constant matrices.
Abstract: Known polynomial interpolation methods to polynomial matrices are generalized to obtain new algorithms for the computation of the inverse of such matrices. The algorithms use numerically stabilizable manipulations of constant matrices. Among the three methods investigated Lagrange's interpolation seems especially suitable for the purpose. >
TL;DR: In this paper, simple proofs are provided for two properties of a new multivariate polynomial interpolation scheme, due to Amos Ron and the author, and a formula for the interpolation error is derived and discussed.
TL;DR: It is found that, if the sum of the degrees of the irreducible curves on which the interpolation points are chosen is small compared to the degree of the interpolating polynomial, then the problem becomes singular.
Abstract: The authors derive a number of results on sufficient conditions under which the 2-D polynomial interpolation problem has a unique or nonunique solution. It is found that, if the sum of the degrees of the irreducible curves on which the interpolation points are chosen is small compared to the degree of the interpolating polynomial, then the problem becomes singular. Similarly, if there are too many points on any of the irreducible curves on which the interpolation points are chosen, then the interpolation problem runs into singularity. Examples of geometric distribution of interpolation points satisfying these conditions are shown. The examples include polynomial interpolation of polar samples, and samples on straight lines. The authors propose a recursive algorithm for computing 2-D polynomial coefficients for the nonsingular case where all the interpolation points are chosen on lines passing through the origin. The result is applied to the problem of nonuniform frequency sampling design for 2-D FIR filter design, and a few examples of such design are shown. >
TL;DR: The results show that of all the interpolation schemes via fast sinusoidal transforms, the proposed scheme is the most promising.
TL;DR: In this article, a method for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions is presented.
Abstract: We develop methods for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions which are uniform, in general. These sets are generated by building up a net of lines and by placing points on these lines which satisfy interlacing properties for univariate spline spaces.
TL;DR: In this paper, an orthogonal trigonometric basis in the space is constructed whose degrees have a growth rate of O(n 2 ) in the orthogonometric space.
Abstract: An orthogonal trigonometric basis in the space is constructed whose degrees have growth rate .
10 May 1992
TL;DR: In this article, a straightforward recursive and global approach for the solution of rational interpolation problems is described, based on a simple and well known matrix identity, namely, the Schur reduction procedure, and exploits connections with structured matrices.
Abstract: Describe a straightforward recursive and global approach for the solution of rational interpolation problems. The derivation is based on a simple and well known matrix identity, namely, the Schur reduction procedure, and exploits connections with structured matrices. The authors use the interpolation data to construct a convenient structure and then apply a recursive triangularization procedure. This leads to a transmission-line cascade of first-order J-lossless sections that makes evident the interpolation property. State-space descriptions for each section and for the entire cascade are given. >
27 Aug 1992
TL;DR: Subjective evaluation of the image interpolation technique aims at optimizing the parameter values of the algorithm, as well as comparing the new algorithm to existing interpolation techniques.
Abstract: Previous work at the Institute for Perception Research has resulted in a new model for representing images, called a polynomial transform. This transform is perceptually relevant since it mimics properties of the early stages of human vision such as localization and decomposition of luminance changes into specific basic patterns, i.e., localized polynomials. The transform also has interesting signal processing properties, some of which will be illustrated in this paper. Image interpolation is an example of how polynomial transforms can be used for image restoration. It is derived that the polynomial transform coefficients of a blurred and sampled image are related by a linear transformation to the polynomial transform coefficients of the original image. By inverting this transformation, we can obtain deblurring and interpolation. This inversion is based on the assumption that the image can be locally approximated by a low-order polynomial description. By adopting a fixed degree for this a priori polynomial description, we obtain non-adaptive interpolation algorithms. The performance of the algorithm can be further improved by varying the degree of this a priori polynomial description depending on whether the image region is locally uniform or non- uniform. Especially in the presence of noise, this adaptivity is usually very important. It is shown how such space-variant image processing can be easily described and implemented using polynomial transforms. Subjective evaluation of the image interpolation technique aims at optimizing the parameter values of the algorithm, as well as comparing the new algorithm to existing interpolation techniques. Some results of this evaluation are presented.
23 Mar 1992
TL;DR: Wavelet-based lowpass and bandpass interpolation schemes that are exact for certain classes of signals including polynomials of arbitrarily large degree are discussed and it turns out that the Fourier transform of the lowpass interpolatory function is also (a positive) interpolatoryfunction.
Abstract: Wavelet-based lowpass and bandpass interpolation schemes that are exact for certain classes of signals including polynomials of arbitrarily large degree are discussed. The interpolation technique is studied in the context of wavelet-Galerkin approximation of the shift operator. A recursive dyadic interpolation algorithm makes it an attractive alternative to other schemes. It turns out that the Fourier transform of the lowpass interpolatory function is also (a positive) interpolatory function. The nature of the corresponding interpolating class is not well understood. Extension to the case of multiplicity M orthonormal wavelet bases, where there is an efficient M-adic interpolation scheme, is also given. >
IBM1
TL;DR: A randomized algorithm is presented that interpolates a sparse polynomial in polynometric time in the bit complexity model and can be applied to approximate polynomials that can be approximated by sparse poynomials.
Abstract: We present a randomized algorithm that interpolates a sparse polynomial in polynomial time in the bit complexity model. The algorithm can be also applied to approximate polynomials that can be approximated by sparse polynomials (the approximation is in the L2 norm).
TL;DR: In this article, a new interpolation method for a 1D signal via the fast Hartley transform (FHT) is presented and a new 2D interpolation algorithm for a 2D signal by using the 1-D FHT is suggested.
Abstract: The revised interpolation method for a 1-D signal via the fast Hartley transform (FHT) is presented and a new interpolation algorithm for a 2-D signal by using the 1-D FHT is suggested. Results show that the proposed algorithm is superior to the FFT method both in accuracy and in computational time for interpolating discrete bandlimited 2-D signals.
TL;DR: In this paper, a method which combines quadrature with trigonometric interpolation for singular integral equations on closed curves is proposed, which is shown to be stable in L 2 provided the operator A is invertible in L 1.
Abstract: A method which combines quadrature with trigonometric interpolation is proposed for singular integral equations on closed curves. For the case of the circle, the present method is shown to be equivalent to the trigonometric ?-collocation method together with numerical quadrature for the compact term, and is shown to be stable inL 2 provided the operatorA is invertible inL 2. The results are extended to arbitraryC ? curves, to give a complete error analysis in the scale of Sobolev spacesH s . In the final section the case of a non-invertible operatorA is considered.
TL;DR: It is shown here that an alternative construction is possible, by developing an umbral interpolation formula which generalises the classical Newton forward interpolations formula, and results in a simple algorithmic procedure for computing χo(G;x).
TL;DR: For certain regular knot systems (geometric or regular meshes, tensor product grids), Neville-Aitken algorithms are derived explicitly and can be extended in a simple way to arbitrary (k+1)-pencil lattices as recently introduced by Lee and Phillips.
Abstract: This is the second part of a note on interpolation by real polynomials of several real variables. For certain regular knot systems (geometric or regular meshes, tensor product grids), Neville-Aitken algorithms are derived explicitly. By application of a projectivity they can be extended in a simple way to arbitrary (k+1)-pencil lattices as recently introduced by Lee and Phillips. A numerical example is given.
TL;DR: An exact and practical method for determining the number, location, and multiplicity of all real zeros of the trigonometric polynomials and an efficient method for the calculation of the coefficients of a corresponding algebraic polynomial is stated.
Abstract: An exact and practical method for determining the number, location, and multiplicity of all real zeros of the trigonometric polynomials is described. All computations can be performed without loss of accuracy. lThe method is based on zero isolation techniques for algebraic polynomials. An efficient method for the calculation of the coefficients of a corresponding algebraic polynomial is stated. The complexity of trigonometric zero isolation depending on the degree and the coefficient size of the given trigonometric polynomial is analyzed. In an experimental evaluation, the performance of the method is compared to the performance of recently developed numeric techniques for the approximate determination of all roots of trigonometric polynomials. The case of exponential or hyperbolic polynomials is treated in an appendix.
TL;DR: In this paper, a nonlinear transformation of the wave function variation coordinates along with the construction of a global interpolating function is presented, which is constructed for each MCSCF iteration in such a way that it reproduces certain known behavior of the exact energy function.
Abstract: A new method of MCSCF wave function optimization is presented. This method is based on a nonlinear transformation of the wave function variation coordinates along with the construction of a global interpolating function. This interpolating function is constructed for each MCSCF iteration in such a way that it reproduces certain known behavior of the exact energy function. It reproduces exactly the energy, gradient, and hessian at the expansion point, at an infinite number of isolated points, and at points on the surfaces of an infinite number of nested multidimensional balls within the wave function variational space. The optimization of the wave function correction parameters on this interpolating function does not require integral transformations or density matrix constructions, although one-index transformation and transition density matrix techniques may be used if desired. The nonlinear coordinate transformations, along with the necessary derivatives, are computed with simple matrix operations, and require onlyO(Norb3) effort. The new method differs from previous optimization methods in several respects. (1) It reproduces certain behavior of the exact energy function that is not displayed by previous approaches. (2) The orbital-state coupling is included explicitly via the partitioned orbital hessian matrix. (3) The minimization of the approximate energy function is simpler than with previous similar approaches. (4) The treatment of redundant orbital rotations is straightforward, since the exact and approximate energy functions display the same qualitative behavior with respect to these wave function variations. (5) Finally, the present method may be implemented as a simple extension to essentially any existing second-order MCSCF code, the required changes being localized within a rather small part of the overall iterative procedure. Examples of the convergence of the new method are presented, along with numerical demonstrations of some of the relevant features of the exact and interpolated energy functions.
TL;DR: A block 2-D decomposition and a new block LU matrix factorization based on a Newton approach are presented for solving quickly and efficiently polynomial or exponential2-D interpolation problems.
Abstract: A block 2-D decomposition and a new block LU matrix factorization based on a Newton approach are presented for solving quickly and efficiently polynomial or exponential 2-D interpolation problems. The sample grids under consideration are described by the product representation (x/sub 0/, x/sub 1/, . . ., x/sub n/) x(y/sub 0/, y/sub 1/, . . ., y/sub m/), where the x grid and the y-grid are not necessarily uniformly spaced. The attractive features of the method are the inherent efficient parallelism, the reduced computational requirements needed for the LU decomposition, and the capability of implementation of 1-D fast and accurate algorithms. The proposed method can be used for modeling 2-D discrete signals, designing 2-D FIR filters, 2-D Fourier matrix factorization, 2-D DFT, etc. >
01 Jan 1992
TL;DR: In this paper, the authors present a survey of recently established inequalities in classical polynomial interpolation theory and their applications in numerical computation, including boundary value problems for ordinary differential equations.
Abstract: This paper surveys recently established inequalities in classical polynomial interpolation theory. Besides their applications in numerical computation these inequalities are of immense value in the study of boundary value problems for ordinary differential equations.
TL;DR: In this paper, it was shown that the relative maxima of the Lebesgue function are strictly decreasing from the outside towards the middle of the interval [0, n], and upper and lower bounds, and an asymptotic expansion, were obtained for the smallest maximum when n is odd.
Abstract: Properties of the Lebesgue function associated with interpolation at the equidistant nodes , are investigated. In particular, it is proved that the relative maxima of the Lebesgue function are strictly decreasing from the outside towards the middle of the interval [0, n], and upper and lower bounds, and an asymptotic expansion, are obtained for the smallest maximum when n is odd.
TL;DR: A successive approximation of the interpolation functions is presented, which is suited to the applications such as the multiplex communication of the images of both directions.
Abstract: This paper presents a comprehensive discussion of the approximation of the n-dimensional wave f(X) using the sampled values of the output wave obtained by exciting a series of time-invariant linear circuits by the wave f(X). It is assumed that the approximate wave h(X) of f(X) is given by the sum of sample values of the output wave multiplied by certain n-dimensional waves. For simplicity, n-dimensional waves to be multiplied with the sample values are called the interpolation functions.
The set of sample points treated in this paper is defined as a subset obtained by sampling periodically the vertices of the n-dimensional parallelepipeds placed periodically in the space Rn. Such a set of sampling points includes the most of the typical arrangements of the sampling points, such as the hexagonal and the octagonal lattices on the two-dimensional space.
It is assumed that the sample values contain statistically independent errors such as the observation error and/or the quantization error. Moreover, it is assumed that the interpolation functions have the supports which are parallel-translations of each other.
First, it is assumed that the functional forms of these interpolation functions may be different. Further, a set of n-dimensional waves is considered where the corresponding spectrum has the weighted p-norms smaller than the prescribed positive constant. The standard deviations of the difference between f(X) and their approximations are considered. As the measure of the approximation error, the upper limit of the standard deviation obtained by varying the original waves over the given set of waves is adopted.
In the following sections it is shown that the interpolation functions minimizing the forementioned measure of error can be expressed as the parallel-translations of a finite number of functions. Further, in special cases, the interpolation functions have the discrete orthogonality. Since the measure of error is a convex function of the interpolation functions, it is ensured that the global optimum is obtained easily by using the ordinary numerical optimization. For some special cases, the concrete expression for the optimal interpolation functions are derived. Considering the approximation system in the reverse direction, where the linear circuits first passing the input wave are exchanged with the interpolation filters, it is shown that the new interpolation functions also minimize the same measure of error. As a direct consequence, a successive approximation of the interpolation functions is presented, which is suited to the applications such as the multiplex communication of the images of both directions.