Showing papers on "Trigonometric interpolation published in 1996"
87 citations
TL;DR: In this article, generalized Lorentz-Zygmund spaces are defined and interpolation theorems for quasilinear operators on such spaces, using weighted Hardy inequalities, are obtained.
Abstract: We define generalized Lorentz-Zygmund spaces and obtain interpolation theorems for quasilinear operators on such spaces, using weighted Hardy inequalities. In the limiting cases of interpolation, we discover certain scaling property of these spaces and use it to obtain fine interpolation theorems in which the source is a sum of spaces and the target is an intersection of spaces. This yields a considerable improvement of the known results which we demonstrate with examples. We prove sharpness of the interpolation theorems by showing that the constraints on parameters are necessary for the interpolation theorems.
58 citations
TL;DR: A multiresolution analysis of nested subspaces of trigonometric polynomials of Hermite interpolation on a dyadic partition of nodes on the interval [0,2π].
Abstract: The aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials. The pair of scaling functions which span the sample spaces are fundamental functions for Hermite interpolation on a dyadic partition of nodes on the interval [0,2π]. Two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions. Together with the corresponding dual functions, these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences. Consequently, this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment.
34 citations
TL;DR: This work clarifies the connections between two apparently unrelated approaches to bandlimited interpolation by showing that, in a certain sense, they are the dual of each other.
Abstract: We clarify the connections between two apparently unrelated approaches to bandlimited interpolation by showing that, in a certain sense, they are the dual of each other. The advantages of recognizing this duality are discussed.
31 citations
18 Nov 1996
TL;DR: Three different polynomial-based interpolation methods which can be efficiently implemented using the so-called Farrow (1988) structure are compared.
Abstract: There is a trend of using digital receivers where the sampling of the received signal is not synchronized to the incoming data symbols. In these receivers, timing adjustment must be done after sampling using interpolation. There are many different interpolation methods which can be used. We compare three different polynomial-based interpolation methods which can be efficiently implemented using the so-called Farrow (1988) structure. Methods to be compared are the linear and cubic Lagrange interpolation and a new interpolation method proposed earlier by the authors.
25 citations
TL;DR: In this paper, necessary and sufficient conditions for the existence of solutions to discrete time-variant interpolation problems of Nevanlinna-Pick and Nudelman type were derived.
Abstract: Necessary and sufficient conditions are derived for the existence of solutions to discrete time-variant interpolation problems of Nevanlinna-Pick and Nudelman type. The proofs are based on a reduction scheme which allows one to treat these time-variant interpolation problems as classical interpolation problems for operator-valued functions with operator arguments. The latter ones are solved by using the commutant lifting theorem.
17 citations
01 Dec 1996-Compel-the International Journal for Computation and Mathematics in Electrical and Electronic Engineering
TL;DR: In this article, a finite element (FE) analysis of saturation effects in a squirrel cage electrical machine is presented, which includes the FE equations of the electromagnetic field, the equations which define the connection of windings, and the mechanical equation.
Abstract: Presents a method for the finite element (FE) analysis of saturation effects in a squirrel‐ cage electrical machine. The proposed mathematical model includes the FE equations of the electromagnetic field, the equations which define the connection of windings, and the mechanical equation. Applies an approach based on a simultaneous solution of these equations, paying special attention to the movement simulation. Applies the time‐stepping method with a fixed grid, independent of the rotar position. In the method the motional effects are simulated by trigonometric interpolation of the results for the previous time step.
17 citations
TL;DR: A set of points on the unit disk of the Euclidean plane is given, which admits unique Lagrange interpolation, and they form an example of natural lattices of Chung and Yao.
Abstract: A set of points on the unit disk of the Euclidean plane is given, which admits unique Lagrange interpolation. The points have rotational symmetry and they form an example of natural lattices of Chung and Yao [2]. Properties of Lagrange interpolation with respect to these points are studied.
15 citations
TL;DR: The main result of this paper shows that this condition characterizes all configurations of Schoenberg-Whitney type such that in every neighborhood of T inK there must exist a configuration of T which admits unique Lagrange interpolation.
Abstract: Lagrange interpolation by finite-dimensional spaces of multivariate spline functions defined on a polyhedral regionK in ℝ
k
is studied. A condition of Schoenberg-Whitney type is introduced. The main result of this paper shows that this condition characterizes all configurationsT inK such that in every neighborhood ofT inK there must exist a configuration
$$\tilde T$$
which admits unique Lagrange interpolation.
13 citations
TL;DR: In this article, the embedding functions of an intermediate space A into a Banach couple (A0,A1) are defined as its embedding constants into the couples (1αA0;A1, 1&# 946;A2;A3), and properties and interrelations of different intermediate spaces, including real interpolation spaces, are studied.
Abstract: The embedding functions of an intermediate space A into a Banach couple (A0,A1) are defined as its embedding constants into the couples (1αA0,1βA1), ∀α,β>0. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach couple to obtain new interpolation theorems.
13 citations
TL;DR: In this article, the Lagrange interpolant of / : R −*• C in the zeros of Ga(Tz) where r > 0 is an entire function of exponential type r. The mean convergence of LTja(f'r) to / as r −> oo is studied.
Abstract: Let JQ, be the Bessel function of the first kind of order a > —1. Then Goi(z) := z~Ja(z) is an entire function whose zeros are all real. We note that under appropriate conditions, the Lagrange interpolant of / : R —*• C in the zeros of Ga(Tz) where r > 0 is an entire function of exponential type r. We denote it by LT,a(/;2) and study the mean convergence of LTja(f'r) to / as r —> oo. We obtain a theorem which is analogous to two well-known results, one due to J. Marcinkiewicz and another due to R. Askey. Some of the lemmas which we need for our proof of the theorem are results of independent interest; for example, Lemma 13 is an extension of the Whittaker-Shannon sampling theorem.
TL;DR: In this article, the authors generalize the univariate divided difference to a multivariate setting by considering linear combinations of point evaluations that annihilate the null space of certain differential operators.
TL;DR: In this paper, the smallest value of that can occur in a non-negative trigonometric polynomial with nonnegative integer coefficients such that (resp., ) for all natural numbers is known.
Abstract: We denote by (resp., ) the smallest value of that can occur in a non-negative trigonometric polynomial with non-negative integer coefficients such that (resp., ). We prove that for all natural numbers
TL;DR: Use of a new notion of multivariate divided difference leads to a short proof of a formula by Sauer and Xu for the error in interpolation, by polynomials of total degree?
Abstract: Use of a new notion of multivariate divided difference leads to a short proof of a formula by Sauer and Xu for the error in interpolation, by polynomials of total degree? n in d variables, at a 'correct' point set.
TL;DR: A trigonometric polynomial that is positive on a set of small measure is constructed in this article, and several new inequalities in the geometry of numbers are proved using the properties of this polynomial.
Abstract: A trigonometric polynomial that is positive on a set of small measure is constructed. Several new inequalities in the geometry of numbers are proved using the properties of this polynomial.
13 Mar 1996
TL;DR: Analysis of proposed unitary trigonometric (cosine, sine and combined sine-cosine) transforms is performed, and efficient algorithms for their computation are developed.
Abstract: In this paper, we introduce a new class of discrete parametric trigonometric transforms. We establish the conditions of unitarity of the discrete parametric trigonometric transform matrices, by which we construct a wide range of orthonormal transform matrices. Analysis of proposed unitary trigonometric (cosine, sine and combined sine-cosine) transforms is performed, and efficient algorithms for their computation are developed.© (1996) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: In this article, the problem of approximating function in a general domain in one and two dimensions using piecewise polynomial interpolation is considered and an error estimator is proposed.
Abstract: : We consider the problem of approximating function in a general domain in one and two dimensions using piecewise polynomial interpolation. We propose an error estimator and show how to adaptively determine the interpolation degree. Numerical examples are given. (AN)
TL;DR: It is shown that both schemes of interpolation using Chebyshev polynomials relate to the types I and II discrete cosine transform (DCT-I and DCT-II), respectively, and can be computed using existing fast algorithms for the DCT.
TL;DR: Subdivision cardinal interpolation schemes that preserve functions of positive type were shown to be related to orthonormal multiresolutions in this paper, where the interpolating function is the solution to a certain optimization problem.
Abstract: Subdivision cardinal interpolation schemes that preserve functions of positive type are shown to be related to orthonormal multiresolutions. The interpolating function is the solution to a certain optimization problem, and this makes it possible to derive error estimates, in particular for Lagrange iterative interpolation schemes.
TL;DR: In this paper, the convergence of Hermite-Fejer interpolation in LP with p > 2 is investigated and a necessary and sufficient condition for such convergence for all continuous functions is given.
Abstract: Weighted LP mean convergence of Hermite-Fejer interpolation based on the zeros of orthogonal polynomials with respect to the weight |x|2α+1(l — x2)β(α, β > — 1) is investigated. A necessary and sufficient condition for such convergence for all continuous functions is given. Meanwhile divergence of Hermite-Fejer interpolation in LP with p > 2 is obtained. This gives a possible answer to Problem 17 of P. Turân [J. Approx. Theory, 29(1980), p. 40].
01 Jan 1996
TL;DR: In this article, the norms of Hermite interpolation operators on some function spaces were studied and the Lagrange interpolation operator was investigated in the context of function spaces with functions.
TL;DR: In this article, the authors used the Runge-Kutta-Hǔta method to find the period doubling bifurcations in nonlinear T-periodic forced oscillators.