Showing papers on "Trigonometric interpolation published in 1990"
Book•
01 Jan 1990
TL;DR: In this paper, the authors present a formal notation for solving homogeneous and non-homogeneous interpolation problems for matrix functions with J-unitary values on the Imaginary Axis or Unit Circle.
Abstract: I Homogeneous Interpolation Problems with Standard Data.- 1. Null Structure for Analytic Matrix Functions.- 2. Null Structure and Interpolation Problems for Matrix Polynomials.- 3. Local Data for Meromorphic Matrix Functions.- 4. Rational Matrix Functions.- 5. Rational Matrix Functions with Null and Pole Structure at Infinity.- 6. Rational Matrix Functions with J-Unitary Values on the Imaginary Line.- 7. Rational Matrix Functions with J-Unitary Values on the Unit Circle.- II Homogeneous Interpolation Problems with Other Forms of Local Data.- 8. Interpolation Problems with Null and Pole Pairs.- 9. Interpolation Problems for Rational Matrix Functions Based on Divisibility.- 10. Polynomial Interpolation Problems Based on Divisibility.- 11. Coprime Representations and an Interpolation Problem.- III Subspace Interpolation Problems.- 12. Null-Pole Subspaces: Elementary Properties.- 13. Null-Pole Subspaces for Matrix Functions with J-Unitary Values on the Imaginary Axis or Unit Circle.- 14. Subspace Interpolation Problems.- 15. Subspace Interpolation with Data at Infinity.- IV Nonhomogeneous Interpolation Problems.- 16. Interpolation Problems for Matrix Polynomials and Rational Matrix Functions.- 17. Partial Realization as an Interpolation Problem.- V Nonhomogeneous Interpolation Problems with Metric Constraints.- 18. Matrix Nevanlinna-Pick Interpolation and Generalizations.- 19. Matrix Nevanlinna-Pick-Takagi Interpolation.- 20. Nehari Interpolation Problem.- 21. Boundary Nevanlinna-Pick Interpolation.- 22. Caratheodory-Toeplitz Interpolation.- VI Some Applications to Control and Systems Theory.- 23. Sensitivity Minimization.- 24. Model Reduction.- 25. Robust Stabilizations.- Appendix. Sylvester, Lyapunov and Stein Equations.- A.1 Sylvester equations.- A.2 Stein equations.- A.3 Lyapunov and symmetric Stein equations.- Notes.- References.- Notations and Conventions.
642 citations
TL;DR: In this paper, the Fourier transform was used to analyze the variational framework for multivariate interpolation and obtained error estimates of arbitrarily high order for a class of interpolation methods that includes multiquadrics.
Abstract: We continue an earlier study of certain spaces that provide a variational framework for multivariate interpolation. Using the Fourier transform to analyze these spaces, we obtain error estimates of arbitrarily high order for a class of interpolation methods that includes multiquadrics
556 citations
TL;DR: In this article, the authors provide a monotone map of polynomial spaces from which interpolation to arbitrary data given at the points in the set is possible and uniquely so.
Abstract: We provide a map
which associates each finite set Θ in complexs-space with a polynomial space πΘ from which interpolation to arbitrary data given at the points in Θ is possible and uniquely so. Among all polynomial spacesQ from which interpolation at Θ is uniquely possible, our πΘ is of smallest degree. It is alsoD- and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with eachgq∈Θ, there is associated a polynomial space PΘ, and, for given smoothf, a polynomialq∈Q is sought for which
$$p(D)(f - q)(\theta ) = 0, \forall p \in P_\theta , \theta \in \Theta $$
.
233 citations
Book•
01 Oct 1990
TL;DR: The most significant results of Lagrange interpolation theory in the last forty years are given in this paper, including lower estimates, fine and rough theory, interpolatory proofs of Jackson and Teliakovski-Gopengauz theorems, lebesgue function, Lebesgue constant, Bernstein and Erdos conjecture on the optimal nodes, almost everywhere divergence, Hermite-Fejer type and lacunary interpolation.
Abstract: This book gives a systematic survey on the most significant results of interpolation theory in the last forty years. It deals with Lagrange interpolation including lower estimates, fine and rough theory, interpolatory proofs of Jackson and Teliakovski-Gopengauz theorems, Lebesgue function, Lebesgue constant of Lagrange interpolation, Bernstein and Erdos conjecture on the optimal nodes, the almost everywhere divergence of Lagrange interpolation for arbitrary system of nodes, Hermite-Fejer type and lacunary interpolation and other related topics.
178 citations
TL;DR: In this article, it was shown that reachability and observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational (bitangen- tial) interpolation problem.
152 citations
TL;DR: A solution to the problem of how to construct some function on R with range also a collection of ‘fuzzy values’, based on the fundamental and well-known polynomial interpolation theorm of Lagrange is presented.
78 citations
TL;DR: In this paper, the error induced by the approximation of a function by an interpolation function which is a combination of algebraic and first-order trigonometric polynomials is analyzed.
37 citations
TL;DR: In this article, the authors generalized the results of Van Barel and Bultheel (1990) for rational interpolation to the case of vector data given at arbitrary points that may coincide or not, and gave a recursive algorithm that does not need a reordering of the given interpolation data to overcome a singularity in the interpolation table.
35 citations
TL;DR: In this article, a novel interpolation method using the type I discrete cosine transform (DCT-I) is introduced, where the original definition of the DCT is modified to suit this application.
Abstract: A novel interpolation method, using the type I discrete cosine transform (DCT-I), is introduced. The original definition of the DCT-I has been modified to suit this application. Three options for the modified DCT-I are proposed.
34 citations
TL;DR: In this paper, a method for discrete representation of signals consisting of a cascade of Chebyshev nonuniform sampling (CNS) followed by the discrete cosine transform (DCT) is presented.
Abstract: A method for discrete representation of signals consisting of a cascade of Chebyshev nonuniform sampling (CNS) followed by the discrete cosine transform (DCT) is presented. It is proven that the considered signal samples and the coefficients of the corresponding Chebyshev polynomial finite series are essentially a discrete cosine transform pair. A method for fast computation of the coefficients of the optimum interpolation formula (which minimizes the maximum instantaneous error) is provided. If the signal g(t) is band-limited and has a finite energy, the condition of convergence for interpolation can be deduced. >
19 citations
TL;DR: In this article, the problem of solving the interpolation problem of analysis, presented by Vo P. Potapov, is connected with objects of j-theory, such as the multiple j-elementary matrix-valued function with pole 0 on the boundary of the unit circle.
Abstract: The method of solving the interpolation problem of analysis, presented by Vo P. Potapov [i, 2], the foundation of which consists in the adequacy of the interpolation problem for the assignment of definite objects of the j-theory of analytic matrix-valued functions, allows us to connect also this problem with objects of j-theory, such as the multiple j-elementary matrix-valued function with pole ~0 on the boundary of the unit circle (!~r01 = i) and the Blaschke--Potapov product of binomial j-elementary factors.
01 Jan 1990
TL;DR: In this article, the authors give a brief description of the notion of vertex splines and discuss their applications to interpolation of discrete data with or without constraints, but they do not have much to offer to the general theory, they will be satisfied with the study of those compactly supported ones with each support containing at least one common vertex.
Abstract: It is well known that B-splines in one variable play a central role in the theory of spline functions. However, although there are various generalizations of the notion of B-splines to the multi-variable setting in the literature, very little is known at this writing on the structure and theory of all compactly supported smooth piecewise polynomial functions on a preassigned grid partition Δ in ℝ s , s > 1, unless Δ is perfectly regular. While we don’t have much to offer to the general theory, we will be satisfied with the study of those compactly supported ones with each support containing at least one common vertex and with the interior of the support containing at most one vertex of Δ. These functions are called vertex splines. The objective of this presentation is to give a brief description of the notion of vertex splines and to discuss their applications to interpolation of discrete data with or without constraints.
TL;DR: A quantitative version of a classical result of S.N. Bernstein concerning the divergence of Lagrange interpolation polynomials based on equidistant nodes is presented.
Abstract: A quantitative version of a classical result of S.N. Bernstein concerning the divergence of Lagrange interpolation polynomials based on equidistant nodes is presented. The proof is motivated by the results of numerical computations.
TL;DR: The algorithm for the computation of the divided differences is shown to be numerically stable and does not require equidistant points, precomputation, or the fast Fourier transform, and can be very useful for very high-order interpolation.
Abstract: We present parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms use parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. Forn+1 given input pairs, the proposed interpolation algorithm requires only 2 [log(n+1)]+2 parallel arithmetic steps and circuit sizeO(n
2), reducing the best known circuit size for parallel interpolation by a factor of logn. The algorithm for the computation of the divided differences is shown to be numerically stable and does not require equidistant points, precomputation, or the fast Fourier transform. We report on numerical experiments comparing this with other serial and parallel algorithms. The experiments indicate that the method can be very useful for very high-order interpolation, which is made possible for special sets of interpolation nodes.
TL;DR: In this article, a Newton-type algorithm is given for computing an interpolating rational function with prescribed poles with no more than O(m 2 n + n 2 m ), where m and n are the numbers of interpolation points in the x - and y -direction, respectively.
TL;DR: In this article, a matricial approach and with the guideline of a result due to Stenger (1968) are shown to be also necessary and expressed in terms of the unisolvence of certain univariate interpolation problems.
TL;DR: In this article, a matrix polynomial of Mcmillan degree n −1 which satisfies n tangential Lagrange interpolation conditions is constructed explicitly in realization form, and a corollary is a type of Euclidean algorithm for matrix poynomials and McMillan degree.
TL;DR: In this paper, the authors studied a kind of interpolation problems on a given nodal set by trigonometric polynomials of order n and entire functions of exponential type.
Abstract: In this paper, we study a kind of interpolation problems on a given nodal set by trigonometric polynomials of order n and entire functions of exponential type according as the nodal set is
$$\left\{ {\frac{{2k\pi }}{n}} \right\}_{k = 0}^{n - 1} or \left\{ {\frac{{2k\pi }}{\sigma }} \right\}_{k = - \infty }^{ + \infty } $$
respectively We established some equivalent conditions and found the explicit forms of some interpolation functions on the interpolation problems As a special case, the explicit forms of fundamential functions of (0,m)-interpolat on by trigonometric case or entire functions case (in B2
σ) respectively, if they exist, may follow from our results Besides, we also considered the convergence of the interpolation functions at above stated
TL;DR: In this article, an analytic solution of a time-optimal problem for the oscillatory system is given, where the spectrum is defined by a special system of trigonometric polynomials (canonical variables) and studied toeplitz determinants in these variables.
Abstract: An analytic solution of a time-optimal problem for the oscillatory system is given, where the spectrum . Introducing a special system of trigonometric polynomials (canonical variables) and studying Toeplitz determinants in these variables, the authors obtain equations for determining the control time, as well as the points and surfaces of switching the optimal control. The solution thus obtained is, on the other hand, the solution of a trigonometric moment problem on the smallest possible interval in the form of a function of a (–1,1)-moment sequence. The question of local equivalence of linear time-optimal problems is considered for systems with a one-dimensional control. Bibliography: 6 titles.
TL;DR: In this article, an upper estimate for the variance of the number of real zeros of the random trigonometric polynomial g 1 cosθ was provided.
Abstract: This paper provides an upper estimate for the variance of the number of real zeros of the random trigonometric polynomial g1cosθ
TL;DR: In this article, a simple new procedure of interpolation with symmetric positive-real matrices is presented, which can be considered as a matrix extension of the Youla-Saito scalar interpolation theory.
Abstract: Employing Hazony's theory of the cascade synthesis of passive n -ports, a simple new procedure of interpolation with symmetric positive-real matrices is presented. The problem related to interpolation with rational lossless positive-real matrices is also discussed. The results of the paper can be considered as a matrix extension of the Youla-Saito scalar interpolation theory.
TL;DR: In this article, the derivatives of a function at equidistant nodes can be calculated numerically using formulae based on the classical Lagrange and Hermite interpolation, using the REDUCE language for algebraic manipulation.
Abstract: The derivatives of a function at equidistant nodes can be calculated numerically using formulae based on the classical Lagrange and Hermite interpolation. Such formulae are derived here using the REDUCE language for algebraic manipulation. They are available for 11 nodes as FORTRAN subroutines which can be used on a personal computer. As an example, the derivatives of a sinusoidal function are calculated. The results show an excellent precision, consistently better than the upper bounds of the remainder terms.
TL;DR: Asymptotically precise estimates for the deviation, in the L 1-norm, of interpolation polynomials with equally-spaced nodes from certain classes of functions are obtained in this paper.
Abstract: Asymptotically precise estimates are obtained for the deviation, in the L1-norm, of interpolation polynomials with equally-spaced nodes from certain classes of functions.
TL;DR: In this paper, the authors studied harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in Ck, and showed that each of the following properties holds for all the intermediate quasorms, if it holds for given boundary spaces: (1) being a normed space, (2) being Hilbert space; (3) satisfying the triangle inequality by the rth power of the quinorm; (4) being uniformly convex; and (5) being uniform smooth.
Abstract: This paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in Ck . It is shown, among other things, that each of the following properties holds for all the intermediate quasinormed spaces, if it holds for all given boundary spaces: (1) being a normed space; (2) being a Hilbert space; (3) satisfying the triangle inequality by the rth power of the quasinorm; (4) being uniformly convex; and (5) being uniformly smooth. As a principal tool, the notion of a harmonic set valued function (a generalization of analytic multifunction) is introduced and studied.
Dissertation•
05 Dec 1990
TL;DR: In this paper, the authors studied the duality relation between the complex interpolation method and the nonlinear PDE of second order for families of finite-dimensional normed spaces where G is open and bounded in Ck.
Abstract: The paper continues the study of one of the complex interpolation methods for families of finite-dimensional normed spaces {Cn , II • liz} zEG ' where G is open and bounded in Ck . The main result asserts that (under a mild assumption on the datum) the norm function (z, w) -+ IIwll; belongs to some anisotropic Sobolew class and is characterized by a nonlinear PDE of second order. The proof uses the duality theorem for the harmonic interpolation method (obtained earlier by the author). A new, simpler, proof of this duality relation is also presented in the paper.
01 Jan 1990
TL;DR: In this article, the authors construct convergent interpolating sequences to analytic functions in the disc, with interpolating points on the circle, and apply this to harmonic identification of linear constant dynamical systems.
Abstract: We construct convergent interpolating sequences to analytic functions in the disc, with interpolating points on the circle. We apply this to harmonic identification of linear constant dynamical systems.