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Trigonometric interpolation
About: Trigonometric interpolation is a research topic. Over the lifetime, 2041 publications have been published within this topic receiving 34376 citations.
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TL;DR: Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable and deserves to be known as the standard method of polynometric interpolation.
Abstract: Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.
1,177 citations
Book•
09 Oct 1975
TL;DR: In this paper, the authors present a generalization of the trigonometric interpolation polynomial of Lagrange to a function of exponential type and show that the resulting function can be represented as a linear space.
Abstract: 1. Preparatory Information.- 1.1. The Spaces C(?) and Lp(?).- 1.2. Normed Linear Spaces.- 1.3. Properties of the Space Lp(?).- 1.4. Averaging of Functions According to Sobolev.- 1.5. Generalized Functions.- 2. Trigonometric Polynomials.- 2.1. Theorems on Zeros. Linear Independence.- 2.2. Important Examples of Trigonometric Polynomials.- 2.3. The Trigonometric Interpolation Polynomial of Lagrange.- 2.4. The Interpolation Formula of M. Riesz.- 2.5. The Bernstein's Inequality.- 2.6. Trigonometric Polynomials of Several Variables.- 2.7. Trigonometric Polynomials Relative to Certain Variables.- 3. Entire Functions of Exponential Type, Bounded on ?n.- 3.1. Preparatory Material.- 3.2. Interpolation Formula.- 3.3. Inequalities of Different Metrics for Entire Functions of Exponential Type.- 3.4. Inequalities of Different Dimensions for Entire Functions of Exponential Type.- 3.5. Subspaces of Functions of Given Exponential Type.- 3.6. Convolutions with Entire Functions of Exponential Type.- 4. The Function Classes W, H, B.- 4.1. The Generalized Derivative.- 4.2. Finite Differences and Moduli of Continuity.- 4.3. The Classes W, H, B.- 4.4. Representation of an Intermediate Derivate in Terms of a Derivative of Higher Order and the Function. Corollaries.- 4.5. More on Sobolev Averages.- 4.6. Estimate of the Increment Relative to a Direction.- 4.7. Completeness of the Spaces W, H, B.- 4.8. Estimates of the Derivative by the Difference Quotient.- 5. Direct and Inverse Theorems of the Theory of Approximation. Equivalent Norms.- 5.1. Introduction.- 5.2. AuDroximation Theorem.- 5.3. Periodic Classes.- 5.4. Inverse Theorems of the Theory of Approximations.- 5.5. Direct and Inverse Theorems on Best Approximations. Equivalent H-Norms.- 5.6. Definition of B-Classes with the Aid of 0) over Functions of Exponential Type.- 8.8. Decomposition of a Regular Function into Series Relative to de la Vallee Poussin Sums.- 8.9. Representation of Functions of the Classes Bp?r in Terms of de la Vallee Poussin Series. Null Classes (1 ? p ? ?).- 8.10. Series Relative to Dirichlet Sums (1 ?).- 9. The Liouville Classes L.- 9.1. Introduction.- 9.2. Definitions and Basic Properties of the Classes Lpr and pr.- 9.3. Interrelationships among Liouville and other Classes.- 9.4. Integral Representation of Anisotropic Classes.- 9.5. Imbedding Theorems.- 9.6. Imbedding Theorem with a Limiting Exponent.- 9.7. Nonequivalence of the Classes Bpr and Lpr.- Remarks.- Literature.- Index of Names.
831 citations
TL;DR: The error bounds show that the polynomial interpolation on a d-dimensional cube, where d is large, is universal, i.e., almost optimal for many different function spaces.
Abstract: We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points.
697 citations
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 article, the authors give a simple and explicit condition on {z,} under which the interpolation is possible with a bounded function f(z) if we allow {w,} to be an arbitrary bounded sequence, the condition is also a necessary one.
Abstract: be possible for a given sequence of points {zv}, I z, 1, and an analytic function f(z) in I z i < 1, I f(z) I _ 1. The result is, however, very implicit ancd gives in a concrete situation very little help in deciding if the interpolatioln is possible or not. The object of the present paper is to give a simple and explicit condition on {z,} under which the interpolation (1. 1) is possible with a bounded function f(z). If we allow {w,} to be an arbitrary bounded sequence, the condition is also a necessary one. It should be observed that even the existence of any infinite such sequence {zv} is non-trivial; this problem was suggested by R. C. Buck and constructions of such examples have also recentlybeen given by G-leason and Newman (unpublished). The proof of the main theorem depends on a reformulation of problem (1.1) which is presented in section 2. It is essentially included in a result by Garabedian [2]; since the discussion there is quite general and the proof complicated, we have included a complete and simple proof here. Section 3 contains an inequality of the Schwarz type, which is the crucial step in our proof. This is then completed in section 4. The last section is devoted to an application to the ideal structure in the algebra of bounded analytic functions.
583 citations