Showing papers on "Trigonometric interpolation published in 2002"
TL;DR: Quadratic trigonometric polynomial curves with a shape parameter are presented and can yield tight envelopes for the quadratic B-spline curves and can be closer to the given control polygon than the quadruple B- Spline curves.
107 citations
TL;DR: In this article, the authors investigated trigonometrical polynomials associated with f∈Lip(α,p) to approximate f in Lp-norm to the degree of O(n−α), where n−α is the number of nodes.
87 citations
82 citations
TL;DR: In this paper, interpolation theorems for Sobolev spaces of functions on nonsmooth domains with vanishing trace on a part of the boundary are proved for functions with vanishing traces.
Abstract: Interpolation theorems are proved for Sobolev spaces of functions on nonsmooth domains with vanishing trace on a part of the boundary.
59 citations
TL;DR: In this article, a general family of interpolation methods is introduced which includes, as special cases, the real and complex methods and also the so-called ± or G1 and G2 methods defined by Peetre and Gustavsson-Peetre.
40 citations
TL;DR: In this article, the bitangential interpolation problem with a finite number of interpolation nodes for a multivariable analogue of the Schur class consisting of matrix-valued analytic functions on the ball is solved via a generalized functional calculus with operator argument, thereby generalizing in a compact way the simple, first-order interpolation conditions considered for this class of functions in earlier work.
25 citations
TL;DR: In this article, the p th derivative of a linear rational trigonometric interpolant written in barycentric form is calculated in terms of sets of points for which the interpolant converges exponentially towards the interpolated function.
Abstract: We present here formulae for calculating the p th derivative of a linear rational trigonometric interpolant written in barycentric form. We give sets of interpolating points for which the interpolant converges exponentially towards the interpolated function.
22 citations
TL;DR: This work considers multivariate interpolation at arbitrary points and provides effective interpolation formulas by using simple ridge polynomials which essentially possess the nature of univariate polynomial in computation, and applies the results to design a computational algorithm for training feedforward neural networks.
19 citations
10 Dec 2002
TL;DR: This paper extends a previously proposed solver for scalar Nevanlinna-Pick interpolation problems with degree constraint to the ones including derivative constraints, finding any real rational interpolant with a degree bound by solving an optimization problem of the same type as encountered in the problem without derivative constraint.
Abstract: This paper extends a previously proposed solver for scalar Nevanlinna-Pick interpolation problems with degree constraint to the ones including derivative constraints. The solver computes any real rational interpolant with a degree bound by solving an optimization problem of the same type as encountered in the problem without derivative constraint. A robust homotopy continuation method, previously devised by the second author for the problem without derivative constraint, can be applied to solve the new optimization problem.
17 citations
TL;DR: The geometric characterization introduced by Chung and Yao, which provides simple Lagrange formulae, is here analyzed for interpolation points lying on a line, a conic or a cubic.
17 citations
TL;DR: An indefinite generalization of Nudel'man's problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data.
Abstract: An indefinite generalization of Nudel'man's problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides known results on existence criteria for Pick-Nevanlinna and Caratheodory-Fejer interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel'man's problem.
TL;DR: The poisedness problem for a bivariate interpolation introduced by B. Bojanov and Y. Xu is solved and the poisedness of several general univariate Birkhoff interpolation problems is proved to which the above problem is reduced.
01 Jan 2002
TL;DR: The trigonometric polynomial B-spline as discussed by the authors is a new uniform spline curve, which can be used as an efficient new model for geometric design in the fields of CAD/CAM.
Abstract: This paper presents a new kind of uniform spline curve, named trigonometric polynomialB-splines, over space Ω = span{sint, cost, tk-3,tk-4,…,t,1} of which k is an arbitrary integerlarger than or equal to 3. We show that trigonometric polynomial B-spline curves have many similarV properties to traditional B-splines. Based on the explicit representation of the curve we have also presented the subdivision formulae for this new kind of curve. Since the new spline can include both polynomial curves and trigonometric curves as special cases without rational form, it can be used as an efficient new model for geometric design in the fields of CAD/CAM.
07 Aug 2002
TL;DR: This paper considers the case where the signal is assumed to be bounded and proposes an approximate solution that leads to closed form expressions for the interpolator coefficients based upon an extension of the Lagrange interpolation formula.
Abstract: For oversampled band-limited signals, min-max optimal interpolators have been proposed under assumptions upon either the signal to be interpolated itself (e.g. finite energy) or its Fourier transform. In this paper, we consider the case where the signal is assumed to be bounded. For band-limited signals this is a more realistic assumption that is weaker than those previously considered. We propose an approximate solution that leads to closed form expressions for the interpolator coefficients. It relies upon an extension of the Lagrange interpolation formula.
01 Jan 2002
TL;DR: In this paper, a generalized algorithm for the division with remainder of polynomials in several variables is presented for the construction of standard bases for polynomial ideals with respect to arbitrary grading structures.
Abstract: Based on a generalized algorithm for the division with remainder of polynomials in several variables, a method for the construction of standard bases for polynomial ideals with respect to arbitrary grading structures is derived. In the case of ideals with finite codimension, which can be viewed upon as a polynomial interpolation problem, an explicit representation for the interpolation space of reduced polynomials can be given.
01 Jan 2002
TL;DR: In this paper, the authors rephrase the Abstract Interpolation Problem (AIP) in terms of unitary scattering systems and give a complete solution to this more general AIP under an additional assumption regarding the data scale ρ 0.
Abstract: In Section 1 we recall the setting and solution of the Abstract Interpolation Problem (AIP) from [1]. In Section 2 we rephrase the AIP in terms of unitary scattering systems rather than in terms of unitary colligations. This allows us to give up the orthogonality assumption on the data scales and to formulate a more general setting of the AIP that corresponds to interpolation problems for harmonic functions also. (The original formulation of the AIP corresponded naturally to interpolating analytic functions only.) In Section 3 we give a complete solution to this more general AIP under an additional assumption regarding the data scale ρ0. Solutions are the spectral functions of the feedback coupling with respect to the scale ρ0. In Section 4 we give up the additional assumption of Section 3 regarding the data scale ρ0 and define the scale ρ associated with any feedback coupling by means of the corresponding wave operator and develop the appropriate modification of the results of Section 3. In Section 5 a remark is given on the feedback coupling of the scattering systems. We plan to demonstrate applications of this approach to the General Commutant Lifting problem at another occasion.
13 May 2002
TL;DR: A unified theory for arithmetic transform of a variety of discrete trigonometric transforms is proposed and it is shown that the interpolation method determines the transform to be computed.
Abstract: In this paper, we propose a unified theory for arithmetic transform of a variety of discrete trigonometric transforms. The main contribution of this work is the elucidation of the interpolation process required in arithmetic transforms. We show that the interpolation method determines the transform to be computed. Several kernels were examined and asymptotic interpolation formulae were derived. Using the arithmetic transform theory, we also introduce a new algorithm for computing the discrete Hartley transform.
24 Jun 2002
TL;DR: In this paper, both the Lagrange interpolation of analytic functions on [-1, 1] and the trigonometric interpolation on [email protected], [p] with period [emailprotected] are discussed.
TL;DR: It is proved that there exists a residual set of functions in the disk algebra, such that the Lagrange interpolation polynomials of each of these functions form a dense subset of the space of all holomorphic functions defined on the unit disk.
Journal Article•
Journal Article•
TL;DR: A new real time interpolation algorithm for complex parametric curve, including high order polynomial curve, Bezier curve, B spline curve, NURBS curve, etc, was developed, which is based on Gauss Legendre quadrature andPolynomial interpolation.
Abstract: A new real time interpolation algorithm for complex parametric curve, including high order polynomial curve, Bezier curve, B spline curve, NURBS curve, etc, was developed, which is based on Gauss Legendre quadrature and polynomial interpolation Firstly, the curve arc length is calculated by using piecewise Gauss Legendre quadrature Then, the curve is divided into small sub intervals according to the parameter value, and a table of relationship between sub parameter value and sub arc length is created Lastly, the parameter value of each interpolation cycle is calculated by using the polynomial interpolation The strategy of smooth motion control and the calculation of declaration point's parameter were also analyzed The proposed algorithm can extend the CNC's trajectory control function, simplify the part program, and improve machining precision
Journal Article•
TL;DR: In this paper, a class of piecewise trigonometric polynomial curves of degree m(m=1,2,3) is presented, which can be used to generate ellipses conveniently.
Abstract: A class of piecewise trigonometric polynomial curves of degree m(m=1,2,3) is presented in this paper. Weighted trigonometric polynomial curves are given by using a shape parameter. Analogous to the cubic B-spline curves, each trigonometric polynomial curve segment is generated by four consecutive control points. For equidistant knots, the given trigonometric polynomial curves of degree m are C 2m-1 continuous. The construction methods of an open and a closed trigonometric polynomial curves are described. The given curves can be used to generate ellipses conveniently. The comparisons between the trigonometric polynomial curves and the cubic B-spline curves are given. By choosing m or the shape parameter, the trigonometric polynomial curve can approach to the given control polygon in a different way. Therefore, the construction method of the trigonometric polynomial curves is simple and useful for curve design.
07 Aug 2002
TL;DR: The problem of signal restoration when P of every N samples in a discrete time system are uniformly decimated is considered and the condition - and thus the noise sensitivity - of the restoration process is analyzed.
Abstract: We consider the problem of signal restoration when P of every N samples in a discrete time system are uniformly decimated. The degraded signal is an aliased form of the original signal. The aliasing can, in certain cases, be unraveled by application of multiplicative discrete time trigonometric polynomials followed by filtering. The filter output is the restored discrete time signal. Conditions required for this restoration are presented. The condition - and thus the noise sensitivity - of the restoration process is also analyzed.
TL;DR: In this article, two-sided and one-sided residue interpolation problems with various symmetries were studied for matrix-valued Hardy functions with conformal conjugate involutions of the unit disk.
TL;DR: In this paper, the maximal convergence and uniqueness spaces for Abel-Goncharov interpolation problems with nodes of interpolation (either arbitrary complex or real) in classes defined by a sequence of majorants of the nodes are found.
Abstract: In the scale of the growth types of entire functions defined in terms of certain comparison functions the maximal convergence and uniqueness spaces are found for Abel-Goncharov interpolation problems with nodes of interpolation (either arbitrary complex or real) in classes defined by a sequence of majorants of the nodes.
TL;DR: An approach to shape-preserving approximation based on interpolation space theory is presented and the corresponding approximation result related to the intersection property of the cone of nonnegative functions with respect to the couple is proved.
TL;DR: The authors obtained asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials in the metric of the space L on classes of convolutions of periodic functions admitting a regular extension into a fixed strip of the complex plane.
Abstract: We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials in the metric of the space L on classes of convolutions of periodic functions admitting a regular extension into a fixed strip of the complex plane.
TL;DR: In this article, the complexity of solving the dual problem is shown to be almost independent of the number of constraints, provided that an appropriate preprocessing has been done, and the results can be extended to other curves of the complex plane (real axis, imaginary axis), to nonnegative matrix polynomials and to interpolation constraints on the derivatives.