Showing papers on "Trigonometric interpolation published in 1982"

On Lagrange and Hermite interpolation in Rk

Abstract: A method for the construction of a set of data of interpolation in several variables is given. The resulting data, which are either function values or directional derivatives values, give rise to a space of polynomials, in such a way that unisolvence is guaranteed. The interpolating polynomial is calculated using a procedure which generalizes the Newton divided differences formula for a single variable.

Analysis of the combined finite element and Fourier interpolation

Abstract: We consider Lagrange interpolation involving trigonometric polynomials of degree ?N in one space direction, and piecewise polynomials over a finite element decomposition of mesh size ?h in the other space directions. We provide error estimates in non-isotropic Sobolev norms, depending additively on the parametersh andN. An application to the convergence analysis of an elliptic problem, with some numerical results, is given.

P-stable andP[α, β]-stable integration/interpolation methods in the solution of retarded differential-difference equations

Abstract: The equationu(t)=pu(t)+qu(t-τ) is presented as an archetype (scalar) equation for assessing the quality of integrator/interpolator pairs used to solve retarded differential-difference equations. The relationships ofP-stability andP[α, β]-stability, defined with respect to this archetype equation, to stability and order of multistep integrators and to passivity and order of Lagrange interpolators are developed. Composite multistep integrators and composite Lagrange interpolators are considered as a means of obtaining high order pairs stable for all step-sizes over a large portion, if not all, of the (p, q)-domain on which the archetype equation is stable.

Asymptotic expansion of the Lebesgue constants associated with polynomial interpolation

Abstract: An infinite asymptotic expansion is obtained for the Lebesgue constants associated with the polynomial interpolation at the zeros of the Chebyshev polynomials. The error due to truncation is shown to be bounded in absolute value by, and of the same sign as, the first neglected term.

Some improvements to the design programs for equiripple FIR filters

Abstract: Some methods are proposed which try to improve the execution of FIR filter design programs based on the Remez algorithm. The number of evaluations of the Lagrange interpolation formula needed for finding the extrema of the error function can be reduced by searching for the zeros of the derivative. The derivative of the Lagrange interpolation polynomial can be computed together with the Lagrange interpolation itself with little additional effort. The precision of the evaluation of the Lagrange interpolation can be improved by utilizing all the points resulting from the Remez algorithm. A couple of other minor improvements are given, too.

On a Special Laurent-Hermite Interpolation Problem

Abstract: We present a recursive algorithm for the construction of a rational approxi mation for a given Laurent series which in a certain sense interpolates at the zeros of its numerator. If certain symmetry conditions are satisfied, the algorithms of Nevanlinna-Pick and Schur are found as special cases. We give also an interpolation of the algorithm as a coupled recursion for reproducing kernels of indefinite inner product spaces defined with the aid of the given Laurent series. In the symmetric case, the approximation can be given a least squares interpretation. The interpolation points can then be chosen in an o+ptimal way. A numerical example of the latter problem is given.

Interpolation of completely monotone functions

Abstract: Intervals in which Lagrange interpolation polynomials converge pointwise to the interpolated function are specified for a family of functions comprising all completely monotone functions.

Convergence of lacunary trigonometric interpolation on equidistant nodes

Abstract: Recently SHARMA, SMITH and TZIMBALARIO [11] have given the necessary and sufficient condition when the problem of (0, m~ . . . . , mg) interpolation by trigono2kn metric polynomials on the nodes (k--0, 1 . . . . . n l ) is uniquely solvable n (or regular) where m~, ms . . . . , mq are given positive integers. A different proof of this result was given earlier by CAVARETTA, SHARMA and VAROA [1]. In order to state the result precisely we shall consider trigonometric polynomials of the class oq M and ~M,, (e=0 or 1). We shall say that T(O) E~'u if

Algorithm fast fourier transforms with recursively generated trigonometric functions

Abstract: New recursive formulae for trigonometric functions generation suitable for FFT algorithms are given. Evaluation of one pair of trigonometric coefficients thus requires 2 multiplications and 2 additions only. Speed comparisons of various radices 2, 4 and 8 FFT FORTRAN realizations were performed. An efficient FORTRAN IV program for one-dimensional complex FFT based on radix 8 algorithm with recursively generated trigonometric coefficients is supplied.

On the interpolation function for classes of sampling theorems with non‐uniform sampling points

Abstract: A time waveform which is absolutely square integrable and whose Fourier spectrum is identically zero above a certain finite frequency is called a band-limited waveform. This paper discusses a sampling theorem for a band-limited waveform based on a set of sampling points. The waveform structure is such that a finite number of sampling points are periodically repeated. The following results are obtained. An interpolation formula which is sufficiently general is established for the sampling theorem of the discussed type. The necessary and sufficient condition is derived for the interpolation function in order that the interpolation formula apply to any band-limited waveform. By this theorem, required interpolation functions can systematically be determined by the Fourier expansion coefficient, with respect to the angular frequency, of a finite number of two-variable functions of time and angular frequency, called the generating variable. Then a special case of sampling theorem is considered, where the interpolation filter determined in correspondence to the interpolation function is time-invariant, deriving the necessary and sufficient condition for the Fourier spectrum of the interpolation function (i.e., the frequency characteristics of the interpolation filter). Some detailed expressions are discussed for the interpolation function, which can easily make the Fourier spectrum continuous or smooth.