Showing papers on "Trigonometric interpolation published in 1987"

The pseudospectral method: Comparisons with finite differences for the elastic wave equation

Abstract: The pseudospectral (or Fourier) method has been used recently by several investigators for forward seismic modeling. The method is introduced here in two different ways: as a limit of finite differences of increasing orders, and by trigonometric interpolation. An argument based on spectral analysis of a model equation shows that the pseudospectral method (for the accuracies and integration times typical of forward elastic seismic modeling) may require, in each space dimension, as little as a quarter the number of grid points compared to a fourth‐order finite‐difference scheme and one‐sixteenth the number of points as a second‐order finite‐difference scheme. For the total number of points in two dimensions, these factors become 1/16 and 1/256, respectively; in three dimensions, they become 1/64 and 1/4 096, respectively. In a series of test calculations on the two‐dimensional elastic wave equation, only minor degradations are found in cases with variable coefficients and discontinuous interfaces.

Uniqueness for trigonometric series

Abstract: We give examples of trigonometric series with slowly decreasing coefficients which sum to zero almost everywhere.

On Aitken-Neville Formulae for Multivariate Interpolation

Abstract: In a paper in J.S.I.A.M., 8, (1960), p.33–42, Thacher and Mil ne showed how the solution of certain polynomial interpolation problems in R s can be constructed from the solutions of s+1 simpler problems. In the present paper we extend the method showing that the number of basic problems to be used depends on the distribution of the points and on the chosen interpolation space, which here is not necessarily a polynomial space.

On interpolation systems and h-reducible interpolation problems

Abstract: In this paper the techniques of interpolation systems in IR 2 are used in the study of the unisolvence in P m (x, y) of interpolation problems of Hermite type in two variables. In the case of H-reducible incidence functions, this kind of problem has been studied by J. R. Busch, using a different method. Here, the relation between the two techniques is established and it is proved that the complete reduction of an incidence function implies the equivalence of the data, denoted by z, to those associated with a certain interpolation system of order m. We also give an affirmative answer to a conjecture posed by J. R. Busch.

On the evaluation of two-dimensional correlation sequences

Abstract: An approach to the evaluation of two-dimensional (2-D) correlation sequences and complex integrals is discussed. It is demonstrated that the system of rational function equations given by Hwang can be solved by making use of the unit circle polynomial interpolation. Thus, a general computer algorithm becomes available for handling the problem.

An investigation of dual eigenvalue systems using trigonometric polynomials and differential geometry

Abstract: The paper suggests that trigonometric polynomials provide a useful way of investigating elastic structures buckling in two modes simultaneously, when they are introduced into the associated potential energy function. They are then used to study a parameter of Huseyin-Mandadi and to demonstrate connections between the structural behaviour of imperfect systems and the differential geometry of the energy surface of the perfect model. The Huseyin-Mandadi parameter is shown to have importance in both structural mechanics and differential geometry.

Interpolation Between Dimensions

Abstract: A method is presented which allows one to interpolate between integer dimensions in a translationally invariant fashion using finite size transfer matrices. For the ferromagnetic Ising model an exact solution is presented for interpolation between isolated spins and d = 1, and finite strip calculations are presented for interpolation between d= 1 and d= 2. A possible application of this interpolation scheme is the numerical study of quantum spin systems.

The optimum interpolatory approximation of waves using time limited interpolation functions

Abstract: An investigation has been conducted about the interpolatory approximation of waveforms using a set of sample points in the form of groups of K generally non-equidistant sample points which are arranged periodically over a time axis First, we consider a set Γ of waveforms f(t) whose Fourier spectra F(ω) are bandwidth limited in |ω| ≤ ω1 and which satisfy for some positive weight W(ω) The interpolation function, which minimizes the supremum of the absolute error between the original waveforms f(t) and the corresponding approximation waveforms over Γ, is derived Here the approximate waveforms are taken to be the total sum of the values at the sample points mentioned above multiplied by the interpolation functions time-limited to a finite time-band centering at the corresponding sample points A choice of time limitation of the interpolation function discussed above is fundamental for realization of interpolation functions by a FIR filter If displacement along the time axis is ignored, the optimal interpolation functions mentioned above are divided into K groups and it is shown that they satisfy the orthogonality for a finite sum Next, using the concept of an inner product expression of a quasi-bilinear form, an extension of the above concept is made and the optimal interpolation function is derived The extension includes a case in which sample values contain statistical errors and a case in which the original wave is reproduced using sample values of linearly transformed waveforms