Showing papers on "Trigonometric interpolation published in 1976"
TL;DR: In this paper, it was shown that if a continuous function satisfies some growth conditions, then the corresponding Lagrange interpolation process converges in every Lp (1 < p < ∞) provided that the weight function is chosen in a suitable way.
105 citations
8 citations
TL;DR: In this paper, the Hermite interpolation formula was employed to obtain a best possible bound for the number of zeros of p-adic exponential polynomials, if the polynomial is small at sufficiently many points.
Abstract: By employing a precise form of the Hermite interpolation formula we obtain a best possible bound for the number of zeros of p-adic exponential polynomials. As companion to this quantitative result we give a best possible bound on the coefficients, if the exponential polynomial is small at sufficiently many points.
8 citations
TL;DR: In this paper, the convergence and summability of simple and multiple trigonometric Fourier series is studied, and the convergence is shown to be convergent and summable for both simple and complex series.
Abstract: This paper studies the problem of the convergence and summability of simple and multiple trigonometric Fourier series.Bibliography: 21 titles.
8 citations
TL;DR: In this paper, the cardinal trigonometric spline interpolation at the nodes to data of power growth is shown to be not unique, and an extremal property for its restriction to $[0, ε ] is shown.
Abstract: About a decade ago Schoenberg introduced trigonometric splines which are related to the differential operator $\Delta m = D(D^2 + 1^2 ) \cdots (D^2 + m^2 )$. Here we introduce the cardinal trigonometric splines and show that cardinal trigonometric interpolation at the nodes to data of power growth is not unique. We also study trigonometric Euler splines and an extremal property for its restriction to $[0,\eta ]$. We prove a similar result for cardinal L-splines of Micchelli.
7 citations
6 citations
TL;DR: In this paper, the possibility of solving the following problems with a given set of Muntz polynomials on a real interval is demonstrated: (i) approximation of a continuous function by a copositive MUND polynomial, (ii) approximation by a comonotone MUNDO, and (iii) interpolation by piecewise monotone mUNDO.
Abstract: The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of \"Muntz polynomials\" on a real interval is demonstrated: (i) approximation of a continuous function by a \"copositive\" Muntz polynomial; (ii) approximation of a continuous function by a \"comonotone\" Muntz polynomial; (iii) approximation of a continuous function with a monotone fcth difference by a Muntz polynomial with a monotone fcth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs. The following related questions have been settled: I iMonotone Approximation). Let fix) he a continuous function with the property that the /th difference u¿f> 0 on [0, 1] where / is some nonnegative integer. Must there be for a given e > 0 a corresponding polynomial p(x) with p0)(x) > 0 on [0, 1] such that ||/-p|| = sup \\f(x)-p(x)\\ 0 must there be a corresponding polynomial p(x) that has the same monotonicity as fix) on each of the intervals (*,_!, xj), i = 1, 2,.... k, and such that ||/-p|| < e? Received by the editors October 31, 1973. AMS (MOS) subject classifications (1970). Primary 41A05, 41A10, 41A30, 41A25.
6 citations
TL;DR: In this paper, a special form of the Birkhoff interpolation problem is investigated and an existence theorem for certain types of interpolation which reduces to a theorem of Meir and Sharma for $(0,2)$ interpolation by $C^3 $ piecewise quintics.
Abstract: A special form of the Birkhoff interpolation problem is investigated. We prove an existence theorem for certain types of interpolation which, in a particular case, reduces to a theorem of Meir and Sharma for $(0,2)$ interpolation by $C^3 $ piecewise quintics. The method of proof enables us to obtain $L_\infty $-estimates for the error in interpolating smooth functions. These error bounds are shown to be sharp by means of a Baire category argument.
5 citations
TL;DR: A Neville-like algorithm for the computation of the value of the trigonometric interpolation polynomial is presented, including an Algol-procedure.
Abstract: Zur Bestimmung des Wertes des trigonometrischen Interpolationspolynoms wird ein Neville-artiger Algorithmus, auch als Algol-Programm, angegeben. A Neville-like algorithm for the computation of the value of the trigonometric interpolation polynomial is presented, including an Algol-procedure.
5 citations
TL;DR: In this paper, a new interpolation formula for bivariate functions is presented, together with a brief discussion of its properties, including some error analysis, its relationship to other interpolation formulas, and an indication of how it can be generalized.
4 citations
TL;DR: In this article, it was shown that for any S 2n+1-point Gauss interpolation formula, r = [ n/2] + 1, r−1 of the nodes must lie within the interval [a, b], and the remaining node (which may or may not be in [a and b]) must be real.
TL;DR: In this paper, the impact of a symmetrical array geometry, the use of a quantized stored cosine function, the exploitation of Digital Fourier transform algorithms and the application of trigonometric interpolation in the computation of array patterns is discussed.
Abstract: The impact of a symmetrical array geometry, the use of a quantized stored cosine function, the exploitation of Digital Fourier transform algorithms and the application of trigonometric interpolation in the computation of array patterns is discussed. Careful selection of parameters permits sampling the array pattern only 6% above the theoretical Nyquist limit. Reconstruction of array patterns showing −20, −30, and −40‐dB relative interpolation errors are presented. A saving of 8000: 1 in computation time over direct “brute force” array pattern computation is illustrated for a hypothetical array. [Research supported by the Office of Naval Research and the Advanced Research Projects Agency.]
TL;DR: The interpolation formula representation and the kernels associated with the Discrete Fourier Transform (DFT) approach to the interpolation of periodic signals are obtained by viewing the interpolations process as a filtering operation on a properly defined sequence.
TL;DR: In this paper, an interpolation method based on double Fourier expansion yields better results than the conventional linear interpolation, in particular the high-wavenumber part of the spectrum is very well reproduced.
Abstract: A crucial step in the calculation of spectra from objective analyses is the transformation of the analysis to the latitude-longitude grid by some interpolation method. By analysing a synthetic two-dimensional function of known spectral content, it is shown that an interpolation method based on double Fourier expansion yields better results than the conventional linear interpolation. In particular, the minus-third- power profile of the high-wavenumber part of the spectrum is very well reproduced.
TL;DR: In this paper, the Fourier-Jacobi series converges uniformly to a function f(x) on any segment [a, b] ⊂(1; 1).
Abstract: Let α>−1 and Β > −1. Then a function f(x), continuous on the segment [−1; 1], exists such that the sequence of Lagrange interpolation polynomials constructed from the roots of Jacobi polynomials diverges almost everywhere on [−1; 1], and, at the same time, the Fourier-Jacobi series of function f(x) converges uniformly to f(x) on any segment [a; b] ⊂(1; 1).