On polynomial interpolation at the points of a geometric progression

1988 ◽  
pp. 391-403 ◽  
Author(s):  
I. J. Schoenberg
Keyword(s):  
Polynomial Interpolation ◽  
Geometric Progression

On polynomial interpolation at the points of a geometric progression

1981 ◽  
Vol 90 (3-4) ◽  
pp. 195-207 ◽  
Author(s):  
I. J. Schoenberg
Keyword(s):  
Quadrature Formula ◽  
Polynomial Interpolation ◽  
Theta Functions ◽  
Important Case ◽  
Divided Differences ◽  
Geometric Progression ◽  
Close Connection ◽  
Numerical Examples ◽  
Interpolation Series

SynopsisThis note pursues two aims: the first is historical and the second is factual.1. We present James Stirling's discovery (1730) that Newton's general interpolation series with divided differences simplifies if the points of interpolation form a geometric progression. For its most important case of extrapolation at the origin. Karl Schellbach (1864) develops his algorithm of q-differences that also leads naturally to theta-functions. Carl Runge (1891) solves the same extrapolation at the origin, without referring to the Stirling-Schellbach algorithm. Instead, Runge uses “Richardson's deferred approach to the limit” 20 years before Richardson.2. Recently, the author found a close connection to Romberg's quadrature formula in terms of “binary” trapezoidal sums. It is shown that the problems of Stirling, Schellbach, and Runge, are elegantly solved by Romberg's algorithm. Numerical examples are given briefly. Fuller numerical details can be found in the author's MRC T.S. Report #2173, December 1980, Madison, Wisconsin. Thanks are due to the referee for suggesting the present stream-lined version.

Polynomial interpolation at points of a geometric mesh on a triangle

1988 ◽  
Vol 108 (1-2) ◽  
pp. 75-87 ◽  
Author(s):  
S. L. Lee ◽  
G. M. Phillips
Keyword(s):  
Polynomial Interpolation ◽  
Geometric Progression ◽  
One Dimension ◽  
Two Dimensional ◽  
Geometric Mesh ◽  
Lagrange Form

SynopsisIn an earlier paper [8], I. J. Schoenberg discussed polynomial interpolation in one dimension at the points of a geometric progression, which was originally proposed by James Stirling. In the present paper, these ideas are generalised to two-dimensional polynomial interpolation at the points of a geometric mesh on a triangle. A Lagrange form is obtained for this interpolating polynomial and an algorithm is derived for evaluating it efficiently.

scholarly journals Profiled Horn Antenna with Wideband Capability Targeting Sub-THz Applications

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 412
Author(s):  
Jay Gupta ◽  
Dhaval Pujara ◽  
Jorge Teniente
Keyword(s):  
Hermite Polynomial ◽  
Polynomial Interpolation ◽  
Horn Antenna ◽  
Submillimeter Wavelengths ◽  
Thz Applications ◽  
Millimeter And Submillimeter Wavelengths ◽  
Better Than

This paper proposes a wideband profiled horn antenna designed using the piecewise biarc Hermite polynomial interpolation and validated experimentally at 55 GHz. The proposed design proves S11 and directivity better than −22 dB and 25.5 dB across the entire band and only needs 3 node points if compared with the well-known spline profiled horn antenna. Our design makes use of an increasing radius and hence does not present non-accessible regions from the aperture, allowing its fabrication with electro erosion techniques especially suitable for millimeter and submillimeter wavelengths.

Theoretical bit error floor analysis of 16‐QAM OFDM signal with channel estimation using polynomial interpolation

2016 ◽  
Vol 10 (3) ◽  
pp. 254-265 ◽  
Author(s):  
Vincent Savaux ◽  
Alexandre Skrzypczak ◽  
Yves Louët
Keyword(s):  
Channel Estimation ◽  
Polynomial Interpolation ◽  
Error Floor ◽  
Ofdm Signal
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