Re: approximation by series expansion

rembremading <rembremading@xxxxxxx> writes:

@Robert Israel

Sorry, for my delayed response. Time shift between Germany and Vancouver
seems to be considerable.
And sorry for my inexactness. But your interpretation was right.
I wanted to approximate
1/(a-x) = 1/(a-cos(theta)) in the region x=[0..1], where a>1 is a
parameter.
And I wanted a reasonable good approximation for arbitrary a in the entire
interval of x.
The taylor expansion usually fails on one end of the intervall and when you

fix it to both ends it fails in the middle ...
Chebyshev approximation is realy quite good. (I remeber, that I did
consider
it before)
However I will face the problem that I have to evaluate the integral for
the
coefficient.
I will have to do this in each step of my numerical simulation, since a
changes everytime.
But maybe I can use this Clenshaw-Curtis quadrature formulae for the
coefficients (http://en.wikipedia.org/wiki/Clenshaw-Curtis_quadrature).
Then I can even get a simple algebraic expression in terms of parameter a.
Up to now I don't understand how I can calculate the "Nyquist frequency"
for
my problem, that is what N will be in this formula on wikipedia.

I have a formula for the coefficients (obtained with the help of Maple):

c_j = 1/pi int_{-pi}^pi 2/(2 a - 1 - cos(t)) cos(j t) dt
= 2/(-1+a)^(1/2)/a^(1/2)*((2*a-1)*((-1+a)*a)^(1/2)+2*a^2-2*a)^j*
((-8*a+1+8*a^2)*((-1+a)*a)^(1/2)+8*a^3+4*a-12*a^2)^(-j)
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • A small numerical error of A. Einstein
    ... I seem to have discovered a small numerical error in Einstein's ... and then to approximate the coefficients: ... coefficient, -0.186, representing the slope of a linear approximation. ...
    (sci.physics.research)
  • Re: DCT and DWT common spectral features
    ... LH3 subband from this example you would get some approximation of the input image. ... which set of DCT coefficients (taken ... then i would consider those two sets of coefficients to capture "simmilar" information about an image. ...
    (comp.compression)
  • Re: erf function in C
    ... > probability) as part of their math libraries? ... * guarantee the error is less than one ulp for erf. ... we use rational approximation to approximate ... * Coefficients for approximation to erf in ...
    (comp.lang.c)
  • Re: approximation by series expansion
    ... The taylor expansion usually fails on one end of the intervall and when you ... Chebyshev approximation is realy quite good. ... coefficient. ... But maybe I can use this Clenshaw-Curtis quadrature formulae for the ...
    (sci.math)
  • Re: Zero Padding in radix 2 FFT
    ... the DFT is a trapezoidal-rule ... > approximation for the integral. ... for arbitrary real coefficients a_k and b_k. ... DFT coefficients only approximate the Fourier series coefficients? ...
    (comp.dsp)