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The use of the Chebyshev norm is indicated in many cases where the residuals of the fit are known not to follow a Gaussian distribution, in particular for all approximations of an empirical nature, where residuals are dominated by the inadequacy of the approximation rather than the errors of the measurements being approximated. Programs performing fits using the Chebyshev norm are usually more time consuming than least squares fit programs, but can be found in some program libraries. A specific application to track fitting can be found in [James83].
Html9/3/2006 14:14:15 Chebyshev Polynomials next up previous contents index Next: Chi-Square Distribution Up: No Title Previous: Chebyshev Norm Chebyshev Polynomials For the Chebyshev polynomials of the first kind are defined by In particular, for is a Fourier series in . A Chebyshev series in x, . Terms , etc. can be ignored (for ) as long as is smaller than the error one can tolerate. The truncated series can be computed by the recursion formula ( which is numerically stable for Horner's Rule) .
A few important ones are listed below. The convolution of two two normal distributions with zero mean and variances distribution with zero mean and variance is a normal . distributions with f1 and f2 degrees of freedom is a The convolution of two and distribution with f1 +f2 degrees of freedom. The convolution of two Poisson distributions with parameters parameter . html (1 of 4)9/3/2006 14:14:41 and is a Poisson distribution with Convolution The convolution of an exponential and a and a normal distribution is approximated by another exponential distribution.