Thread: Generating functions for multi-dimensional Chebyshev Polynomials?

Hi all, the title pretty much says it all. Does anyone know the generating functions for multi-dimensional Chebyshev Polynomials? Does that question even make sense?

I'm a physicists and I'm the first to admit my maths skills are not as good as they should be. I use Chebyshev Polynomials from time to time as arbitrary fitting functions, i.e. by numerically searching for the co-efficients of the first N Chebyshev's to best fit some unknown function to data. However I want to generalise this to multi-dimensional problems, but I can't see an obvious unique way to do this. I don't really know how to derive the 1D case (I just read the generating function out of a textbook...) so I don't know it it even makes sense to speak of them as functions of more than one variable?

Any advice or feedback would be appreciated.

Originally Posted by bogdanovist

Hi all, the title pretty much says it all. Does anyone know the generating functions for multi-dimensional Chebyshev Polynomials? Does that question even make sense?

I'm a physicists and I'm the first to admit my maths skills are not as good as they should be. I use Chebyshev Polynomials from time to time as arbitrary fitting functions, i.e. by numerically searching for the co-efficients of the first N Chebyshev's to best fit some unknown function to data. However I want to generalise this to multi-dimensional problems, but I can't see an obvious unique way to do this. I don't really know how to derive the 1D case (I just read the generating function out of a textbook...) so I don't know it it even makes sense to speak of them as functions of more than one variable?

Any advice or feedback would be appreciated.

You need to construct a set of orthogonal polynomials with repect to an inner product appropriate to the problem at hand. Then the fitting process is analagous to computing the first few Fourier coefficients.

Alternativly try googling with a search string like "polynomials dimensional chebyshev"

CB