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In article <1992Dec2.170529.24897@news2.cis.umn.edu> 
 pires@milli.cs.umn.edu (Luiz S. Pires) writes:
>
> Please forgive me if these are trivial questions. I am not a mathematician.
>
> When approximating a function with a polynomial, should one use the
> least squares approximating polynomial or a Chebyshev polynomial?
>
> Does L.S. minimize overall error and Chebyshev minimize maximum error?
>
> Which is more commonly used in approximating transcendental functions?
>
> Also, when are other methods (eg Rational Chebyshev approximation) used?
> Any good references?
>
	The idea of "least squares" is the same as orthogonal projection--
"the shortest distance between a point and a line..."  This is tied up
with geometry, ie a metric [inner product].  The Chebyshev polynomials
form an orthonormal basis [q.v.] for the L^2 metric;  other functions
form orthonormal bases for other metrics.

	I cannot guess what you mean by "least squares approximating
polynomial," but approximation implies a notion of distance; best 
approximation(s) follow from that.