The Relative Sizes of the Terms in Chebyshev and other Ultraspherical Expansions

Lanczos (1952) claims that expansion of a function in a seriesof Chebyshev polynomials is usually superior to expansion ina series of ultraspherical polynomials for any possible non-zerovalue of the parameter . We look for conditions on the functionunder which this superiority can be proved and measured; weachieve some success on the assumption that is positive, butthe results for negative are less satisfactory.