The minimality properties of Chebyshev polynomials and their lacunary series

By considering four kinds of Chebyshev polynomials, an extended set of (real) results are given for Chebyshev polynomial minimality in suitably weighted Hölder norms on [–1,1], as well as (L ) minimax properties, and best L 1 sufficiency requirements based on Chebyshev interpolation. Finally we establish best L p , L and L 1 approximation by partial sums of lacunary Chebyshev series of the form i=0 a i b i(x) where n (x) is a Chebyshev polynomial and b is an odd integer 3. A complete set of proofs is provided