Chebychev coefficients in approximation of powers of \(x\) (Q1813617)

The author makes use of studies by himself and others on the behavior of generalized polynomials and in particular, oscillating generalized polynomials, in order to obtain estimates on \(E_ n(\alpha)\) and \(E_ n'(\alpha)\) the degree of approximation of \(x^{\alpha}\), \(\alpha>0\), by means of polynomials of degree \(\leq n\) and by means of such polynomials which vanish at \(x=0\), respectively. The estimates are given by means of the coefficients of the Chebyshev polynomials \(T_{2n}\) and \(T_{2n+1}\). If \(\text{coef}(u,v)\) denotes the coefficient of \(x^ u\) in \(T_ v\), then it is shown that for \(\alpha\in[k,k+1]\) we have \(E_ n(\alpha)<E_ n'(\alpha)<1/\text{coef}(2k,2n)\) and if \(\alpha\in[k+1\over 2,k+1]\) then \(E_ n(\alpha)<E_ n'(\alpha)<1/\text{coef}(2k+1,2n+1)\). See earlier work of the author [J. Approximation Theory 23, 163-174 (1978; Zbl 0386.41004)].