The Lp minimality and near-minimality of orthogonal polynomial approximation and integration methods

It is known that the Chebyshev polynomials of the first and second kinds are minimal in Lp on [−1,1] with respect to appropriate weight functions, namely certain powers of 1−x2, for 1≤p≤∞. These properties are here exploited in two applications. First, convergence and optimality properties are established for a "complete" Chebyshev polynomial expansion method for the determination of indefinite integrals. Second, conjectures are derived concerning the near-minimality of the Laguerre polynomials L n ±1/2 (2β ×) for β≃1 with respect to appropriate exponentially weighted Lp norms on [0,∞).