The expected L-p norm of random polynomials

The results of this paper concern the expected L-p norm of random polynomia ls on the boundary of the unit disc (equivalently of random trigonometric p olynomials on the interval [0, 2 pi]). Specifically, for a random polynomia l [GRAPHICS] let parallel toq(n)parallel to (p)(p) = integral (2 pi)(o)\q(n)(theta)\(p)d the ta/(2 pi). Assume the random variables X-k; k greater than or equal to 0, are independ ent and identically distributed, have mean 0, variance equal to 1 and, if p >2, a finite p(th) moment E(\X-k\(p)). Then E(parallel toq(n)parallel to (p)(p))/n(p/2) --> Gamma (1+p/2) and E(parallel toq(n)((r))parallel to (p)(p))/n((2r+1)p/2) --> (2r+1)(-p/2) Gam ma (1+p/2) as n --> infinity In particular if the polynomials in question have coefficients in the set { +1, -1} (a much studied class of polynomials), then we can compute the expe cted L-p norms of the polynomials and their derivatives E(parallel toq(n)parallel top)/n(1/2) --> (Gamma (1+p/2))(1/p) and E(parallel toq(n)((r))parallel top)/n((2r+1))/2 --> (2r+1)(-1/2) (Gamma (1p/2))(1/p). This complements results of Fielding in the p := 0 case, Newman and Byrnes in the p := 4 case, and Littlewood et al. in the p = infinity case.