RIESZ-SUMMABILITY OF MULTIPLE HERMITE SERIES

Let E(lambda)(alpha)integral(y)=Sigma(1-mu(k)/lambda(alpha)a(k) phi k(y), mu(k)<lambda be the Riesz means of order alpha of the multiple Hermite series of the function f. In this Note we expose the following results: (i) The convergence E(lambda)(alpha)f-->f as lambda-->infinity into the space L(m)(1) = L(m)(1)(R(n)), n greater than or equal to 2, with a norm integral(1+\x\)(m)f(x)\dx if alpha>(n-1)/2 and -2 alpha-2/3 less than or equal to m less than or equal to 2 alpha+1-n. These conditions on alpha and m are optimal. (ii) The convergence E(lambda)(alpha)f(y)-->f(y) for alpha>(n-1)/2 onto the Lebesgue set of the function f from the spaces: L(m)(1)(m greater than or equal to-2 alpha-2/3, its dual L(m) infinity (m<2 alpha+1-n) and L(p)=L(p)(R(n)) (1 less than or equal to p<infinity). (iii) The localization principle for alpha greater than or equal to(n- 1)/2 into the spaces: L(m)(1)(m>-2 alpha 2/3), L(m) infinity(m<2 alpha+1-n) and L(P) (1 less than or equal to p<infinity).