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The primary purpose of this paper is to prove inequalities of the type      | | σ  n ( f , x ) W ( x ) | | p ⩽  C | | f ( x ) W ( x ) | | p ||{\sigma _n}(f,x)W(x)|{|_p} \leqslant C||f(x)W(x)|{|_p}       where      σ  n ( f , x ) {\sigma _n}(f,x)       is the      n n      th      ( C , 1 ) (C,1)       mean of the Laguerre or Hermite series of      f , W ( x ) f, W(x)       is a suitable weight function of particular form,      C C       is a constant independent of      f ( x ) f(x)       and      n n      , and the norm is taken over      ( 0 , ∞  ) (0,\infty )       in the Laguerre case and      ( −  ∞  , ∞  ) ( - \infty ,\infty )       in the Hermite case for      1 ⩽  p ⩽  ∞  1 \leqslant p \leqslant \infty      . Both necessary and sufficient conditions for these inequalities to remain valid are determined. For      p &gt; ∞  p &gt; \infty       and      f ( x ) W ( x ) ∈  L p f(x)W(x) \in {L^p}      , mean summability results showing that      lim n →  ∞  | | [ σ  n ( f , x ) −  f ( x ) ] W ( x ) | | p = 0 \lim \nolimits _{n \to \infty } ||[{\sigma _n}(f,x) - f(x)]W(x)|{|_p} = 0       are derived by use of the appropriate density theorems. Detailed proofs are presented for the Laguerre expansions, and the analogous results for Hermite series follow as corollaries.

Mean Cesàro summability of Laguerre and Hermite series