Contributions to the theory of Hermitian series III. Meanvalues

Einar Hille

International Journal of Mathematics and Mathematical Sciences, 1980, vol. 3, 1-15

Abstract:

Let f ( z ) be holomorphic in the strip − σ < y < σ < ∞ and satisfy the conditions for having an expansion in an Hermitian series f ( z ) = ∑ n = 0 ∞ f n h n ( z ) , h n ( z ) = ( π 1 2 2 n n ! ) − 1 2 e − 1 2 z 2 H n ( z ) , absolutely convergent in the strip. Two meanvalues 𝔐 k ( f ; y ) = { π − 1 2 ∫ − ∞ ∞ e − k x 2 | f ( x + i y ) | 2 d z } 1 2 , k = 0 , 1. are discussed, directly using the condition on f ( z ) or via the Hermitian series. Integrals involving products h m ( x + i y ) h n ( x − i y ) are discussed. They lead to expansions of the mean squared in terms of Laguerre functions of y 2 when k = 0 and in terms of Hermite functions h n ( 2 1 2 i y ) when k = 1 . The sumfunctions are holomorphic in y . They are strictly increasing when | y | increases.

Date: 1980
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:869379

DOI: 10.1155/S0161171280000294

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