International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 3, Pages 407-421
Contributions to the theory of Hermitian series III. Meanvalues
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Received 9 March 1979
Copyright © 1980 Einar Hille. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be holomorphic in the strip and satisfy the conditions for having an expansion in an Hermitian series absolutely convergent in the strip. Two meanvalues are discussed, directly using the condition on or via the Hermitian series. Integrals involving products are discussed. They lead to expansions of the mean squared in terms of Laguerre functions of when and in terms of Hermite functions when . The sumfunctions are holomorphic in . They are strictly increasing when increases.