Contributions to the theory of Hermitian series III. Meanvalues
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∞fnhn(z), hn(z)=(π122nn!)−12e−12z2Hn(z),absolutely convergent in the strip. Two meanvalues 𝔐k(f;y)={π−12∫−∞∞e−kx2|f(x+iy)|2dz}12, k=0,1.are discussed, directl...
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| Format: | Article |
| Language: | English |
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Wiley
1980-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171280000294 |
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| author | Einar Hille |
| author_facet | Einar Hille |
| author_sort | Einar Hille |
| collection | DOAJ |
| description | 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∞fnhn(z), hn(z)=(π122nn!)−12e−12z2Hn(z),absolutely convergent in the strip. Two meanvalues 𝔐k(f;y)={π−12∫−∞∞e−kx2|f(x+iy)|2dz}12, k=0,1.are discussed, directly using the condition on f(z) or via the Hermitian series. Integrals involving products hm(x+iy)hn(x−iy) are discussed. They lead to expansions of the mean squared in terms of Laguerre functions of y2 when k=0 and in terms of Hermite functions hn(212iy) when k=1. The sumfunctions are holomorphic in y. They are strictly increasing when |y| increases. |
| format | Article |
| id | doaj-art-7c8cd1bd60f243c58214cad14f5c6d7f |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1980-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-7c8cd1bd60f243c58214cad14f5c6d7f2025-08-20T02:07:24ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251980-01-013340742110.1155/S0161171280000294Contributions to the theory of Hermitian series III. MeanvaluesEinar Hille08862 La Jolla Scenic Drive N., La Jolla, California 92037, USALet 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∞fnhn(z), hn(z)=(π122nn!)−12e−12z2Hn(z),absolutely convergent in the strip. Two meanvalues 𝔐k(f;y)={π−12∫−∞∞e−kx2|f(x+iy)|2dz}12, k=0,1.are discussed, directly using the condition on f(z) or via the Hermitian series. Integrals involving products hm(x+iy)hn(x−iy) are discussed. They lead to expansions of the mean squared in terms of Laguerre functions of y2 when k=0 and in terms of Hermite functions hn(212iy) when k=1. The sumfunctions are holomorphic in y. They are strictly increasing when |y| increases.http://dx.doi.org/10.1155/S0161171280000294 |
| spellingShingle | Einar Hille Contributions to the theory of Hermitian series III. Meanvalues International Journal of Mathematics and Mathematical Sciences |
| title | Contributions to the theory of Hermitian series III. Meanvalues |
| title_full | Contributions to the theory of Hermitian series III. Meanvalues |
| title_fullStr | Contributions to the theory of Hermitian series III. Meanvalues |
| title_full_unstemmed | Contributions to the theory of Hermitian series III. Meanvalues |
| title_short | Contributions to the theory of Hermitian series III. Meanvalues |
| title_sort | contributions to the theory of hermitian series iii meanvalues |
| url | http://dx.doi.org/10.1155/S0161171280000294 |
| work_keys_str_mv | AT einarhille contributionstothetheoryofhermitianseriesiiimeanvalues |