Strictly Hermitian Positive Definite Functions
| dc.creator | Pinkus, Allan | |
| dc.date | 2004-04-01 | |
| dc.date.accessioned | 2026-07-07T05:06:58Z | |
| dc.date.available | 2026-07-07T05:06:58Z | |
| dc.description | Let H be any complex inner product space with inner product <, >. We say that f : C -->C is Hermitian positive definite on H if the matrix $$(f(<z^r,z^s>))_{r,s=1}^n \eqno(*)$$ is Hermitian positive definite for all choice of z^1,...,z^n in H, all n. It is strictly Hermitian positive definite if the matrix (*) is also non-singular for any choice of distinct z^1,...,z^n in H. In this article we prove that if dim H >= 3, then f is Hermitian positive definite on H if and only if $$f(z) = \sum_{k,m =0}^\infty b_{k,m} z^k \oz^m \eqno(**)$$ where \oz is the conjugate of z, b_{k,m}>= 0, all k,m in Z_+, and the series converges for all z in C. We also prove that f of the form (**) is strictly Hermitian positive definite on any H if and only if the set $$J={(k, m) : b_{k,m}> 0}$$ is such that (0,0) is in J, and every arithmetic sequence in Z intersects the values {k-m : (k,m)\in J} an infinite number of times. | |
| dc.description | 25 pages | |
| dc.identifier | https://arxiv.org/abs/math/0404013 | |
| dc.identifier | http://arxiv.org/abs/math/0404013 | |
| dc.identifier | Journal d'Analyse Math. 94 (2004), 293-318. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70677 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 43A35 | |
| dc.title | Strictly Hermitian Positive Definite Functions | |
| dc.type | text |