Strictly Hermitian positive definite functions

Abstract

LetH be any complex inner product space with inner product <·,·>. We say thatf: ℂ→ℂ is Hermitian positive definite onH if the matrix

$$(f(< z^r ,z^s ))_{r,s = 1}^n $$

(1)

is Hermitian positive definite for all choice ofz ¹,…,z ⁿ inH for alln. It is strictly Hermitian positive definite if the matrix (*) is also non-singular for any choice of distinctz ¹,…,z ⁿ inH. In this article, we prove that if dimH≥3, thenf is Hermitian positive definite onH if and only if

$$f(z) = \sum\limits_{k,\ell = 0}^\infty {b_{k.\ell ^{z^k z^{ - \ell } ,} } } $$

(1)

whereb _k,l ≥0 for allk, l in ℤ, and the series converges for allz in ℂ. We also prove thatf of the form (**) is strictly Hermitian positive definite on anyH if and only if the setJ={(k,l):b _k,l >0} is such that (0,0)∈J, and every arithmetic sequence in ℤ intersects the values {k−l: (k, l)∈J} an infinite number of times.