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On differentiability and analyticity of positive definite functions
We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels by purely algebraic methods. Our main results show thatExpand
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A linear algebraic approach to holomorphic reproducing kernels in Cn
Abstract Let Ω ⊆ C n be a domain and k be a holomorphic reproducing kernel on Ω . By the Moore–Aronszajn characterization, every finite matrix k ( Z i ,  Z j ) is positive semidefinite. We show that,Expand
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Positive-Definiteness, Integral Equations and Fourier Transforms
We show that positive definite kernel functions k(x, y), if continuous and integrable along the main diagonal, coincide with kernels of positive integral operators in L2(R). Such an operator is shownExpand
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Eigenvalue distribution of Mercer-like kernels
We study eigenvalues of positive definite kernels of L2 integral operators on arbitrary intervals. Assuming integrability and uniform continuity of the kernel on the diagonal, we show that theExpand
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Positive definite matrices and differentiable reproducing kernel inequalities
Abstract Let I ⊆ R be a interval and k : I 2 → C be a reproducing kernel on I. By the Moore–Aronszajn theorem, every finite matrix k ( x i , x j ) is positive semidefinite. We show that, as a directExpand
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Positive-definiteness and integral representations for special functions
We characterize a holomorphic positive definite function $f$ defined on a horizontal strip of the complex plane as the Fourier-Laplace transform of a unique exponentially finite measure onExpand
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Propagation of regularity and positive definiteness: a constructive approach
We show that, for positive definite kernels, if specific forms of regularity (continuity, Sn-differentiability or holomorphy) hold locally on the diagonal, then they must hold globally on the wholeExpand
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Real and complex variable positive definite functions
In this paper we present an overview of the implications of our previously derived results for positive definite kernels on the general theory of positive definite functions. We begin by exploringExpand
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