On differentiability and analyticity of positive definite functions |
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Authors: | Jorge Buescu AC Paixão |
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Institution: | a Dep. Matemática, FCUL and CMAF, Portugal b Área Departamental de Matemática, ISEL, Portugal |
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Abstract: | 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 that the global behavior of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin: if a single one of these vanishes then the function is constant; if they are all non-zero and satisfy a natural growth condition, the function is real-analytic and consequently extends holomorphically to a maximal horizontal strip of the complex plane. |
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Keywords: | Positive definite functions Inequalities Analytic functions |
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