Abstract
The highly important totally positive kernels of the form K(x, y) = f(x−y) where x, y traverse the real line (or the integers), Schoenberg called Pólya frequency (PF) functions (sequences). The two prime examples of PF functions are the normal function
where γ is a positive parameter, and the (truncated) exponential function
where λ is a free positive parameter. In a remarkable series of papers, Schoenberg (see his review paper (1953) [48*]) set the basis of the theory of Pólya frequency functions, and established the fundamental representation theorems. Earlier works of Pólya, Laguerre and Schur aided these developments; their concern was the approximation of functions by polynomials with only real zeros.
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Karlin, S. (1988). Pólya frequency functions and sequences. In: de Boor, C. (eds) I. J. Schoenberg Selected Papers. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-0433-1_18
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DOI: https://doi.org/10.1007/978-1-4899-0433-1_18
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