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Pólya operators. I: Total positivity. (English) Zbl 0533.34024

It is well known [cf. S. Karlin, Total positivity. Vol. I (1968; Zbl 0219.47030) Chap. 10] that certain ordinary linear differential operators have a Green’s kernel which is totally positive. In the present paper, a class of operators is investigated for which the Green’s kernel and the solutions biorthonormal to the boundary conditions can be uniquely arranged into a joint kernel (the extended Green’s kernel) which is totally positive. The boundary conditions for which this result holds true are derived from an incidence matrix satisfying Pólya’s condition [I. J. Schoenberg, J. Math. Anal. Appl. 16, 538-543 (1966; Zbl 0156.287)].
As an application it is proved that up to a normalization the Green’s kernel is compact and that its limiting functions are two of the basic solutions of the Pólya operator.

MSC:

34B27 Green’s functions for ordinary differential equations
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
47E05 General theory of ordinary differential operators
26A51 Convexity of real functions in one variable, generalizations
34L99 Ordinary differential operators
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References:

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