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Von Neumann indices and classes of positive definite functions

### Abstract

With view to applications, we establish a correspondence between two problems: (i) the problem of finding continuous positive definite extensions of functions F which are defined on open bounded domains Ω in , on the one hand; and (ii) spectral theory for elliptic differential operators acting on Ω (constant coefficients). A novelty in our approach is the use of a reproducing kernel Hilbert space computed directly from (Ω, F), as well as algorithms for computing relevant orthonormal bases in .

© 2014 AIP Publishing LLC

Received 03 March 2014
Accepted 27 August 2014
Published online 16 September 2014

Acknowledgments:
The co-authors thank the following for enlightening discussions: Professors Sergii Bezuglyi, Dorin Dutkay, Paul Muhly, Myung-Sin Song, Wayne Polyzou, Gestur Olafsson, Robert Niedzialomski, and members in the Math Physics seminar at the University of Iowa.

Article outline:

I. INTRODUCTION
II. THE REPRODUCING KERNEL HILBERT SPACE
III. TYPE I V.S. TYPE II EXTENSIONS
IV. MERCER OPERATORS
V. THE CASE OF *F*(*x*) = *e* ^{−|x|}, |*x*| < 1
A. The self-adjoint extensions *A* _{θ}⊃ − *iD* _{ F }
B. The operator *T* _{ F }
C. The Spectra of the s.a. extensions *A* _{θ}⊃ − *iD* _{ F }
D. Examples of type 2 extensions
E. The unitary groups
F. Harmonic ONBs in the RKHS : Complex exponentials
VI. ELLIPTIC POSITIVE DEFINITE FUNCTIONS *F*
A. A characterization of for elliptic *F* in terms of the first Sobolev space of (0, *a*)
B. If *F* is elliptic, then the operator *D* _{ F } has indices (1, 1)
C. If *F* is elliptic, then its distribution derivative has a Dirac discontinuity at *x* = 0
D. If *F* is elliptic, then there is an associated system of linear conditions for the kernel functions for at the two endpoints 0 and *a*
E. Two examples *F* = *e* ^{−|x|}, and *F* = 1 − |*x*|, on the respective intervals, are elliptic
F. Translation representation for the unitary one-parameter groups *U*(*t*) in
VII. EXTENSION FROM FINITE OR COUNTABLY INFINITE SUBSETS
VIII. SOME ONBS IN

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/content/aip/journal/jmp/55/9/10.1063/1.4895462

2014-09-16

2016-09-26

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