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Extensions of Positive Definite Functions pp 151–169Cite as

Green’s Functions

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2160))

Abstract

The focus of this chapter is a detailed analysis of two specific positive definite functions, each one defined in a fixed finite interval, centered at x = 0. Rationale: The examples serve to make explicit some of the many connections between our general theme (locally defined p.d. functions and their extensions), on the one hand; and probability theory and stochastic processes on the other.

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References

  1. I. Armendáriz, A. Cuevas, R. Fraiman, Nonparametric estimation of boundary measures and related functionals: asymptotic results. Adv. Appl. Probab. 41 (2), 311–322 (2009). MR 2541178 (2010k:62137)

    Google Scholar 

  2. D. Alpay, P.E.T. Jorgensen, Stochastic processes induced by singular operators. Numer. Funct. Anal. Optim. 33 (7–9), 708–735 (2012). MR 2966130

    Google Scholar 

  3. D. Alpay, P. Jorgensen, D. Levanony, A class of Gaussian processes with fractional spectral measures. J. Funct. Anal. 261 (2), 507–541 (2011). MR 2793121 (2012e:60101)

    Google Scholar 

  4. D. Alpay, D. Levanony, On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. Potential Anal. 28 (2), 163–184 (2008). MR 2373103 (2009g:46043)

    Google Scholar 

  5. D. Alpay, D. Levanony, Linear stochastic systems: a white noise approach. Acta Appl. Math. 110 (2), 545–572 (2010). MR 2610579 (2011b:93120)

    Google Scholar 

  6. F. Aurzada, On the one-sided exit problem for fractional Brownian motion. Electron. Commun. Probab. 16, 392–404 (2011). MR 2831079 (2012g:60125)

    Google Scholar 

  7. C.J.K. Batty, Vector-valued Choquet theory and transference of boundary measures. Proc. Lond. Math. Soc. (3) 60 (3), 530–548 (1990). MR 1044310 (91f:46061)

    Google Scholar 

  8. T. Hida, Brownian Motion. Applications of Mathematics, vol. 11 (Springer, New York, 1980). Translated from the Japanese by the author and T. P. Speed. MR 562914 (81a:60089)

    Google Scholar 

  9. Z.-C. Hu, W. Sun, Balayage of semi-Dirichlet forms. Can. J. Math. 64 (4), 869–891 (2012). MR 2957234

    Google Scholar 

  10. N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Mathematical Library, vol. 24 (North-Holland, Amsterdam, 1989). MR 1011252 (90m:60069)

    Google Scholar 

  11. A.M. Jaffe, O.E. Lanford III, A.S. Wightman, A general class of cut-off model field theories. Commun. Math. Phys. 15, 47–68 (1969). MR 0253682 (40 #6896)

    Google Scholar 

  12. P.E.T. Jørgensen, A uniqueness theorem for the Heisenberg-Weyl commutation relations with nonselfadjoint position operator. Am. J. Math. 103 (2), 273–287 (1981). MR 610477 (82g:81033)

    Google Scholar 

  13. P.E.T. Jorgensen, E.P.J. Pearse, A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks. Isr. J. Math. 196 (1), 113–160 (2013). MR 3096586

    Google Scholar 

  14. P.E.T. Jorgensen, E.P.J. Pearse, Multiplication operators on the energy space. J. Operator Theory 69 (1), 135–159 (2013). MR 3029492

    Google Scholar 

  15. Y. Movshovich, Inferior mean of Cantor boundary measures. J. Math. Anal. Appl. 386 (2), 780–795 (2012). MR 2834786

    Google Scholar 

  16. E. Nelson, Kernel functions and eigenfunction expansions. Duke Math. J. 25, 15–27 (1957). MR 0091442 (19,969f)

    Google Scholar 

  17. S. Teleman, Topological Properties of the Boundary Measures. Studies in Probability and Related Topics (Nagard, Rome, 1983), pp. 457–463. MR 778242

    Google Scholar 

  18. F. Treves, Book review: pseudodifferential operators and spectral theory. Bull. Am. Math. Soc. (N.S.) 19 (1), 343–346 (1988). MR 1567695

    Google Scholar 

  19. F. Trèves, Basic Linear Partial Differential Equations (Dover, Mineola, NY, 2006). Reprint of the 1975 original. MR 2301309 (2007k:35004)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Iowa, Iowa City, Iowa, USA

    Palle Jorgensen

  2. Department of Mathematics, Wright State University, Dayton, Ohio, USA

    Steen Pedersen

  3. Mathematics Department, Hampton University, Hampton, Virginia, USA

    Feng Tian

Authors
  1. Palle Jorgensen
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  2. Steen Pedersen
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  3. Feng Tian
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Jorgensen, P., Pedersen, S., Tian, F. (2016). Green’s Functions. In: Extensions of Positive Definite Functions. Lecture Notes in Mathematics, vol 2160. Springer, Cham. https://doi.org/10.1007/978-3-319-39780-1_7

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