Abstract
In this chapter, we identify extensions of the initially give positive definite (p.d.) functions F which are associated with operator extensions in the RKHS \(\mathcal{H}_{F}\) itself (Type I), and those which require an enlargement of \(\mathcal{H}_{F}\), Type II. In the case of \(G = \mathbb{R}\) (the real line) some of these continuous p.d. extensions arising from the second construction involve a spline-procedure, and a theorem of G. Pólya, which leads to p.d. extensions of F that are symmetric around x = 0, and convex on the left and right half-lines. Further these extensions are supported in a compact interval, symmetric around x = 0.
“Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either.”—Ian Stewart, Preface to second edition of What is Mathematics?
by Richard Courant and Herbert Robbins
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space (Dover, New York, 1993). Translated from the Russian and with a preface by Merlynd Nestell, Reprint of the 1961 and 1963 translations, Two volumes bound as one. MR 1255973 (94i:47001)
N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis. Translated by N. Kemmer (Hafner, New York, 1965). MR 0184042 (32 #1518)
M. Ahsanullah, M. Maswadah, A.M. Seham, Kernel inference on the generalized gamma distribution based on generalized order statistics. J. Stat. Theory Appl. 12 (2), 152–172 (2013). MR 3190274
N. Dunford, J.T. Schwartz, Linear Operators. Part II. Wiley Classics Library (Wiley, New York, 1988). Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication. MR 1009163 (90g:47001b)
T.N.E. Greville, I.J. Schoenberg, A. Sharma, The behavior of the exponential Euler spline S n (x; t) as n → ∞ for negative values of the base t, in Second Edmonton Conference on Approximation Theory (Edmonton, Alta, 1982), CMS Conference Proceedings, vol. 3 (American Mathematical Society, Providence, RI, 1983), pp. 185–198. MR 729330 (85c:41017)
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)
P.E.T. Jorgensen, Analytic continuation of local representations of symmetric spaces. J. Funct. Anal. 70 (2), 304–322 (1987). MR 874059 (88d:22021)
D. Kundu, A. Manglick, Discriminating between the Weibull and log-normal distributions. Nav. Res. Logist. 51 (6), 893–905 (2004). MR 2079449 (2005b:62033)
M.G.M. Khan, D. Rao, A.H. Ansari, M.J. Ahsan, Determining optimum strata boundaries and sample sizes for skewed population with log-normal distribution. Commun. Stat. Simul. Comput. 44 (5), 1364–1387 (2015). MR 3271013
M. Krein, Concerning the resolvents of an Hermitian operator with the deficiency-index (m, m). C. R. (Doklady) Acad. Sci. URSS (N.S.) 52, 651–654 (1946). MR 0018341 (8,277a)
U. Küchler, S. Tappe, Bilateral gamma distributions and processes in financial mathematics. Stoch. Process. Appl. 118 (2), 261–283 (2008). MR 2376902 (2008k:60109)
S.H. Lin, Comparing the mean vectors of two independent multivariate log-normal distributions. J. Appl. Stat. 41 (2), 259–274 (2014). MR 3291234
A.A. Litvinyuk, On types of distributions for sums of a class of random power series with independent identically distributed coefficients. Ukr. Mat. Zh. 51 (1), 128–132 (1999). MR 1712765 (2000h:60022)
E. Nelson, Analytic vectors. Ann. Math. (2) 70, 572–615 (1959). MR 0107176 (21 #5901)
E. Nelson, Topics in Dynamics. I: Flows. Mathematical Notes (Princeton University Press, Princeton, NJ, 1969). MR 0282379 (43 #8091)
J. Neunhäuserer, Absolutely continuous random power series in reciprocals of Pisot numbers. Stat. Probab. Lett. 83 (2), 431–435 (2013). MR 3006973
G. Pólya, Remarks on characteristic functions, in Proceedings of First Berkeley Conference on Mathematical Statistics and Probability (1949), pp. 115–123
I.J. Schoenberg, Interpolating splines as limits of polynomials. Linear Algebra Appl. 52/53, 617–628 (1983). MR 709376 (84j:41019)
F. Trèves, Basic Linear Partial Differential Equations (Dover, Mineola, NY, 2006). Reprint of the 1975 original. MR 2301309 (2007k:35004)
J. von Neumann, Über adjungierte Funktionaloperatoren. Ann. Math. (2) 33 (2), 294–310 (1932). MR 1503053
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Jorgensen, P., Pedersen, S., Tian, F. (2016). Type I vs. Type II Extensions. In: Extensions of Positive Definite Functions. Lecture Notes in Mathematics, vol 2160. Springer, Cham. https://doi.org/10.1007/978-3-319-39780-1_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-39780-1_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-39779-5
Online ISBN: 978-3-319-39780-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)