Abstract
While the present chapter is made up of examples, we emphasis that the selection of examples is carefully chosen;—chosen and presented in such a way that the details involved, do in fact cover and illustrate general and important ideas, which in turn serve to bring out the structure of more general theorems (in the second half of our monograph.) Moreover, the connection from the examples to more general contexts will be mentioned inside the present chapter, on a case-by-case basis. We further emphasize that some of the examples have already been used to illustrate key ideas in Chaps. 1 and 2 above. Below we flesh out the details. And our present examples will be used again in Chaps. 6 through 11 below, dealing with theorems that apply to any number of a host of general settings.
It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances.
—Hannah Arendt, The Life of the Mind (1971), p. 7.
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References
D.E. Dutkay, P.E.T. Jorgensen, Fourier duality for fractal measures with affine scales. Math. Comp. 81 (280), 2253–2273 (2012). MR 2945155
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)
J. Du, On non-zero-sum stochastic game problems with stopping times. ProQuest LLC, Ann Arbor, MI. Ph.D. Thesis, University of Southern California (2012). MR 3103657
P. Jorgensen, S. Pedersen, F. Tian, Translation representations and scattering by two intervals. J. Math. Phys. 53 (5), 053505, 49 (2012). MR 2964262
P. Jorgensen, F. Tian, Unbounded operators, lie algebras, and local representations, in Operator Theory (Springer, Basel, 2014), pp. 1–21
P.D. Lax, R.S. Phillips, Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume. Trans. Am. Math. Soc. 289 (2), 715–735 (1985). MR 784011 (86f:11045)
P.D. Lax, R.S. Phillips, Scattering Theory, 2nd edn. Pure and Applied Mathematics, vol. 26 (Academic, Boston, MA, 1989). With appendices by Cathleen S. Morawetz and Georg Schmidt. MR 1037774 (90k:35005)
E. Nelson, Analytic vectors. Ann. Math. (2) 70, 572–615 (1959). MR 0107176 (21 #5901)
S. Pedersen, F. Tian, Momentum operators in the unit square. Integr. Equ. Oper. Theory 77 (1), 57–88 (2013). MR 3090164
W. Rudin, Functional Analysis. McGraw-Hill Series in Higher Mathematics (McGraw-Hill, New York, 1973). MR MR0365062 (51 #1315)
W. Rudin, Fourier Analysis on Groups. Wiley Classics Library (Wiley, New York, 1990). Reprint of the 1962 original, A Wiley-Interscience Publication. MR 1038803 (91b:43002)
K. Schmüdgen, On commuting unbounded selfadjoint operators. I. Acta Sci. Math. (Szeged) 47 (1–2), 131–146 (1984). MR 755571 (86b:47045)
K. Schmüdgen, On commuting unbounded selfadjoint operators. III. Manuscripta Math. 54 (1–2), 221–247 (1985). MR 808690 (87h:47061)
K. Schmüdgen, A note on commuting unbounded selfadjoint operators affiliated to properly infinite von Neumann algebras. II. Bull. Lond. Math. Soc. 18 (3), 287–292 (1986). MR 829589 (87g:47079)
K. Schmüdgen, On commuting unbounded selfadjoint operators. IV. Math. Nachr. 125, 83–102 (1986). MR 847352 (88j:47026)
K. Schmüdgen, J. Friedrich, On commuting unbounded selfadjoint operators. II. Integr. Equ. Oper. Theory 7 (6), 815–867 (1984). MR 774726 (86i:47032)
A. Sokol, An elementary proof that the first hitting time of an open set by a jump process is a stopping time, in Séminaire de Probabilités XLV. Lecture Notes in Mathematics, vol. 2078 (Springer, Cham, 2013), pp. 301–304. MR 3185919
F. Tian, On commutativity of unbounded operators in Hilbert space. Ph.D. thesis, University of Iowa (2011)
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Jorgensen, P., Pedersen, S., Tian, F. (2016). Examples. In: Extensions of Positive Definite Functions. Lecture Notes in Mathematics, vol 2160. Springer, Cham. https://doi.org/10.1007/978-3-319-39780-1_4
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