In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the padic numbers. The padic numbers contain the padic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp, w). The real analogue of the padic integers is the interval 1,1], and a probability measure w on it gives rise to a special basis for L2( 1,1], w)  the orthogonal polynomials, and to a Markov chain on "finite approximations" of 1,1]. For special (gamma and beta) measures there is a "quantum" or "qanalogue" Markov chain, and a special basis, that within certain limits yield the real and the padic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the padic group GLn(Zp), and between its real (and complex) analogue the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and padic Fourier transform and between the real and padic (local unramified part of) Tate thesis, and Weil explicit sums.
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