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Extra resources for A tutorial on Principal Component Analysis

Example text

The numbers a, and w, are called the Jacobi-Szego parameters of p. w,, n 2 0. (2) It can be easily checked that A, = JRIPn(x)12dp(z). Assume that the sequence satisfies the condition that infn>O - A" :/ > 0. Define a complex Hilbert space rp by with norm I\ . 1) given by + Let @, = (0,. , O , l , O , . ) with 1 in the (n 1)st component. -I = 0 by convention and an's and w,'s are the Jacobi-Szego parameters of p. It can be easily shown that the operators a+ and a- are adjoint t o each other. The Hilbert space F p together with the operators {a+, a-, a o } is called the interacting Fock space associated with the measure p.

In the ”anti-commutative’’ case we found the commutation relation (38). From Eq. (38) we get that in the ”anticommutative” case the causally normally ordered form of a Fermi white noise equations is the same as in the Bose case. ) was visiting at Centro Vito Volterra. V. 1 for scientific schools. Appendix A. t Ut = 1- i I’ dt’(Dbi, + Dtbr)Uti . Gt = 1+ i 1 dt’(Dbi, + Dtbtl)ctj satisfies t Consider the iterated series U ( N )for the solution of the integral equation with the initial condition u(0)= and relation 1 41 and the same series for 6 with the initial condition 6p)= 1.

Let a, be the leading coefficient of Qn(x) and let P,(x) = &,(%)/a,. Then the polynomials {P,(x)}are the orthogonal polynomials satisfying Equation (1) for the measure p. 1 is satisfied. Case 1: h ( z ) = ez measure Gaussian Poisson gamma negative binomial I polynomials Hermite Charlier Laguerre Meixner 46 Case 2: h ( z )= (1 - z)' measure polynomials uniform arcsine semi-circle beta-type Legendre Chebyshev of 1st kind Chebyshev of 2nd kind Gegenbauer The above polynomials are derived from the power series expansion of the resulting generating functions.