By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recovering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalism. The key property on which we rely is the fact that the Fourier transforms of the retarded and advanced functions in the energy variable have to be the "unique Carlsonian analytic interpolations" of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete Matsubara imaginary energies, respectively in the upper and lower half-planes. Starting from the Fourier coefficients regarded as "data set", we then develop a method based on the Pollaczek polynomials for constructing explicitly their analytic interpolations.
We study the action of Radon-type integral transformations on the one-sheeted hyperboloid X d -i and on the associated complex quadric XjfLi-Classes of invariant Volterra kernels and perikernels with moderate growth are transformed into subspaces of holomorphic functions whose Fourier-Laplace analysis has been treated in Part L 1991 Mathematics Subject Classification: 44A10, 44A12, 44A35, 33A90. II -Horocycles and Radon-Abel transformations on the (real and complexified) one-sheeted hyperboloidsThe main purpose of the present work, narnely the connection between the harmonic analysis on the sphere and on the one-sheeted hyperboloid in general dimension d, will be achieved in Part III äs a set of results which we have called "Theorem FG" (see our general introduction in Part I). The latter has been fully established for the case d = 2 (circle and hyperbola) in Part I ( See. 1-4). For the general case d > 3, we shall propose a method which reduces the proof of the theorem FG to results on holomorphic functions of a single variable, established in Part I (See. 1-3) under the name of Property (FG) 0 . This reduction will require the use of: i) a Radon-type integral transformation ^d acting on the invariant Volterra kernels K on the one-sheeted hyperboloid XA-I [FaV].ii) the extension of the latter, denoted by ^j c) acting on invariant perikernels 3C on the corresponding complex quadric Xjf-i·The result of the action of the integral transformation ^j c) on a perikernel Jf on Xjfl i will be a function/(0) of a single complex variable enjoying the properties described Brought to you by | University of Arizona Authenticated Download Date | 7/8/15 1:49 AM
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It is proven that for each given two-field channel -called the "t−channel"with (off-shell) "scattering angle" Θ t , the four-point Green's function of any scalar Quantum Fields satisfying the basic principles of locality, spectral condition together with temperateness admits a Laplace-type transform in the corresponding complex angular momentum variable λ t , dual to Θ t . This transform enjoys the following properties: a) it is holomorphic in a half-plane of the form Re λ t > m, where m is a certain "degree of temperateness" of the fields considered, b) it is in one-to-one (invertible) correspondence with the (off-shell) "absorptive parts" in the crossed two-field channels, c) it extrapolates in a canonical way to complex values of the angular momentum the coefficients of the (off-shell) t−channel partial-wave expansion of the Euclidean four-point function of the fields. These properties are established for all space-time dimensions d + 1 with d ≥ 2.
In this paper we consider the problems of object restoration and image extrapolation, according to the regularization theory of improperly posed problems. In order to take into account the stochastic nature of the noise and to introduce the main concepts of information theory , great attention is devoted to the probabilistic methods of regularization. The kind of the restored continuity is investigated in détail; in particular we prove that, while the image extrapolation présents a Hôlder type stability, the object restoration bas only a logarithmic continuity.
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