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Register a new userAnalytic Continuation of Generalized Trigonometric Functions
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Pisheng Ding
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Via a unified geometric approach, a class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning commutation with rotation, continuation beyond the domain of univalence, and periodicity.
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P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We continue the studies of Moutard-type transform for generalized analytic functions started in our previous paper: arXiv:1510.08764. In particular, we suggest an interpretation of generalized analytic functions as spinor fields and show that in the framework of this approach Moutard-type transforms for the aforementioned functions commute with holomorphic changes of variables.
We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point and additional analyticity properties. Within the class of functions analytic on a common Riemann surface Omega, with a common rate of growth and a common Maclaurin polynomial, we prove an optimality result on their reconstruction at arbitrary points in Omega, and find a procedure to attain it. This procedure uses the uniformization theorem; the optimal reconstruction errors depend only on the conformal distance to the origin. A priori knowledge of Omega is rigorously available for functions often encountered in analysis (such as solutions of meromomorphic ODEs and classes of PDEs). It is also available, rigorously or conjecturally based on numerical evidence, for perturbative expansions in quantum mechanics, statistical physics and quantum field theory, and in other areas in physics. For a subclass of such functions, we provide the optimal procedure explicitly. These include the Borel transforms of the linear special functions. We construct, in closed form, the uniformization map and optimal procedure for the Borel plane of nonlinear special functions, tronquee solutions of the Painleve equations P_I--P_V. For the latter, $Omega$ is the covering of CZ by curves with fixed origin, modulo homotopies. We obtain some of the uniformization maps as rapidly convergent limits of compositions of elementary maps. Given further information about the function, such as is available for the ubiquitous class of resurgent functions, significantly better approximations are possible and we construct them. In particular, any chosen one of their singularities can be eliminated by specific linear operators which we introduce, and the local structure at the chosen singularity can be obtained in fine detail.
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P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We continue studies of Moutard-type transforms for the generalized analytic functions started in arXiv:1510.08764, arXiv:1512.00343. In particular, we show that generalized analytic functions with the simplest contour poles can be Moutard transformed to the regular ones, at least, locally. In addition, the later Moutard-type transforms are locally invertible.
Bayesian parametric analytic continuation (BPAC) is proposed for the analytic continuation of noisy imaginary-time Greens function data as, e.g., obtained by continuous-time quantum Monte Carlo simulations (CTQMC). Within BPAC, the spectral function is inferred from a suitable set of parametrized basis functions. Bayesian model comparison then allows to assess the reliability of different parametrizations. The required evidence integrals of such a model comparison are determined by nested sampling. Compared to the maximum entropy method (MEM), routinely used for the analytic continuation of CTQMC data, the presented approach allows to infer whether the data support specific structures of the spectral function. We demonstrate the capability of BPAC in terms of CTQMC data for an Anderson impurity model (AIM) that shows a generalized Kondo scenario and compare the BPAC reconstruction to the MEM as well as to the spectral function obtained from the real-time fork tensor product state impurity solver where no analytic continuation is required. Furthermore, we present a combination of MEM and BPAC and its application to an AIM arising from the ab initio treatment of SrVO$_3$.
By a famous result, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation a.e. on $T$. More can be said if the spectrum of the associated inner function has holes on $T$. Then the functions of the invariant subspaces even extend analytically through these holes. We will discuss the situation in weighted backward shift invariant subspaces. The results on analytic continuation will be applied to consider some embeddings of weighted invariant subspaces into their unweighted companions. Such weight
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