Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about x_0 converges to the function in some neighborhood for every x_0 in its domain. It is important to note that it's a neighborhood and not just at some point x_0 , since every differentiable function has at least a tangent line at every point, which is its Taylor series of order 1. So just having a polynomial expansion at singular points is not enough, and the Taylor series must also converge to the function on points adjacent to x_0 to be considered an analytic function. As a counterexample see the Fabius function. Definitions

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In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifolds by Seeley in 1969, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors (2014).

American Mathematical Society2016

Eigenfunction expansions of ultradifferentiable functions and ultradistributions

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In this paper we give a global characterisation of classes of ultradi_erentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifold by Seeley [See69], and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors.

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Le carré de la fonction des diviseurs

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