Eigenfunction expansions of ultradifferentiable functions and ultradistributions in $$\mathbb R^n$$ R n

Abstract

Abstract. We obtain a characterization of S {Mp} {Mp} (R n ) and S(Mp) (R n ), the general Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for the Fourier coefficients of their elements with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. Moreover, we show that the eigenfunctions of such operators are absolute Schauder bases for these spaces of ultradifferentiable functions. Our characterization extends earlier results by Gramchev et al. (Proc. Amer. Math. Soc. 139 (2011), 4361-4368) for Gevrey weight sequences. It also generalizes to R n recent results by Dasgupta and Ruzhansky which were obtained in the setting of compact manifolds. IntroductionBack in 1969 Seeley characterized [18] real analytic functions on a compact analytic manifold via the decay of their Fourier coefficients with respect to eigenfunction expansions associated to a normal analytic elliptic differential operator. In recent times, this result by Seeley has attracted much attention and has been generalized in several directions. In a recent article [6], Dasgupta and Ruzhansky extended Seeley's work and achieved the eigenfunction expansion characterization of DenjoyCarleman classes of ultradifferentiable functions, of both Roumieu and Beurling type, and the corresponding ultradistribution spaces on a compact analytic manifold. See also [5] for Gevrey classes on compact Lie groups.Such results have also a global Euclidean counterpart. In this setting, it is natural to consider differential operators of Shubin type, that is, differential operators with polynomial coefficientsIn [9] Gramchev, Pilipović, and Rodino used this type of operators to give an analogue to Seeley's result for some classes of Gelfand-Shilov spaces. The aim of this paper is to extend the results from [9] by supplying a characterization of the general Gelfand-Shilov spaces S {Mp} (R n ) = S {Mp} {Mp} (R n ) and(Mp) (R n ) of ultradifferentiable functions of Roumieu and Beurling type [3,4,7,8,15]. Our characterization is as follows. We refer to Section 2 for the notation. Note that if P is globally elliptic and normal (P P * = P * P ), then there is an orthonormal basis of L 2 (R n ) consisting of eigenfunctions of P . Properties of

The nuclearity of Gelfand–Shilov spaces and kernel theorems

“…The study of nuclearity for spaces of type S goes back to Mityagin [20], and has recently captured much attention [4,5,6,11,25]; particularly, in connection with applications to microlocal analysis of pseudo-differential operators and the convolution theory for generalized functions. We mention that in some cases nuclearity becomes a straightforward consequence of sequence space representations provided by eigenfunction expansions with respect to various PDO [8,18,31]. However, such representations are not available for all Gelfand-Shilov spaces.…”

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations

“…The obtained characterisations of Komatsu classes found their applications, for example for the well-posedness problems for weakly hyperbolic partial differential equations [8]. The spaces of coefficients of eigenfunction expansions in R n with respect to the eigenfunctions of the harmonic oscillator have been analysed in [9] , and the corresponding Komatsu classes have been investigated in [26]. The original Komatsu spaces of ultradifferentiable functions and ultradistributions have appeared in the works [11][12][13] by Komatsu (see also Rudin [18]), extending the original works by Roumieu [17].…”

Ultradistributional boundary values of harmonic functions on the sphere

“…Note that eigenfunction expansions of ultradistributions on compact analytic manifolds have recently been investigated in [8,9] with the aid of pseudodifferential calculus (cf. [28] for the Euclidean global setting). However, our approach here is quite different and is rather based on explicit estimates for partial derivatives of solid harmonics and spherical harmonics that are obtained in Section 3.…”