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

Abstract. We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.

We prove that an ultradistribution is rotation invariant if and only if it coincides with its spherical mean. For it, we study the problem of spherical representations of ultradistributions on R n . Our results apply to both the quasianalytic and the non-quasianalytic case.

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