Abstract
In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our paper [Trans. Amer. Math. Soc. 368 (2016), pp.8481-8498]. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mappings between these spaces on compact manifolds.
Highlights
In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our paper [Trans
If E is a positive elliptic pseudo-differential operator on a compact manifold X without boundary and λj denotes its eigenvalues in the ascending order, smooth functions on X can be characterised in terms of their Fourier coefficients: (1.1) f ∈ C∞(X) ⇐⇒ ∀N ∃CN : |f (j, k)| ≤ CN λ−j N for all j ≥ 1, 1 ≤ k ≤ dj, where f (j, k) = f, ekj L2 with ekj being the kth eigenfunction corresponding to the eigenvalue λj; see (2.1)
In [4] we extended such characterisations to Gevrey classes and, more generally, to Komatsu classes of ultradifferentiable functions and the corresponding classes of ultradistributions
Summary
Let X be a closed C∞-manifold of dimension n endowed with a fixed measure dx. We first recall an abstract statement from [5, Theorem 2.1] giving rise to the Fourier analysis on L2(X). For f ∈ H, we denote f (j, k) := (f, ekj )H and let f (j) ∈ Cdj denote the column of f (j, k), 1 ≤ k ≤ dj. (iii) If in addition all ekj are in the domain of T ∗, for each l ∈ N0 there exists a matrix σ(l) ∈ Cdl×dl such that for all f ∈ H∞ we have. Hj = span {ekj }dkj=1, and we have dj = dim Hj. Here we will consider H = L2(X) for a compact manifold X with Hj being the eigenspaces of an elliptic positive pseudo-differential operator E. We denote by Ψν+e(X) the space of positive elliptic pseudo-differential operators on order ν > 0 on M. The Fourier coefficients of f ∈ L2(X) with respect to the orthonormal basis {ekj } are denoted by (2.1).
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