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

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. Contents 1. Introduction 81 2. Fourier analysis on compact manifolds 83 3. Sequence spaces and sequential linear mappings 85 4. Tensor representations for Komatsu classes and their α-duals 86 5. Beurling class of ultradifferentiable functions and ultradistributions 98 References 99

Eigenfunction expansions of ultradifferentiable functions and ultradistributions

“…The present paper is a continuation of our papers [4] and [5]. In [5], we analysed the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds.…”

“…The present paper is a continuation of our papers [4] and [5]. In [5], we analysed the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds. We also described the tensor structure of sequential mappings on spaces of Fourier coefficients and characterised their adjoint mappings.…”

“…Such characterisations for analytic functions were obtained by Seeley in [25], with a subsequent extension to Gevrey and, more generally, to Komatsu classes, in [4]. The results obtained in [5] do not include the cases of smooth functions on compact manifolds. We will extend the results in [5] to the spaces of smooth functions.…”

“…The results obtained in [5] do not include the cases of smooth functions on compact manifolds. We will extend the results in [5] to the spaces of smooth functions. Moreover in this work, we aim at discussing an abstract analysis of the spaces of smooth type functions generated by basis elements of an arbitrary Hilbert space.…”

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In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our papers [4] and [5]. We prove that these spaces are perfect sequence spaces.

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