Approximation by linear combinations of translates in invariant Banach spaces of tempered distributions via Tauberian conditions

This paper describes an approximation theoretic approach to the problem of completeness of a set of translates of a “Tauberian generator”, which is an integrable function whose Fourier transform does not vanish. This is achieved by the construction of finite rank operators, whose range is contained in the linear span of the translates of such a generator, and which allow uniform approximation of the identity operator over compact sets of certain Banach spaces $(B,{\Vert \cdot \Vert }_B)$. The key assumption is availability of a double module structure on $(B,{\Vert \cdot \Vert }_B)$, meaning the availability of sufficiently many smoothing operators (via convolution) and also pointwise multipliers, allowing localization of its elements. This structure is shared by a wide variety of function spaces and allows us to make explicit use of the Riesz–Kolmogorov Theorem characterizing compact subsets in such Banach spaces.

The construction of these operators is universal with respect to large families of such Banach spaces, i.e. they do not depend on any further information concerning the particular Banach space. As a corollary we conclude that the linear span of the set of the translates of such a Tauberian generator is dense in any such space $(B,{\Vert \cdot \Vert }_B)$. Our work has been inspired by a completeness result of V. Katsnelson which was formulated in the context of specific Hilbert spaces within this family and Gaussian generators.