The main results of this paper concern far reaching generalizations of Wiener's third Tauberian theorem (Thm.29 in his famous book "The Fourier Integral and Certain of Its Applications", published in 1933).

While the classical and well known Tauberian theorem concerns functions which are bounded, and uses integrable kernels (satisfying the Tauberian condition of having a non-vanishing Fourier transform), Wiener's third Tauberian theorem deals with functions having bounded quadratic means (integrals of |f|^2 over intervals [-T,T] are uniformly bounded by the square root of the interval length, i.e. sqrt(2T), and a smaller class of kernels.

In the present paper it is shown that this space of functions can be interpreted as the Banach dual of a certain Banach algebra E2(R), which admits a uniformly bounded family of L^1-normalized stretching operators (similar to L^1, the predual of essentially bounded functions, considered in the classical "first" Tauberian theorem).
It turns out, that the same tools can be used to prove quite similar (and indeed somewhat more general) results for functions having bounded p-means, for general p, p>=1, and for arbitrary dimensions.

There are also further Tauberian theorems involving the class of so-called subordinative operators, i.e. to operators T, such that Tf belongs to the closed linear span of translates of f for each f. It turns out that those operators are convolution operators with suitable distributional kernels.

Cf. also the recent paper for the N.Wiener Centennial.

Keywords: function spaces, Herz spaces, Banach convolution algebras, dyadic decomposition spaces, atomic decomposition.

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