a b s t r a c tA directionally sensitive variant of the short-time Fourier transform is introduced which sends functions on R n to those on the parameter space S n−1 × R × R n . This transform, which is named directional short-time Fourier transform (DSTFT), uses functions in L ∞ (R) as window and is related to the celebrated Radon transform. We establish an orthogonality relation for the DSTFT and explore some operator-theoretic aspects of the transform, mostly in terms of proving a variant of the Hausdorff-Young inequality. The paper is concluded by some reconstruction formulas.

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The problem is, of course, more delicate in the general context of Banach spaces. The importance of the space X constructed in [10] is that every infinite-dimensional subspace of X (and hence X itself) is indecomposable, and this is what allows us to call it a hereditarily indecomposable (H.I.) Banach space. The space X, which is also reflexive, is indeed a development of Schlumprecht's space, which is in turn a Tsirelson-type arbitrarily distortable Banach space, together with the Maurey-Rosenthal space [14]. Tsirelson's space was constructed in [15]. The paper [10] contains some remarkable results about H.I. spaces. First, every bounded linear operator T from a complex H.I. space into itself can be written as T = λI + S, where λ ∈ C, I is the identity operator, and S is a strictly singular operator. Second, an H.I. space is not isomorphic to any proper subspace, and the space cannot be, accordingly, isomorphic to its hyperplanes.