Gabor meets Littlewood--Paley: Gabor expansions in \(L^p({\mathbb R}^d)\) (Q2717579)

Let \(W=(L^\infty, \ell^1)\) denote the amalgam space (consisting of functions on \(R^d\)) defined by the norm \(\|f\|= \sum_{k\in Z^d} \sup_{x\in [0,1]^d}|f(x+k)|\). Denote the Köthe dual by \(\widetilde{W}\). Under the assumption that \(g\in W\) generates a Gabor frame \(\{e^{2\pi i \beta n\cdot t}g(t-\alpha k)\}_{k,n\in Z^d}\) for \(L^2(R^d)\) and that the dual generator \(\gamma\) also belongs to \(W\) it is proved that a function \(f\in \widetilde{W}\) belongs to \(L^p\) for a \(1<p<\infty\) if and only if the Gabor coefficients \(\{\langle f, e^{2\pi i \beta n\cdot t}g(t-\alpha k) \rangle \}_{k,n\in Z^d}\) belong to a certain sequence space \(s^p\), in case of which the natural norm equivalence holds. Thus \(L^p\) can be characterized in terms of the coefficients in a Gabor expansion. It is proved that the same assumptions imply that the partial sums \(\sum_{|k|\leq K} \sum_{|n|\leq N} \langle f, e^{2\pi i \beta n\cdot t}g(t-\alpha k) \rangle e^{2\pi i \beta n\cdot t}\gamma(t-\alpha k) \) of the Gabor expansion of a function \(f\in L^p\) converges to \(f\) in \(L^p\)-norm (the ordering is crucial!). Pointwise convergence is obtained if \(g\) is compactly supported and \(\widehat{g}\in L^1\). Furthermore, the Walnut representation of the frame operator is extended to \(\widetilde{W}\).