Characterization of L(r) Using the Gabor Frame
نویسندگان
چکیده
We characterize L norms of functions on R for 1 < p < ∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L functions converge to the functions almost everywhere and in L for 1 < p < ∞. In L we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L(R) when 1 ≤ p < ∞.
منابع مشابه
Gabor (super)frames with Hermite Functions
We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions Hn. Let h = (H0, H1, . . . , Hn) be the vector of the first n+ 1 Hermite functions. We give a complete characterization of all lattices Λ ⊆ R such that the Gabor system {e2πiλ2th(t − λ1) : λ = (λ1, λ2) ∈ Λ} is a frame for L(R,C). As a corollary we obtain sufficient conditions for a single H...
متن کاملImage processing by alternate dual Gabor frames
We present an application of the dual Gabor frames to image processing. Our algorithm is based on finding some dual Gabor frame generators which reconstructs accurately the elements of the underlying Hilbert space. The advantages of these duals constructed by a polynomial of Gabor frame generators are compared with their canonical dual.
متن کاملCharacterization and computation of canonical tight windows for Gabor frames
Let (gna,mb)n,m∈Z be a Gabor frame for L (R) for given window g. We show that the window h0 = S− 1 2 g that generates the canonically associated tight Gabor frame minimizes ‖g − h‖ among all windows h generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in ...
متن کاملGabor frames by sampling and periodization
By sampling the window of a Gabor frame for L(R) belonging to Feichtinger’s algebra, S0(R), one obtains a Gabor frame for l(Z). In this article we present a survey of results by R. Orr and A.J.E.M. Janssen and extend their ideas to cover interrelations among Gabor frames for the four spaces L(R), l(Z), L([0, L]) and C. Some new results about general dual windows with respect to sampling and per...
متن کاملConcerning the frame of minimal prime ideals of pointfree function rings
Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of continuous real-valued functions on $L$. We study the frame $mathfrak{O}(Min(mathcal{R}L))$ of minimal prime ideals of $mathcal{R}L$ in relation to $beta L$. For $Iinbeta L$, denote by $textit{textbf{O}}^I$ the ideal ${alphainmathcal{R}Lmidcozalphain I}$ of $mathcal{R}L$. We show that sending $I$ to the set of minimal prime ...
متن کامل