Let X be a rearrangement-invariant space. An operator T : X → X is called narrow if for each measurable setA and each ε > 0 there exists x ∈ X with x = χA, ∫ xdμ = 0 and ‖Tx‖ < ε. In particular all compact operators are narrow. We prove that if X is a Lorentz function space Lw,p on [0,1] with p > 2, then there exists a constant kX > 1 so that for every narrow projection P on Lw,p ‖Id− P‖ ≥ kX ....

Let Ω be an open subset of Rn. Denote by LpΩ(R n) the closure in Lp(Rn) of the set of all functions ε ∈ L1(Rn)∩Lp(Rn) whose Fourier transform has compact support contained in Ω. The subspaces of the form LpΩ(R n) are called the spectral subspaces of Lp(Rn). It is easily seen that each spectral subspace is translation invariant; i.e., f(x + a) ∈ LpΩ(R) for all f ∈ L p Ω(R n) and a ∈ Rn. Sufficie...

It is shown that the span of fa i h i b i e i g n i=1 , where fh i g is the Haar system in L p and fe i g the canonical basis of`p , is well isomorphic to a well complemented subspace of L p ; 2 < p < 1. As a consequence we get that there is a rearrangement of the (initial segments of the) Haar system in L p ; 2 < p < 1, any block basis of which is well isomorphic to a well complemented subspac...

This paper presents a new efficient approach for the solution of the lp-lq minimization problem based on the application of successive orthogonal projections onto generalized Krylov subspaces of increasing dimension. The subspaces are generated according to the iteratively reweighted least-squares strategy for the approximation of lp/lq-norms by weighted l2-norms. Computed image restoration exa...

W.T.Gowers proved in [G1] the celebrated dichotomy concerning unconditional sequences and hereditarily indecomposable spaces using Ramsey-type argumentation. In [G2] he generalized the reasoning and showed, as an application, a dichotomy concerning quasiminimal spaces, ie. in which any two infinitely dimensional subspaces contain further two infinitely dimensional subspaces which are isomorphic...

This paper presents a computation of the Vγ dimension for regression in bounded subspaces of Reproducing Kernel Hilbert Spaces (RKHS) for the Support Vector Machine (SVM) regression ǫ-insensitive loss function Lǫ, and general Lp loss functions. Finiteness of the Vγ dimension is shown, which also proves uniform convergence in probability for regression machines in RKHS subspaces that use the Lǫ ...

We prove the following noncommutative version of Lewis’s classical result. Every n-dimensional subspace E of Lp(M) (1 < p < ∞) for a von Neumann algebra M satisfies dcb(E,RC n p ) ≤ cp · n ̨̨ ̨ 1 2 − 1 p ̨̨ ̨ for some constant cp depending only on p, where 1 p + 1 p = 1 and RC p = [Rn∩ Cn, Rn +Cn] 1 p . Moreover, there is a projection P : Lp(M) → Lp(M) onto E with ‖P‖cb ≤ cp · n ̨̨ ̨ 1 2 − 1 p ̨̨ ̨ . We fo...

This paper concerns the dynamics of non-expansive maps on strictly convex finite dimensional normed spaces. By using results of Edelstein and Lyubich, we show that if X = (R, ‖ · ‖) is strictly convex and X has no 1-complemented Euclidean plane, then every bounded orbit of a non-expansive map f : X → X , converges to a periodic orbit. By putting extra assumptions on the derivatives of the norm,...