دانلود رایگان مقاله فضاهای فرعی با نزدیکترین نامتغیر اضافی به داده مشاهده شده

Abstract

Given an arbitrary finite set of data F={f1,…,fm}⊂L2(Rd) we prove the existence and show how to construct a “small shift invariant space” that is “closest” to the data F over certain class of closed subspaces of L2(Rd). The approximating subspace is required to have extra-invariance properties, that is to be invariant under translations by a prefixed additive subgroup of Rd containing Zd. This is important for example in situations where we need to deal with jitter error of the data. Here small means that our solution subspace should be generated by the integer translates of a small number of generators. An expression for the error in terms of the data is provided and we construct a Parseval frame for the optimal space. We also consider the problem of approximating F from generalized Paley–Wiener spaces of Rd that are generated by the integer translates of a finite number of functions. That is finitely generated shift invariant spaces that are translation invariant. We characterize these spaces in terms of multi-tile sets of Rd, and show the connections with recent results on Riesz basis of exponentials on bounded sets of Rd. Finally we study the discrete case for our approximation problem.

5. The discrete case The optimal subspace V ∗ in Theorem 3.1 is the closest to the data F over all subspaces V in the class V M. It is not difficult to see that almost each fiber space JV ∗ (ω) ⊂ 2(Zd) of V ∗ is the closest to the fibers of our data, τ (F)(ω) = {τf1(ω), ..., τfm(ω)} over a certain class of closed subspaces of 2(Zd) that we will call D N . Clearly this class of subspaces is determine by the class V M. So, Theorem 3.1, implies an approximation result in 2(Zd), for a very particular class determined by the extra-invariance. Therefore it is interesting to see if this approximation result in 2(Zd), extends to more general classes. We will define in what follows a very general class D N . The cases coming from the continuous case will be particular cases of our general definition. The proof of the optimality that we obtain is more general and cannot follow from Theorem 3.1