Atomic Systems for Subspaces

Atomic Systems for Subspaces Hans G. Feichtinger and Tobias Werther NuHAG, Dept. of Mathematics University of Vienna, Strudlhofg. 4 A-1090 Wien, Austria

Abstract

of H. {fn } is called a family of local atoms for H0 if there exists a sequence of linear functionals {cn } such that

This paper presents a family of analysis and synthesis systems with frame-like properties for a closed subP 2 2 n |cn (f )| ≤ Ckf k , spaces H0 of a separable Hilbert space H. In contrast (i) ∃ C > 0 with to frames the building blocks for H0 do not neces- (ii) f = P cn (f )fn . n sarily belong to H0 . Thus, the new approach gives us more freedom on the choice of the atomic system for all f in H . 0 than standard frame theory. The motivation of the definition is based on examples arising in sampling We say that the pair {fn , cn } provides an atomic decomposition for H0 . theory.

1

Remark: The functionals {cn } need only to be defined on PNthe smaller space H0 . Note that the partial sums n cn (f )fn can be converging to f from the ”outside” of H0 .

Introduction

Frames can be seen as a generalized basis system. Their redundancy destroys the uniqueness in the series expansion and leaves much freedom for the choice of the coefficients. Frames have been extensively analyzed in the last decade, cf. [1] and references. In contrast to frames there exist systems of functions generating proper subspaces even though they do not belong to them. A well-known example is the system of regular translates of Helms-Thomas kernels for the Paley-Wiener space in case of oversampling, cf. [6].

The expression ”local atom” makes reference to the examples given below. It also might emphasize that the defined properties are restricted to some proper subspace H0 .

An atomic decomposition for H0 is a so-called pseudo-frame as defined by Shidong Li et al. [5] but with additional norm estimates which are crucial for unconditional norm convergence. For practical purposes it might be useful to make H0 smaller in order In the sequel H denotes a Hilbert space with inner to control the upper bounds B and C. product h , i. In order to maintain frame-like properties we make the following definition.

2

Basic Results

Definition 1 Let {fn } in H be a Bessel sequence, P 2 i.e., there exists B > 0 such that n |hf, fn i| ≤ In what follows we state two basic properties of a 2 Bkf k for all f ∈ H and let H0 be a closed subspace family of local atoms. 1

P 2 Proposition 2 Let H0 be a closed subspace of H. n |hf, gn i| ≤ Ckf k for all f in H0 . In order to If {fn } is a family of local atoms for H0 then there show the commutativity of {fn } and {gn } we define exists A > 0 with the following operators: X X |hf, fn i|2 ∀f ∈ H0 . Akf k2 ≤ T1 : l2 −→ H0 , T1 (c) = cn PH0 fn n

n

X

2

T2 : l −→ H0 , T2 (c) = cn gn Proof. From Definition 1 we know that {cn (f )} and n {hf, fn i} are both l2 -sequences. By using CauchyBoth operators are well-defined on H0 and their adSchwarz Inequality we compute for all f in H0 joints are given by !2 X T1∗ : H0 −→ l2 , T1∗ (f ) = hf, fn i, kf k4 = hf, cn (f )fn i n

T2∗ : H0 −→ l2 ,

!2 =

X

cn (f )hf, fn i

From Property (ii) of Definition 1 we get

n

≤

X

|cn (f )|2

n

X

T1 T2∗ = Id = Id∗ = T2 T1∗

|hf, fn i|2

n 2

≤ Ckf k

X

T2∗ (f ) = hf, gn i.

on H0 . This leads us to the second expansion X f = T2 T1∗(f ) = hf, fn ign .

|hf, fn i|2

n

n

The last inequality follows from Property (i) of Definition 1. Setting A = C −1 proves the statement. 2

The lower frame estimate for {gn } follows from Proposition 2. 2

In contrast to frames the series expansion is merely Proposition 2 easily implies that a family of local atoms {fn } for H0 induces the frame {PH0 fn } for H0 defined on the subspace H0 even though the dual syswhere PH0 is the orthogonal projection operator onto tem of {gn } does not necessarily belong to H0 . The less restrictive conditions on {fn } which make the H0 . family not belong to H0 can improve the convergence Next we can prove a dual property for a family of behavior of Expression 1, for example, with respect local atoms. to locality. Note that the coefficients in the second series expression are in general not the minimal norm Proposition 3 Let {fn , cn } be an atomic decompo- least square coefficients. sition for some subspace H0 of H. Then, the family {cn } defines a unique system {gn } in H0 . {gn } is a commutative frame for H0 with respect to {fn }, i.e., 3 Fundamental Examples for all f in H0 X X f= hf, gn ifn = hf, fn ign . (1) The first example is deduced from the general irregular sampling and series expansion theory developed n n by Feichtinger and Gr¨ochenig [3]. Proof. By Riesz Representation Theorem we can identify each functional cn with a function gn Definition 4 Given δ > 0, a set {xn } is called δin H0 . This P gives us immediately the expan- dense in R if the family {xn + [−δ, δ]} covers R. It sion f = n hf, gn ifn and the Bessel inequality is called relatively separated if there is a uniform 2

bound C for the number of points xn in any interval g ∈ Bω2 1 ∩ L1 with gˆ positive on [−ω, ω] and any wellof fixed length. A δ-dense and relatively separated set spread set {xn } with density δ every f ∈ Bω2 has a representation {xn } is called well-spread with density δ. X f= cn (f )Txn g, We denote by Bω2 the space of Paley-Wiener funcn tions with bandwidth 2ω which is a closed subspace of L2 (R). For any ω > 0 there exists δ > 0 such that where T denotes the translation by x . The mapxn n for any well-spread set {xn } with density δ the ex- ping of f to its coefficients is linear and depends conP pression ( n |f (xn )|2 )1/2 defines an equivalent norm tinuously on f in the following sense: on Bω2 [3]. X |cn (f )|2 ≤ Ckf k2 . Proposition 5 [7] Let ω1 , ω be constants with ω1 > n ω > 0. Choose δ > 0 such that 2δω1 < 1. Then, any f ∈ Bω2 (R) can be completely reconstructed from We see immediately that the system {T g} satisxn its sampled values on any δ-dense sequence {xn } and fies the assumptions of Proposition 3 and, therefore, there are functions en ∈ L2 (R) with supp(ˆ en ) ⊆ there exists a frame {h } in B 2 such that for all f in n ω [−ω1 , ω1 ], such that Bω2 (R) X X X f= f (xn )en . (2) f= hf, hn iTxn g = hf, Txn gihn . n

n

It is possible to choose {en } from some weighted L1 -space. Compared to standard frame theory the benefit of the reconstruction in (2) is the good localization property due to the rapid decay of the building blocks en . If {xn } is a well-spread set of density δ < (2ω)−1 then the pair {en , cn } is an atomic decomposition for Bω2 , where cn denotes the pointevaluation functional at xn . Moreover, it can be shown that {en } is a Bessel family for Bω2 and thus a family of local atoms. As a consequence of Proposition 3 we obtain for all f in Bω2 (R) X f= hf, en isn

n

A different situation occurs when starting from a frame-like operator. Let {en } be an arbitrary family of vectors in a Hilbert space H. We denote by F a finite index set and define the operator SF on H by X SF f = hf, en ien . n∈F

SF is self-adjoint and has a finite number of eigenvectors hSF with non-negative eigenvalues λSF . Operators of this kind arise for example as time-frequencylocalization operators in Gabor analysis. For α > 0 we may restrict the operator SF to the finite dimensional subspace Hα defined by

n

where sn is the family of sinc-functions shifted to xn Hα = span{hSF : λSF ≥ α}. which, in this case, constitutes a frame for Bω2 (R) [4]. The coefficients hf, en i are in general different to the minimal norm least square coefficients from the We claim that {en }n∈F is a family of local atoms for Hα , for any α > 0. Indeed, hSF f, f i ≥ αkf k2 on Hα canonical dual frame. and the family {hSF : λSF ≥ α} is an orthonormal The following result can be proved in a similar way basis for H . This motivates the introduction of the α as Proposition 5. self-adjoint operator X Proposition 6 [3] Let ω1 , ω, δ be positive constants λ−1 TF,α f = SF hf, hSF ihSF which ω < ω1 and 2ω1 δ < 1. Then for any function λSF ≥α 3

on H. It follows immediately that TF,α and SF commute and that for all f in Hα

[4] S. Jaffard, “A density criterion for frames of complex exponentials”, Michigan Math. J., 38(1991), pp.559-561.

f = TF,α SF f = SF TF,α f.

[5] S. Li and H. Ogawa, “A theory of pseudoframes with applications”, preprint, 1998.

As a consequence we have for f in Hα X ) f= hf, e˜(F n ien

[6] M. Pawlak and U. Stadtm¨ uller, “Recovering band-limited signals under noise”, IEEE Trans. Info. Theory, 42(1994), pp. 1425-1438.

n∈F

and f=

X

) hf, en i˜ e(F n

[7] T. Werther, “Reconstruction from irregular samples with improved locality”, Master’s thesis, University of Vienna, Dec. 1999.

n∈F

where

(F ) e˜n

= TF,α en .

Note that the operator TF,α can be seen as the pseudo inverse operator of SF with threshold α. If {en } is assumed to be a tight frame for H then SF approximates the identity as F increases and TF,α converges to the identity both in the strong operator topology for α smaller than, say, 1/2.

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Conclusion

The definition of local atoms for subspaces and the basic results have been motivated by the attempt to axiomatize synthesis methods arising in sampling problems. We consider this approach as a possible interface between well-known examples and their axiomatic treatment in a Banach space context.

References [1] O. Christensen, “Frames and the projection method”, Appl. Comput. Harmonic Anal., 1(1993), pp. 50-53. [2] H. G. Feichtinger and K. Gr¨ ochenig, “Iterative reconstruction of multivariate band-limited functions from irregular sampling values”, SIAM J. Math. Anal., 23(1992), pp. 244-261. [3] H. G. Feichtinger and K. Gr¨ ochenig, “Irregular sampling theorems and series expansion of band-limited functions”, J. Math. Anal. Appl., 167(1992), pp. 530-556. 4