We consider translates of functions in along an irregular set of points, that is, —where is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform.
Frames of translates, i.e., frame sequences originated by the shifts of a given, fixed function, are an important mathematical background for Gabor [1,2,3], wavelet [4,5] and sampling theory [6,7]. One generating function is shifted to create the analyzing family of elements, for a sequence These systems play a main role in the theory of shift invariant spaces (SIS) [8,9,10,11], which is very useful for the modeling of problems in signal processing, and is central in approximation. It is also a very active field of research in mathematics, see e.g., [12,13,14,15]. Frames of translates are closely related to frames of exponentials also called Fourier frames [16,17,18].
The regular shifts, e.g., when for some fixed , were studied e.g., in . Investigations of irregular frames of translates, where the set has no regular structure, were done e.g., in [1,6]. But there remain several open questions, in particular how the concept of Gramian function could be defined in the irregular case, which is a fundamental tool for describing properties of regular systems of translates. Also no explicit formulas for the canonical dual of irregular frames of translates were given so far. We do this in the present paper.
The paper is structured as follows: In Section 2 we present an overview over the main results. In Section 3 we summarize basic notations and preliminaries. In Section 4 we generalize the concept of the Gramian function and use it to describe Bessel frame and Riesz properties of irregular shifts. In Section 5 we look at the inter-relation of the frame-related operators for those sequences to those of exponential functions. This permits us to study Bessel, frame and Riesz properties of irregular frames of translates, and furthermore give formulas for the canonical dual, and for equivalent Parseval frames.
2. Main Results
We state here our main results at the beginning of the paper. For notations and definitions pleaser refer to the later sections.
For regular frames of translates the Gramian function is a key concept. For the case of irregular frames of translates, where we have no group structure, we introduce the following notion.
Let We define its pseudo-Gramian function by
Surprisingly, as we will see in Section 4 and Section 5, this definition leads to quite analogous results as in the regular case, where this function is used only in a periodized version. Among them we show the following:
Let such that is compact. In each of the following cases (i.e., (1) or (2)), if one of the following conditions (i.e., (a),(b) or (c)) is fulfilled, the other two are equivalent:
is a frame in .
is a frame in .
There exist such that a.e. on supp.
is a Riesz basis in .
is a Riesz basis in .
There exist such that a.e. on supp.
Note that the condition implies that the frame properties of and coincide.
Theorem 1 is an extension of results in . We prove it using an operator theory based approach. Furthermore we can give an explicit form for the canonical dual of an irregular set of translates, as a nice generalization of results in the regular case:
Let such that is compact. Assume that there exist such that a.e. on and let be a frame sequence in Then the canonical dual of is where
In particular if is an A-tight frame, then with (on ).
Note that the definition of the pseudo-Gramian gives a convenient way of formulating the other results, although they do not depend on this definition per-se.
3. Notation and Preliminaries
Given we denote by the function defined by The operators and are given by and respectively. We write for the Fourier transform, , given by for with the natural extension to
In this paper is a bounded set. We denote
is a closed subspace of which is isomorphic to , and we will identify these two spaces. The Fourier transform maps onto
When we write that is a frame (frame sequence, Bessel sequence or Riesz basis) of we will mean that the set has that property, where stands for the indicator function of E, that is, it is an outer frame, Bessel or Riesz basis . For a given operator O we denote its pseudo-inverse by . Motivated by that, for a function we denote
Throughout this work will be a sequence in For a function f we denote its range by
Let be a separable Hilbert space. A sequence is a frame for if there exist positive constants A and B that satisfy
We denote the optimal bounds by and . If then it is called a tight frame, and if a Parseval frame. If satisfies the right inequality in the above formula, it is called a Bessel sequence.
A sequence is a Riesz basis for if it is complete in and there exist such that for every finite scalar sequence
The constants A and B are called Riesz bounds. A complete sequence is exact if it ceases to be complete when an arbitrary element is removed. Exact frames are precisely the Riesz bases.
We say that is a frame sequence (Riesz sequence) if it is a frame (Riesz basis) for its span.
For a Bessel sequence, its analysis operator is defined by , and its synthesis operator by .
If is a frame of , the frame operator given by is bounded, invertible, self-adjoint and positive. There always exists a dual frame of which is a frame such that
The sequence satisfies (1). It is called the canonical dual frame of and we will denote it by . We distinguish between the operators by subscripts, e.g., is the analysis operator for the set of exponentials and those of the system of translates . For the operators corresponding to the canonical dual we will write respectively .
To every frame a canonical tight frame can be associated, which is the sequence where is the positive square root of
The (bi-infinite) Gram matrixG for is given by for This matrix defines an operator by the standard matrix multiplication on , the set of square summable sequences. It is known, that many properties of the sequence are directly related to properties of this operator, for a summary see (Prop. 4.4, ). For example, that G is bounded, if and only if the sequence forms a Bessel sequence. G is invertible on all of if and only if the sequence is a Riesz sequence.
In this work the operator that consists of a multiplication by a given function will play an important role.
For we can give the following results, which are either known, see for example, Reference  or can be easily proved:
Let ϕ be a measurable function defined in , and consider the multiplication operator on given by .
if and only if is well defined and bounded. In this case .
Assume there exist such that for almost all . Then and . On we have
is boundedly invertible if and only if there exists such that a.e. In this case
4. Gramian Function for Irregular Frames of Translates
In this section we will show some results about sufficient and necessary conditions on the pseudo-Gramian function for Bessel sequence and Riesz basis properties.
As a connection of the pseudo-Gramian function with the Gramian matrix we get the following result.
Let For any system of translates the Gramian matrix G has the entries
As a consequence of Schur’s test, see for example, Reference  and the properties of the Gram matrix we get:
then the system of translates is a Bessel sequence.
This is equivalent to the Gram matrix being included in the Schur-type matrix algebra with no weight, that is, the set of translates forms an intrinsically localized frame [24,25].
On the other hand using necessary conditions for bounded matrices on [26,27] the following can be shown:
If and is a Bessel sequence, then
For Riesz bases we obtain the following result:
Let . If
then the system of translates is a Riesz sequence.
As we have
Therefore, by assumption, the Gram matrix is strictly diagonally dominant, and it is invertible by (Lemma 6.5.4, ), which is the infinite-dimensional version of the Levy-Desplanque theorem. The Gram-matrix is therefore invertible from onto and so the sequence is a Riesz sequence . □
5. The Operator-Based Approach to Irregular Frames of Translates
5.1. Irregular Translates in
From now on we will assume Analogous as in (Theorem 4.1, Prop. 3.6, ) the following can be proved.
Assume is a Bessel sequence of with bound and there exists such that a.e. Then is a Bessel sequence for with bound .
Let be a Bessel sequence for with bound and assume there exists such that a.e. in E. Then is a Bessel sequence for with bound .
The following results generalize Lemmas 7.2.1 and 7.3.2 in .
Let be a Bessel sequence for and assume there exists such that a.e. Let Then converges unconditionally in , converges unconditionally in and
Therefore and if and only if there exists such that .
Following Lemma 1 is a Bessel sequence and so all involved sums converge unconditionally.
As is bounded, the multiplication operator is also bounded and therefore
Let be a Bessel sequence in and assume there exists such that a.e. Let be a frame sequence. Then and so if and only if there exists such that .
In this case . We see from the proof of Lemma 2 that . □
5.2. Relation of Operators
In this section we state the relations of the operators of the set of exponentials and the system of translates.
Let be a Bessel sequence of and assume that there exists such that a.e. Then is a Bessel sequence of and
By Lemma 1 is a Bessel sequence of . For by Lemma 2,
In particular we have that .
Let be a frame of , and assume there exist such that a.e. on . Then is a frame sequence that spans . Furthermore
Moreover and .
By Proposition 1 is invertible on , and therefore is invertible on , and
Note that maps onto and maps onto , with Therefore .
By Lemma 2, and since is a frame for , we know that . Now implies and so .
We have and hence .
Analogously, since , it can be proved that
Furthermore, we can state the following result about the exactness of sequences of translates and sequences of exponentials. Observe that Proposition 2 follows also from the equation in Lemma 3.
Let be a Bessel sequence of
Assume there exists such that a.e.. If is exact, then is exact.
Assume there exist such that a.e. in Then is exact if and only if is exact.
Lemma 3 yields .
If is bounded from above and below, then is invertible and so by Proposition 4, part 2 follows. □
We can now proof one of the theorems stated in Section 2.
(1) Since is bounded from above and from below, is invertible and so by Lemma 3 . If and are surjective as well as and are injective, is bijective and so the other directions are shown.
(2) follows from Corollary 4 applied to and □.
Note that if the completeness assumption is dropped (i.e., we use frame sequences and Riesz sequence) we still have the following:
For upper semi-frames , that is, complete Bessel sequences, similar relations can be given but not equivalencies.
Using Theorem 1, we can generalize Proposition 7.3.6. in  in the following sense:
Let be a frame in . If forms a frame, then is discontinuous and so .
5.3. The Canonical Dual
We arrive to the relation between the canonical duals of frame sequences of exponentials and frame sequences of translates stated in Theorem 2.
As in Proposition 4
From Lemma 3 we obtain:
Assume is an -tight frame of , and that there exist such that a.e. Then is a Bessel sequence of and the frame operator of is
The following Corollary shows the relation between the analysis and synthesis operator of the duals of the shifts and of the exponentials.
Assume that there exist such that a.e. on and let be a frame sequence in Then
Now let us investigate the operator .
Let a.e. Then .
Let form an -tight frame of . Then .
1. With the assumption is a unitary operator. Therefore
2. By Corollary 6, , and so . □
Finally, we can give the following representation for an equivalent Parseval frame to a frame sequence of translates:
Let be a frame for , and assume there exist such that a.e. on . Set
where we denote the canonical tight frame of by . Then forms a Parseval frame with the same span as . The frames and are equivalent, that is, are images of bounded and boudedly invertible operator of each other.
If forms an -tight frame, then forms a tight frame of translates with shifts and generator .
By (Theorem 4.1, ) is a frame sequence. Clearly . For ,
Therefore and . The equivalence follows from the fact that by assumption is a bijection and so .
The result follows from Proposition 4. □
6. Outlook and Conclusions
We have shown that under not-that-strong assumptions, we could, surprisingly, get similar results for the irregular setting as in the regular case, in particular for the (generalization of the) Gramian function and a formula for the canonical dual and Parseval frame.
We focused on the generalization of the sampling set to an irregular set. To handle the problem in full detail within the scope of this paper, we sticked to the bandlimitness assumption. Some of the results shown above are still valid for not band-limited functions, for example the results in Section 5.1 and Lemma 3. In the future an interesting question is what other results can be proved for this case.
We phrased the assumptions on the set of exponentials in a very general way, on purpose. In this way they can be easily combined with results about frames of exponentials and density results, see e.g., (Section 7.6, ).
Conceptualization, P.B. and S.H.; Formal analysis, P.B. and S.H.; Investigation, P.B. and S.H.; Methodology, P.B. and S.H.; Writing—original draft, P.B. and S.H.; Writing—review & editing, P.B. and S.H.
This research was funded by the Austrian Science Fund (FWF) START-project FLAME (Frames and Linear Operators for Acoustical Modeling and Parameter Estimation; Y 551-N13), and the FP7 project PIEF-GA-2008-221090. We also thank MINCyT (Argentina) and BMWF-OeAD-WTZ (Austria) for the support of the bilateral project Shift Invariant Spaces And Its Applications To Audio Signal Processing; AU/10/25. Furthermore we acknowledge the partial support of PIP 112-201501-00589-CO (CONICET) and PROICO 03-1618 (UNSL).
The first author thanks Nora Simovich for help with typing. This publication is supported by the open access funding by the Austrian Science Fund (FWF).
Conflicts of Interest
The authors declare no conflict of interest.
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