^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

We consider Gabor localization operators defined by two parameters, the generating function of a tight Gabor frame , indexed by a lattice , and a domain whose boundary consists of line segments connecting certain points of . We provide an explicit formula for the boundary form , the normalized limit of the projection functional , where are the eigenvalues of the localization operators applied to dilated domains , R is an integer and is the area of the fundamental domain. The boundary form expresses quantitatively the asymptotic interactions between the generating function and the oriented boundary from the point of view of the projection functional, which measures to what degree a given trace class operator fails to be an orthogonal projection. Keeping the area of the localization domain bounded above corresponds to controlling the relative dimensionality of the localization problem.

We start by formulating the main results of this paper. Let

where the convergence of the sum is understood in the unconditional norm sense. Function

is called a Gabor multiplier of localization type, if its symbol b, defined on the lattice Λ, is non-negative and summable. It can be easily verified that Gabor multipliers of localization type are non-negative, trace class, and that

and the operator norm of

The projection functional PF is defined on positive definite, trace class operators T, with their operator norm bounded above by 1, via the formula

where

Φ:

For a Λ domain

where

The projective metaplectic representation defined on

In our first result we describe the invariance properties of PF and BF with respect to the projective metaplectic representation. Although the invariance of PF is standard we include it as well for the sake of completeness.

Theorem 1.1. Let μ be the projective metaplectic representation of

1) for a Gabor multiplier

2) for any Ω a Λ domain,

Our second result is the principal result of the current paper. It describes the limit behavior of the projection functional PF applied to a Gabor multiplier with the symbol of the form

Theorem 1.2. Let

Formula (7) expresses the limit behavior of the projection functional PF in terms of the boundary form BF. It provides a very explicit, quantitative way of describing the interactions between the boundary

The invariance of the projection functional PF and the boundary form BF with respect to the action of the projective metaplectic representation expressed in Theorem 1.1, taken together with the limit result of Theorem 1.2, has its important consequences. We can conclude that no lattice Λ is distinguished, neither from the point of view the value of the limit of

Corollary 1.3. For any lattice

Theorem 1.2, Corollary 1.3 and part 2) of Theorem 1.1, together with their proofs, constitute new contributions of the authors to the topic of Gabor multipliers. Proofs of these results make an essential use of the extended metaplectic representation and the calculus of Gabor multipliers. Lemma 4.3 constitutes the essence of the computational setup dealing with discrete line segments. Lemmas 4.6, 4.7 deal with the reduction process of general lattice domains to unbounded strip domains. They allow the splitting of the boundary of the domain into its line segment components and the manipulation of their positions in the coordinate system.

Many important mathematical theories started in the one-dimensional setup, where a multitude of additional tools is available, and then through various stages of evolution came up into their full form in any finite dimension. This was the case of the representation theory of semi-simple Lie groups, which started with the listing of all irreducible representations of

The principal results of the current paper deal with discrete one-dimensional setup of tight Gabor frames, therefore for the sake of consistency we formulate definitions and reference results only in one dimension. Operators of composition of convolution with g followed by a pointwise multiplication by f, where both functions f and g are defined on the real line

They are commonly called convolution-product operators. Historically three dimensional convolution-product operators played an important role in the study of Schrödinger operators. The Birman-Schwinger principle allows a transition from a Schrödinger operator to a convolution-product operator. Out of that transition it was possible to obtain sharp estimates for the number of bound states of the Schrödinger operator. Classical TF-lo- calization operators are one dimensional convolution-product operators with f and ǧ characteristic functions of intervals, where ǧ is the inverse Fourier transform of g. Their spectral properties were carefully studied many years ago by Landau, Pollak, Slepian and Widom (see [

We are interested in operators with integral kernels of the form (8), where translations constituting the convolution with g are extended to combined actions of translations and modulations of the Schrödinger representation, and applied to a function

where

where

Gabor-Toeplitz operators generalize Fock space Toeplitz operators. The Bargmann transform provides their mutual unitary equivalence, in case the normalized Gaussian is chosen for the generating function. Books by Folland [

and that if

Asymptotic properties, as

integrable,

tral decomposition of

We need to multiply

and the size of their plunge region

are expressed directly via the corresponding quantities of the symbol function, the distribution of b

and the Lebesgue measure of its plunge region

In the passage to the normalized limit it is necessary to assume that the level sets

In the next step operators

expressed in terms of the area of a strip of fixed size R around the boundary

where_{1}, c_{2} it was necessary to assume uniform decay and non-degeneracy of the reproducing kernels obtained out of generating functions ^{c} from the points near their boundaries

The next step in the study of mutual interactions between generating functions and domains of localization from the point of view of estimating the size of the eigenvalue plunge region was accomplished in [^{1} boundary. Symbol

Under an appropriate integrability condition imposed on

Formula (20) expresses the limit behavior of the projection functional in terms of the boundary form. It describes quantitatively the interactions between the boundary

Let us fix the area A and a generating function

Function

For the given generating function

The principal results of the current paper deal with the discrete setup of tight Gabor frames defined on

In the discrete setup the Gabor reproducing formula (10) is substituted by a tight Gabor frame expansion (1)

and the Gabor-Toeplitz operator (11) by a Gabor multiplier (2)

Gabor multipliers were introduced as a phase space analysis tool parallel to Gabor expansions. Both Gabor- Toeplitz operators and Gabor multipliers are currently very actively investigated, mostly from the point of view of their usage in phase space analysis. They were applied as phase space partitioning operators in [

We restrict attention to Gabor multipliers of localization type, i.e. we assume that the symbol b defined on Λ is non-negative and summable. The operator with kernel

a discrete analogue of (9), acting from

Out of Theorems 1.1, 1.2 we were able to conclude Corollary 1.3, expressing the fact that from the point of view of the projection functional PF and the boundary form BF no lattice Λ is distinguished. It would be interesting to isolate those phase space phenomena that make distinction between lattices parameterizing tight Gabor frames. The book by Martinet [

In the continuous case the Wulff shape (22) is the optimal localization domain. There is no analogue of it for lattice domains. We do not know how large is the class of localization domains for which the asymptotic boundary forms exist. We expect that in the general case the problem of existence of normalized limits of projection functionals has to be considered in parallel with the asymptotic properties of the counting function

Books by Christensen [

Let

For any frame of

rise to a discrete reproducing formula. We may renormalize a tight frame

with the convergence of the sum understood in the unconditional norm sense. There is a canonical way of constructing a tight frame out of a frame. For any frame

Condition (24) guarantees that the above sum is unconditionally convergent for any

Gabor frames have the form

Let

Let

is finite. If

Let us recall that a function

Proposition 2.1. If

The first construction of tight Gabor frames in dimension 1 was obtained by Daubechies, Grossmann and Meyer in 1986. Tight Gabor frames were called painless nonortho-gonal expansions back then. The construction produced generating functions with compact support either in position or in momentum and with arbitrary smoothness measured by the number of continuous derivatives, any

nuous from

We start by recalling the definitions of our principal objects of interest. We assume that a tight Gabor frame

where

nerating function

where

Now we are ready to present the proof of Theorem 1.1.

Theorem 1.1. Let

1) for any Gabor multiplier

2) for any Ω a Λ domain,

Proof. The transformation rule 2) of Lemma 4.1, describing the effect of the conjugation by the projective metalplectic representation, shows that 1), i.e. formula (5) holds.

In the remaining part of the proof we deal with 2), i.e. with formula (6)

The group

and for the geometric atoms

where

The linear map

where

The Iwasawa decomposition of

where

We observe that

the normalized vector v orthogonal to

therefore we obtain

The above calculation verifies (32). It also shows (31), since plugging

Let us recall that Gabor multipliers with symbols of the form

Theorem 1.2. Let

Steps of Proof:

Step 1. We make a transition from lattice Λ to

notes the projective metaplectic representation defined on

Once the unitary equivalence is verified it is clear that

We substitute the boundary form

where l_{i}, _{i} is the unit vector or-

thogonal to l_{i} directed outside

Step 2. We use tight Gabor frames versions of Toeplitz and Hankel operators in order to express the projection functional

as the square of the Hilbert-Schmidt norm of the matrix

where

where

Next, we change variables and let them range over the dilated lattice

Step 3. We split the boundary of

and function H representing a given segment of the boundary is a linear function with rational slope of the form

Step 4. We adjust the form of variables x_{1}, x_{2} to the arithmetic form of the slope of H. We represent x_{1}, x_{2} as_{1}, k_{2} are integers and_{1}, r_{1}, k_{2}, r_{2}, expression (41) becomes

Step 5. Invariance of expression (42) with respect to variables y_{1}, y_{2} allows us to substitute the double summation with a single summation. We count the number of repetitions in the representation

where square brackets denote the integer part of a rational number.

Step 6. Form (43) is convenient for making the passage to the limit. It occurs that taking the limit in (43) with respect to R is just the same as performing summation with respect to k. We obtain

Step 7. We write expression (44) in the form involving function

where

Step 8. We interpret expression (45) geometrically. We make use of the structural features of lattice lines and we show that in fact (45) equals

where

distance inside the plane

al strip domain

Step 9. We put together the boundary forms (47) of vertical strip domains corresponding to all line segments of the boundary of

In view of (34), and (36), formula (48) concludes the proof.

Proofs of Steps:

Proof of Step 1. We construct matrix

We conclude that (34) holds. The transition from Λ to

The justification of the transition from Λ to

but we still need to switch to

Proof of Step 2. Since

Lemma 4.2 implies that

where

Proof of Step 3. This is the most tedious step of the proof. For the sake of notational convenience we will use symbol

into pieces with the help of a partition of unity of

We traverse each component of the boundary of Ω according to the orientation, keeping the interior on the right and the exterior on the left, and at each initial lattice point of the boundary segment l_{i} we place a sufficiently small rational line segment s_{i} with its middle being the initial lattice point. We chose s_{i} in such a way that it is transversal to both l_{i} and the boundary segment l_{l} preceding it, and that both exterior and interior angles with l_{i} and l_{l} are smaller than π. We assume that segments s_{i}, i = 1, ∙∙∙, N are so small so that they intersect the boundary _{i} is inside the interior of Ω and the other in the exterior of Ω. Let l_{r} be the segment following l_{i}. We form quadrilaterals W_{i} out of consecutive segments s_{i}, s_{r} attached at the beginning and at the end of l_{i}, and the segments joining their endpoints, both endpoints inside the interior of Ω or both in the exterior of Ω. _{i}. We also assume that the sizes of the transversal segments are so small so that the segments joining the endpoints of s_{i} and s_{r} are contained either in the exterior or in the interior of Ω. We include s_{i} inside W_{i}, but not s_{r}. Adding s_{r} to W_{i} we would obtain a closed quadrilateral, but we need to form a family of pairwise disjoint sets, and s_{r} is already included in W_{r}.

We define a partition of

We observe that

We do not need terms with factors

for

If

Formula (49)-(52) allow us to conclude that only terms of the form

contribute to the limit.

Let us recall that we assumed that each boundary segment l_{k} connects lattice points and does not have lattice points in its interior. We may apply Lemma 4.5 in order to bring each segment l_{k} to the form

graph of H, and the image of _{k} and the bottom portion by a line b_{k}. Lemma 4.6 allows us to substitute the bounded vertical strip domain we have just obtained by a rectangle with the top boundary represented by a horizontal line_{k}, and the bottom boundary represented by a horizontal line_{k}.

In the last step we switch to the unbounded strip domain

with its complement

We have the splitting

We need to show that the normalized sums coming out of terms (54), (55), (56) have zero limits. Lemma 4.6 applies to both (54), (55), but (56) needs to be treated separately. After the change of variables the normalized sum of (56) becomes

In the proof that its limit is zero as

because

Proof of Step 4. With the new representation of

Proof of Step 5. We have

therefore

for some

Proof of Step 6. We move summations over k_{1}, k_{2} in (43) inside and summations over r_{1}, r_{2}, i outside. Our target is to identify convolution kernels defined in terms of variables k_{1}, k_{2} and then, keeping variables r_{1}, r_{2}, i fixed, take the limit with respect to R. _{i} ranges over the integer interval

Lemma 4.4 guaranties that as

tends to

Proof of Step 7. Each number_{1}, r_{2} by summations with t, s, therefore (44) becomes (45).

Proof of Step 8. We fix summation variables k, t of expression (45) and we consider the effect of the summation done with respect to s, i. By Lemma 4.3 we know that function ^{th} point of it, j ≥ 2, with counting done upwards, corresponds to values^{th} point, ^{th} point of the discrete half line equals

Proof of Step 9. It is enough to combine together the outcomes of steps 1 - 8, i.e. all of the intermediate stages of the reduction process.

The proof of Corollary 1.3 follows directly from Theorems 1.1, 1.2.

Corollary 1.3. For any lattice

Proof. It is enough to take

Transformation properties with respect to the metaplectic representation. A comprehensive presentation of the metaplectic representation from the point of view of phase space analysis is contained in Folland’s book [

where

The Schrödinger representation of the group

We write

By Schur’s lemma,

The representations

The extended metaplectic representation provides all affine transformations of the phase space

The extended metaplectic representation is a convenient setup for performing computations involving compositions of the Schrödinger and the metaplectic representations.

Both Gabor multipliers (2) and lattice boundary forms (4) have natural transformation properties with respect to the projective metataplectic representation. These properties are important ingredient of our proofs. We will deduce them out of the the fundamental lemma formulated below.

Lemma 4.1. Let

1)

2)

Proof. Clearly

The above calculation shows that

Symbolic calculus of Gabor multipliers. Let us assume that

Tight frame properties of

is the orthogonal projection onto the range of

It is convenient to describe Gabor multiplier

expresses this relationship quantitatively and it is the main conceptual ingredient of the argument that allows us to write down the projection functional

as the square of the Hilbert-Schmidt norm the matrix

where

Lemma 4.2. Let

Proof. The non-zero eigenvalues of the localization operator

and that

Therefore we obtain

and this finishes the proof.

Lattice slopes of rational lines. For an integer t we define the t-slope of the line

where _{t} defined in (46) is the principal analytic component of the boundary form. It occurs that it may be interpreted as the

t-slope of the rational line

tion shows, that values

Our primary geometric concern are the values of lattice slopes and the frequencies with which they occur. Observe that _{t}. If however _{t} takes precisely two values. The next lemma describes those values and the frequencies with which they occur.

Lemma 4.3. Let m, n be a pair of relatively prime integers. Assume that

values

Proof. We know that R_{t} and S_{t} are equal. It is therefore enough to prove Lemma 4.3 with S_{t} instead of R_{t}. Let

integer and

_{t} takes two values

Approximation to the identity by Fejér’s kernel. In the lemma that follows we quote a well known approximation to the identity property of the Fejér’s kernel. We translate the original property from the group of one dimensional torus to the group of integers. For

Lemma 4.4. [

where

Invariance properties of the restricted kernel K_{R}. Let us recall that kernel

where

Lemma 4.5. Let

becomes

For any

m, n are relatively prime, and the portion of

then we may assume that

Proof. The proof of the first part, the formula for the coordinate change is a straightforward computation which makes use of the fact that

Asymptotic limits of the restricted kernel K_{R}. The following two lemmas are the main technical tools behind the reduction process of general lattice domains to vertical strip domains. In the first lemma we deal with separated supports of variables

Lemma 4.6. Suppose that there is

Then

Proof. For each

Let

where for

Lemma 4.7. Suppose that there are two closed, bounded cones

being segments of rational lines, their apertures smaller than

Then

Proof. We observe that in view of Lemma 4.6 we may substitute cones C_{1}, C_{2} by restricted cones_{1}, C_{2}. We may take the radius

through v and separating C_{1} and C_{2}. Lemma 4.5 allows us to assume that line l has the form

_{1}, C_{2}, and that C_{1} lies above l and C_{2} below l.

If

and kernel F to kernel

where

In the next step we bring the sum (64) to a computable form. We introduce an integer cone

where

Let

Since

This concludes the proof that since the constants M, _{1}, C_{2}, condition Φ guaranties that

The authors would like to thank ESI (Erwin Schrödinger Institute, University of Vienna), where the joint work on this paper began (Thematic Programme: Modern Methods of Time-Frequency Analysis II), and CIRM (Centre international de recontres mathématique, Luminy, Marseille), where the three authors had the chance to complete their work during the period of Hans Feichtinger’s Morlet Chair.

H. G.Feichtinger,K.Nowak,M.Pap, (2015) Asymptotic Boundary Forms for Tight Gabor Frames and Lattice Localization Domains. Journal of Applied Mathematics and Physics,03,1316-1342. doi: 10.4236/jamp.2015.310160