Degree Project
Boundedness of a Class of
Hilbert Operators on
Modulation Spaces
Abstract
Acknowledgements
Contents
1 Introduction 1
2 Preliminaries 3
2.1 Lebesgue Integration and Lp-theory . . . . 6
2.2 Fourier Analysis . . . 8
2.3 The Short-Time Fourier Transform . . . 10
2.4 Operators . . . 13
3 Frame Theory 13 3.1 Frames . . . 14
4 Time-Frequency Analysis and the Hilbert Transform 17 4.1 B-splines . . . 17
4.2 Gabor Frames . . . 19
4.3 Boundedness of Hilbert Operators on Modulation Spaces . . . 20
5 Discussion 30 5.1 Summary . . . 30
1
Introduction
Time-Frequency analysis is the study of and attempt to represent signal phe-nomena in terms of both time and frequency simultaneously, instead of only in terms of time or frequency as in classical Fourier analysis. Such representa-tions provide advantages to classical ones by revealing more information directly about the signal itself, although they warrant more sophisticated mathematical tools and give rise to new difficulties such as the uncertainty principle, which says approximately that one cannot know instantaneous frequencies. Because of this, time-frequency analysis connects a wide variety of mathematical disci-plines, including Fourier analysis, harmonic analysis and Gabor analysis just to mention a few.
For measurable functions f and g defined on Rd we define the short-time
Fourier transform (abbreviated STFT) of f with respect to g as Vgf (x, ω) =
Z
Rd
f (t)g(t − x)e−2πit·ωdt (1) in all points (x, ω) ∈ R2d where the integral in (1) converges. The function
g is known as the ”window function” and we normally require this function to satisfy certain decay properties. If for example the window g has compact support or decays rapidly toward infinity, then the short time Fourier transform will reveal local time-frequency information about f . An example of such a class of functions that will turn out to be useful is the set of Schwartz functions S(Rd) and its associated dual space S0
(Rd), the set of tempered distributions. A fundamental question to consider is which properties hold for the short-time Fourier transform as defined by (1). One such fundamental result states that if the functions f and g belong to L2
(Rd), then the short-time Fourier
transform will be uniformly continuous and just like with the classical Fourier transform there is an inversion formula at our disposal.
Another classical result is the uncertainty principle of Heisenberg-Pauli-Weyl (see [1]) which states that
Z (x − a)2|f (x)|2dx 12Z (ω − b)2| ˆf (ω)|2dω 12 ≥ 1 4π||f || 2 2
where the integration is taken over R, for all a, b ∈ R whenever f ∈ L2
(R). We will in this work take primary interest in the connection between time-frequency representations and the theory of frames. In particular the so called Gabor frames that involve the short-time Fourier transform. In general, a se-quence {fk}∞k=1 in a separable Hilbert space H is a frame for H if there are
constants A, B > 0 such that A||f ||2≤
∞
X
k=1
These sequences serve as generalizations to orthonormal bases and let us de-compose and reconstruct the elements of H as if though we had an orthonormal basis. They therefore allow for the discretization of H.
We will invoke the properties of Gabor frames as we work with the Schwartz class and modulation spaces Mp,q, introducing the class of Hilbert operators
Hb,c= ∞
X
n=−∞
cn(MbnHM−bn− Mb(n+1)HM−b(n+1)) (2)
where H is the classical Hilbert transform, c = {cn}∞n=1 denotes a bounded
sequence of complex numbers and Mω is the modulation operator with ω ∈ R.
The operator in (2) can be understood in terms of Fourier multipliers \ Hb,cf = mb,cfˆ where mb,c= −2i ∞ X n=−∞ cnχ(bn,b(n+1)).
χ(a,b)in is the characteristic function for the interval (a, b) ⊂ R. Letting cn= 1
for all n ≥ 0 and cn = −1 for all n < 0 we recover the Hilbert transformation
in (2).
The primary goal of this thesis is to present a proof of a boundedness result about the operator class (2) in Theorem 4.16 and also provide a path towards its generalization through Proposition 4.14.
An essential element in the proofs of each of these results are the piece-wise polynomial functions known as B-splines, obtained iteratively of higher and higher order through convolution starting from the indicator function χ[−1
2, 1 2].
Their advantage is their compact support and the decay of their Fourier trans-forms, which increases with the order of the B-spline.
Another essential component in the proof of the boundedness of the operator class in (2) over the modulation spaces Mp,q is Theorem 4.11 that characterizes
the elements of this same space by providing two norm equivalences between the Mp,q-norm and the discrete `p,q sequence space norm of the Gabor coefficients.
Namely that
||f ||Mp,q ||hf, MβnTαkgi||`p,q
whenever f ∈ Mp,q and g ∈ M1 is such that {M
βnTαkg}k,n∈Z is a Gabor
frame for L2 with given parameters α, β > 0. This result allows us transfer
the question about the boundedness of the operator class in Mp,q to instead
a question about the boundedness of the Gabor coefficients of the elements of Mp,q with respect to the `p,q-norm. A procedure that ultimately allows us to
the rest of the work. In Section 3 we introduce and study the basic structural properties of frames that will serve us further on in the text. In Section 4 we finally arrive at and prove the main theorem of the work and proceed to prove a result relating to its generalization.
2
Preliminaries
To begin let us establish some notational conventions used in this work, although most of the notation is likely to be recognized by the reader or be apparent from the context. For example, sets will be denoted with capital letters and Z, R and C will refer to the set of integers, real numbers and complex numbers respectively (excluding infinity). Whereas the letter d is to be understood as a positive integer and will refer to dimension. We write
Rd= R × R × · · · × R
where the Cartesian product is repeated d − 1 times. We will primarily be concerned with functions mapping Rd
into C and we will mostly use the letters x and t for elements in Rd writing
x = (x1, x2, . . . , xd)
t = (t1, t2, . . . , td)
or simply x and t in the case of d = 1. As the notation may become quite complicated when working in d dimensions a tool that serves to simplify the appearance of many expressions is known as multi-indexing. For this we will mostly use the Greek letters α and β.
Given two real valued functions f (ξ) and g(ξ), the notation f . g will mean that there exists C > 0 such that f (ξ) ≤ Cg(ξ) for all ξ in some specified set. In addition, the notation f := g will mean that f is set to equal g by definition. Definition 2.1. An ordered n-tuple of non-negative integers α = (α1, α2. . . , αn)
is called a multi-index when the following formal rules are understood α! = α1!α2! · · · αn! |α| = α1+ α2+ · · · + αn Dα= ( ∂ ∂x1 )α1( ∂ ∂x2 )α2· · · ( ∂ ∂xn )αn xα= xα1 1 x α2 2 · · · x αn n , x ∈ R n.
C(Rd) will denote the set of all complex valued continuous functions on
Rd and for a positive integer n we let Cn(Rd) and C∞(Rd) denote the sets of complex valued functions on Rd that are n respectively infinitely many times
continuously differentiable. That is to say: Cn(Rd) = {f : Rd→ C : Dα
and
C∞(Rd) = {f : Rd→ C : Dα
f ∈ C(Rd), |α| < ∞}, where α is a multi-index.
In order to make sense of ideas such as openness, closeness, continuity and convergence in spaces more abstract than C or Rd mathematicians have
ab-stracted certain essential properties that ought to remain for these concepts to have meaning. These properties are found whenever the space under consid-eration, call it X, is equipped with a topology (usually denoted τ ). Let ρ(X) denote the power set of X, then we have the following definition.
Definition 2.2. τ ⊂ ρ(X) is said to be a topology to X if (1) ∅, X ∈ τ
(2) τ is closed under finite intersections (3) τ is closed under arbitrary unions.
The couple (X, τ ) is called a topological space. Each A ⊂ τ is called an open set and if a point x ∈ A, then A is called a neighborhood of x.
Definition 2.3. Let X and Y denote topological spaces with respective topolo-gies τxand τy. A mapping f : X → Y is said to be continuous at a point x ∈ X
if there for all neighborhoods V ∈ τy of f (x) exists a neighborhood U ∈ τx of x
such that f (U ) ⊂ V .
Since we are often interested in sets equipped with vector space structure, we are motivated to introduce the concept of topological vector spaces. Let K denote either C or R.
Definition 2.4. (X, τ ) is called a topological vector space if there are binary operations + : X × X → X and ξ : K × X → X that satisfy the vector space axioms and are continuous. For x, x1, x2∈ X and α ∈ K we write
+(x1, x2) = x1+ x2
for vector addition and
ξ(α, x) = αx for scalar multiplication.
Example 2.5. Familiar examples of topological vector spaces include: (1) Rd
and Cd;
(2) C(Rd) and C∞(Rd).
Definition 2.6. Let (X, d) denote a metric space.
(1) A sequence {xk}∞k=1 is called a Cauchy sequence in X if ∀ > 0 ∃N > 0 :
d(xm, xn) < whenever m, n ≥ N .
(2) X is said to be a complete metric space if every Cauchy sequence in X converges to some x ∈ X.
We are now ready to introduce some of the most central spaces treated in this work.
Definition 2.7. Let H and B denote topological vector spaces.
(1) B is called a Banach space if it is complete with topology induced by a norm.
(2) H is called a Hilbert space if it is a Banach space with norm obtained through an inner product.
It follows from the previous definition that the norm of a Hilbert space H is given by
||x||2= hx, xi when h·, ·i is its associated inner product.
When treating infinite series in general, the issue of convergence is one that always has to be considered. The basic ideas of convergence of a sequence or a series are in principle no different in more abstract spaces than in the familiar cases of R or C. However since the convergence of a series may depend on the ordering of the elements over which the corresponding sequence of partial sums are computed and since a series converging with respect to one particular ordering does not necessarily converge to the same value with respect to another we are motivated to consider the circumstances under which the convergence of a series is independent of the order in which those elements are summed. Definition 2.8. Let X denote a topological vector space and let I be a count-able index set for which there is an associated sequence {xi}i∈I in X. Then the
series
X
i∈I
xi
is said to converge unconditionally to x ∈ X if for all permutations I0of I holds
X
i∈I0
xi= x. (3)
The following equivalent condition (Lemma 2.1.1 in [2]) characterizes uncon-ditionally convergent series.
Lemma 2.9. Let {fk}∞k=1 denote a sequence in the Banach space B. Then the
(i) P∞
k=1fk converges unconditionally to f ∈ B;
(ii) ∀ ≥ 0 ∃F ⊂ N which is finite such that f −X k∈I fk <
whenever I ⊂ N is a finite set containing F .
When studying linear mappings of various sorts (such as operators intro-duced below), we will often be concerned with their boundedness or continuity properties. As it turns out, for linear mappings between vector spaces, these are directly connected as the following standard result showcases.
Theorem 2.10. If X, Y are normed vector spaces and T : X → Y is a linear map, then the following conditions are equivalent:
(1) T is continuous; (2) T is continuous at 0; (3) T is bounded.
Proof. See for example Theorem 5.4 of [3].
Definition 2.11. Suppose that X is vector space. Then two norms || · ||1 and
|| · ||2 defined on X are said to be equivalent if there are constants C1> 0 and
C2> 0 such that
C1||x||2≤ ||x||1≤ C2||x||2
for all x ∈ X. In such case we write ||x||1 ||x||2 to indicate the norm
equiva-lence.
2.1
Lebesgue Integration and L
p-theory
Definition 2.12. Let X denote any set. A collection of subsets Σ ⊂ ρ(X) is called a σ − algebra if
(i) X ∈ Σ;
(ii) Σ is closed under complement; (iii) Σ is closed under countable unions.
The couple (X, Σ) is called a measurable space and the elements of Σ are called measurable sets.
Throughout this work, we will treat the case when X = Rdand Y = C. The term ”measurable” will be used interchangeably with ”complex measurable”. Also all integrals will be considered in the Lebesgue sense and dx will denote the Lebesgue measure on Rd.
The concepts of measurability and Lebesgue integration lead naturally to the introduction of the Lp-spaces. For 0 < p ≤ ∞ we define the Lp-norm (which is
a quasi-norm in the case when 0 < p < 1) of a measurable function f through
||f ||p= Z Rd |f |pdx 1p . if p is finite and otherwise
||f ||∞= ess sup
x∈Rd
|f (x)|. With these norms we get the following definition
Lp(Rd) = {f : Rd→ C : f is measurable and ||f||p< ∞}
Remark 2.14. It is a well known fact in analysis that the space L2(Rd) is a Hilbert space when equipped with the inner product h·, ·i defined for f, g ∈ L2(Rd) through
hf, gi = Z
Rd
f (x)g(x)dx.
In this case the Cauchy-Schwarz inequality applies and gives |hf, gi| ≤ Z Rd |f (x)|2dx 12 Z Rd |g(x)|2dx 12 . (4) The inequality in (4) is generalized by H¨older’s inequality involving the con-cept of conjugate exponents.
Definition 2.15. If p ∈ (1, ∞) then the real number q such that p−1+ q−1 = 1 is known as the conjugate exponent to p and the two are called conjugate exponents. Also the couple of 1 and ∞ are considered conjugate exponents. Theorem 2.16 (H¨older’s Inequality). Suppose p and q are conjugate exponents. Then for f ∈ Lp
(Rd) and g ∈ Lq
(Rd) is valid
||f g||1≤ ||f ||p||g||q.
2.2
Fourier Analysis
We define f ∈ C(Rd) to be rapidly decreasing if (1 + |x|2)N|f (x)| is uniformly
bounded with respect to x for each N ∈ N.
Definition 2.17. The Schwartz space S(Rd) is the set of all f ∈ C∞
(Rd) such
that for all N ∈ N, sup |α|≤N sup x∈Rd (1 + |x|2)N|(Dαf )(x)| < ∞. That is to say, S(Rd
) consists of all infinitely differentiable functions on Rdsuch
that it and all of its partial derivatives are rapidly decreasing. Example 2.18. The Gaussians f (x) = e−a|x|2 where x ∈ Rd
belong to S(Rd)
when a > 0. To see this, note that for any multi-index α and any positive integer N, the expression (1 + |x|2)N|(Dαf )(x)|, with f (x) = e−a|x|2
, is such that we may factor out f (x) to obtain a product on the form P (x)e−a|x|2 where P (x) is of at most polynomial growth whereas the other factor declines at an exponential rate. It follows that the expression goes to zero when |x| → ∞ and since it is a continuous expression, it must assume a maximal value on Rd
implying that it is bounded and hence belongs to S(Rd).
Associated to the Schwartz space S is its dual space S0, defined as the set of all continuous linear maps from S to C. The elements of S0 are called tempered distributions, which form a special class of distribution that allow us to extend various transforms to a greater set of objects.
Definition 2.19. Let f ∈ L1
(Rd). Then the Fourier transform of f is defined
as ˆ f (ω) = (F f)(ω) = Z Rd f (x)e−2iπx·ωdx.
Definition 2.20. For measurable functions f, g : Rd → C, the convolution
between the two is defined as (f ∗ g)(x) = R
Rdf (x − y)g(y) dy whenever the
involved integral converges.
For each ω ∈ Rd, we define the character eωto be the function
eω(x) = e2πiω·x.
Note that it ∀x, y ∈ Rd has the property
eω(x + y) = eω(x)eω(y),
mul-tiplicative group of complex numbers with modulus one. Furthermore, (f ∗ eω)(0) = Z Rd f (0 − y)eω(y) dy = Z Rd f (−y)e2πiω·ydy = Z Rd f (y)e−2πiω·ydy = ˆf (ω) so that we obtain ˆ f (ω) = (f ∗ eω)(0).
Remark 2.21. Note that for an arbitrary function f ∈ Lp with p 6= 1, the
given definition of the Fourier transform cannot be applied directly to obtain the Fourier transform ˆf . Instead one can introduce a limit process to define the Fourier Transform, for which we refer to page five of [1].
The Fourier Transforms has the following formal properties with respect to S(Rd).
Proposition 2.22. Let f ∈ S(Rd
), h ∈ Rd and α be a multi-index. Then
(1) F (f(x + h)) = ˆf (ω)e2iπω·h; (2) F (f(x)e−2iπx·h) = ˆf (ω + h); (3) F ((∂x∂ )αf (x)) = (2iπω)αf (ω);ˆ (4) F ((−2iπx)αf (x)) = ( ∂ ∂ω) αf (ω).ˆ
Proof. See for example Proposition 2.1 of chapter 6 in [4].
We recall the inversion formula for Fourier transform on the Schwartz space: Theorem 2.23. For each f ∈ S(Rd)
f (x) = Z Rd ˆ f (ω)e2iπx·ωdω and Z Rd | ˆf (ω)|2dω = Z Rd |f (ω)|2dx. Proof. See for example Theorem 2.4 of chapter 6 in [4]. Theorem 2.24. For f, g ∈ S(Rd):
(1) f ∗ g ∈ S(Rd);
Proof. See for example Theorem 7.8 in [5].
Theorem 2.25 (Hausdorff-Young). Suppose 1 ≤ p ≤ 2 and let p0 be the conju-gate exponent of p. ThenF : Lp(Rd) −→ Lp0(Rd) and
|| ˆf ||p0 ≤ ||f ||p.
Proof. See for example Theorem 1.1.3 in [1].
Remark 2.26. Note that this means that for p = 2 one obtainsF : L2
(Rd) →
L2(Rd) and
|| ˆf ||2≤ ||f ||2.
The following inequality, known as Young’s inequality, is an important result in Fourier and Time-Frequency analysis that will be used in this work.
Proposition 2.27 (Young’s Inequality). Let p, q, r ∈ [1, ∞] be such that1p+1q =
1
r + 1. Suppose also that f ∈ L
p and g ∈ Lq. Then f ∗ g ∈ Lr
(Rd) and
||f ∗ g||r≤ ||f ||p||g||q.
Proof. See for example Theorem 1.2.1 of [1].
2.3
The Short-Time Fourier Transform
Definition 2.28. Let f ∈ Lp(Rd) and let x, ω ∈ Rd, then the translation operator Tx and the modulation operator Mω are defined on Lp(Rd) as follows:
(i) Txf (t) = f (t − x);
(ii) Mωf (t) = eω(t)f (t) = e2iπω·tf (t).
We verify a few properties relating to the short-time Fourier transform: Proposition 2.29. If f ∈ Lp, then for each 1 ≤ p ≤ ∞ the following relations
Now, if we let g = Mωf , we see that ||TxMωf ||pp= ||Txg||pp= Z Rd |Txg(t)|pdt = Z Rd |g(t − x)|pdt = Z Rd |g(s)|pds = ||g||pp= ||Mωf ||pp= ||f || p p
which proves (1). Let s = t − x. Then F (Txf )(ω) = Z Rd f (t − x)e−2πiω·tdt = Z Rd f (s)e−2πiω·(s+x)ds = e−2πiω·x Z Rd f (s)e−2πiω·sds = e−2πiω·xf (ω) = Mˆ −xf (ω),ˆ
which proves (2). To prove (3) we note that F (Mωf )(ξ) =
Z
Rd
e2πiω·tf (t)e−2πiξ·tdt =
Z
Rd
f (t)e−2πi(ξ−ω)·tdt = ˆf (ξ − ω) = Tωf (ξ).ˆ
Corollary 2.30. If f ∈ Lp(Rd), then so are Txf and Mωf and hence also
TxMωf .
Since we have now proved that the fundamental operators map Lp into Lp we can show that they are continuous and linear operators on this same family of spaces.
Proposition 2.31. Tx and Mω are continuous linear operators on Lp(Rd)
Proof. Let f, f1, f2∈ Lp(Rd) and α ∈ C. Then
Tx(f1+ f2)(t) = (f1+ f2)(t − x) = f1(t − x) + f2(t − x) = Txf1(t) + Txf2(t)
and
Tx(αf )(t) = (αf )(t − x) = αf (t − x) = αTxf (t).
Since ||Txf ||p ≤ C||f ||p with C = 1 by Proposition 2.29, it follows that Tx is
Definition 2.32. Let f and g be (complex) measurable functions on Rd. Then the short-time Fourier transform of f with respect to g is denoted Vgf and is
given by
Vgf (x, ω) =
Z
Rd
f (t)g(t − x)e−2πit·ωdt
in all points (x, ω) ∈ R2dwhere the integral in question converges. The function
g is known as the window function.
In practice, one will always require some additional properties of the involved functions in order to ensure that the STFT is sufficiently well-behaved as to be useful. We may for example require that f is locally integrable over Rd and that g be compactly supported and smooth or that both functions belong to the Hilbert space L2
(Rd).
Lemma 2.33. Suppose that f, g ∈ L2
(Rd). Then V
gf is uniformly continuous
on Rd.
Proof. See for example Lemma 3.1.1. in [1].
Remark 2.34. The short-time Fourier transform is a kind of Fourier transform in the classical sense as we have that
F (fTxg)(ω) =
Z
R
f (t)g(t − x)e−2πiω·tdt and if f, g ∈ L2
(Rd), then it is expressible as a scalar product in that same space
through hf, MωTxgi = Z R f (t)e2πiω·tT xg(t)dt = Z Rd f (t)g(t − x)e−2πit·ωdt. To understand the final section of this work we are in need of defining a special class of function spaces that turn out to be of interest in the study of time-frequency analysis, examples of mixed-norm spaces.
Definition 2.35. Let 0 < p, q ≤ ∞. For measurable functions f : R2
→ C we let
fp(y) := ||f (·, y)||Lp
and we define the Lp,q-norm of f by
||f ||Lp,q:= ||fp||Lq.
Lp,q= Lp,q
(R × R) is the set of all measurable functions f : R2→ C such that
||f ||Lp,q < ∞.
Furthermore, we let `p,q = `p,q
(Z × Z) denote the set of all sequences c =
These definitions lead to the introduction of modulation spaces.
Definition 2.36. For 0 < p, q ≤ ∞ and a given g ∈ S(R), we define the set Mp,q= Mp,q
(R) to be the set of all f ∈ S0(R) such that ||f ||Mp,q:= ||Vgf ||Lp,q < ∞
for a given window function g. Whenever p = q we write Mpin place of Mp,p.
2.4
Operators
The study of frames involves the introduction and applications of certain oper-ators (such as the frame operator). Therefore we are in need of a few definitions from operator theory to be able to proceed to the next section.
Definition 2.37. Let B1and B2denote Banach spaces. An operator is a linear
map U : B1→ B2and if there exists a K > 0 such that
||U x||B2≤ K||x||B1 ∀x ∈ B1,
then the operator is said to be continuous. Usually it is apparent from the context with respect to which space the norm is taken so we may write simply || · || in place. We define the operator norm
||U || = sup{||U x|| : x ∈ B1, ||x|| = 1}.
If we are given two operators in U1 : B1 → B2, U2 : B2 → B3 for Banach
spaces B1, B2, B3, then we define the composition U2U1 between the two via
U2U1x = (U2◦ U1)(x) ∀x ∈ B1.
If we instead are given two Hilbert spaces H1, H2and an operator U : H1→ H2,
then there is a unique so called adjoint operator U∗: H2→ H1 such that
hx, U yiH2 = hU
∗x, yi
H1 ∀x ∈ H2 and ∀y ∈ H1.
Also,
(1) if H1 = H2 = H then U is called unitary if U U∗ = U∗U = I where I
denotes the identity operator on H (2) U is called self-adjoint if U = U∗
(3) U is called positive if hU x, xi ≥ 0 ∀x ∈ H.
3
Frame Theory
In this section we introduce the special category of Bessel sequences known as frames and see how they can be used to decompose and reconstruct vector spaces, for which an example is given. At the same time, we introduce the analysis, synthesis and frame operators that are central to the field.
3.1
Frames
Throughout this section we let H denote a Hilbert space.
Definition 3.1. A sequence {fk}∞k=1in H is called a frame for H if there exist
A, B > 0 such that for all f ∈ H holds A||f ||2≤
∞
X
k=1
|hf, fki|2≤ B||f ||2.
A and B are called lower and upper frame bounds respectively. In addition, any sequence satisfying the upper frame bound condition is called a Bessel sequence in H. Furthermore, if A and B coincide, the frame is said to be tight.
Theorem 3.2. If {fk}∞k=1is a Bessel sequence in H, then
P∞
k=1ckfk converges
unconditionally for all sequences {ck}∞k=1∈ ` 2
(N).
Proof. See for example Theorem 3.2.3 and Corollary 3.2.4 in [2].
Therefore it is that any reordering of the elements of the Bessel sequence {fk}∞k=1 does not change the convergence of the series
P∞
k=1ckfk granted that
the elements of {ck}∞k=1 are reordered in the same manner. Hence we can use
an arbitrary index set for Bessel sequences (of which the natural numbers are the simplest), which is useful in the context of Gabor frames.
We define the pre-frame operator (also called synthesis operator) on H with respect to a given frame {fk}∞k=1 as the mapping
T : `2(N) → H, T {ck}∞k=1= ∞
X
k=1
ckfk.
Associated to this operator is its adjoint (called the analysis operator) T∗: H → `2(N), T∗f = {hf, fki}∞k=1.
That together via composition act to form the frame operator S : H → H, Sf = T T∗f =
∞
X
k=1
hf, fki fk.
Note now that if {fk}∞k=1is a frame for H, then for f ∈ H: ∞ X k=1 |hf, fki|2≤ B||f ||2⇔ ∞ X k=1 |hf, fki|2 12 ≤√B||f ||.
That is to say, {hf, fki}k=1∞ ∈ `2(N) so that Theorem 3.2 implies that Sf =
P∞
k=1hf, fki fk converges unconditionally in H.
(i) S is invertible, bounded, self-adjoint and positive.
(ii) {S−1fk}∞k=1 is also a frame for H with bounds A−1, B−1 and frame
op-erator S−1.
Proof. See Lemma 5.1.5 in [2].
Lemma 3.3 tells us that to each frame there is an associated frame {S−1fk}∞k=1
called the canonical dual of {fk}∞k=1. These frames acting together allow us to
uniquely represent each element of H in terms of a series expansion called frame decomposition. This is analogous to how any particular ON-basis of H gives a unique series representation of each element, which motivates us to consider frames as a generalization of the concept of basis in Hilbert spaces.
Theorem 3.4. If {fk}∞k=1 is a frame for H, then
f = ∞ X k=1 hf, S−1f kifk
with unconditional convergence for all f ∈ H. Proof. See theorem 5.1.6 in [2].
Example 3.5. Let {e1, e2} denote any ON-basis of R2. We consider whether
the set {fk}3k=1= {e1, e2, e1+ e2} is a frame for R2 or not. If it is a frame it
should satisfy A||f ||2≤ 3 X k=1 |hf, fki|2≤ B||f ||2
for some constants A, B > 0. By the Cauchy inequality
3 X k=1 |hf, fki|2≤ 3 X k=1 ||fk||2||f ||2= B||f ||2
so that the sequence {fk}3k=1 is a Bessel sequence. Indeed, it follows by the
similar consideration that any finite sequence in a Hilbert space is a Bessel sequence in that same space. For any f ∈ R2, f = ae
which is true for A ≤ 1. Hence it must be that {fk}3k=1is a frame for R 2. Se1= 3 X k=1 he1, fkifk = he1, e1ie1+ he1, e2ie2+ he1, e1+ e2i(e1+ e2) = e1+ 0e2+ e1+ e2 = 2e1+ e2 and Se2= 3 X k=1 he2, fkifk = he2, e1ie1+ he2, e2ie2+ he2, e1+ e2i(e1+ e2) = e2+ e1+ e2 = e1+ 2e2
so that for f = ae1+ be2 for some a, b ∈ R. Therefore,
Sf = S(ae1+ be2) = aSe1+ bSe2= (2a + b)e1+ (a + 2b)e2.
We can represent the frame operator as a matrix: S =2 1 1 2 so that S−1= 1 3 2 −1 −1 2 . We have that S−1e1= 2 3e1− 1 3e2 and S−1e2= − 1 3e1+ 2 3e2. By Lemma 3.3, {S−1fk}3k=1= { 2 3e1− 1 3e2, − 1 3e1+ 2 3e2, 1 3e1+ 1 3e2} is the
4
Time-Frequency Analysis and the Hilbert
Trans-form
In this section we consider the basic properties of the B-spline function class as well as their corresponding Fourier transforms, known as sinc functions. We will prove a result relating to their belonging to the modulation space Mq.
Since these spaces are invariant under Fourier transformation, it follows that the obtained result is valid for sinc functions as well.
We also present some essential results about Gabor frames that allow us to prove the main theorem of this thesis (Theorem 4.16), about the boundedness of a class of operators related to the Hilbert transform.
4.1
B-splines
Definition 4.1. A B-spline of order n is a function Bn : R → R defined
inductively through B1(x) = χ[−1 2, 1 2](x) Bn+1(x) = Bn∗ χ[−1 2,12](x) = Z R Bn(x − t)χ[−1 2,12](t)dt.
Example 4.2. B2(x) can be calculated through
Therefore, B2(x) = 1 + x when − 1 ≤ x ≤ 0 1 − x when 0 < x ≤ 1 0 otherwise = max(0, 1 − |x|).
In general, B-splines become increasingly regular as their order increases although their symmetric support around the origin becomes increasingly wide as the following Proposition from A.9.1 of [2] shows.
Proposition 4.3. ∀n ∈ N, the below properties hold for Bn.
(1) Bn ∈ Cn−2(R) for n ≥ 2 (2) supp(Bn) = [−n2,n2] (3) Bn(x) > 0 ∀x ∈ (−n2,n2) (4) R RBn(x)dx = 1 (5) P n∈ZBn(x − n) = 1 ∀x ∈ R.
(6) Granted a continuous function f : R → C, Z R Bn(x)f (x)dx = Z [−n 2,n2]n f (x1+ x2+ ... + xn)dx1dx2...dxn.
Remark 4.4. We may introduce Nn(x) = Tn
2Bn(x) if we wish to obtain a set
of functions with the same properties as the B-splines instead with support on [0, n] as opposed to [−n2,n2]. These splines are referred to as cardinal B-Splines. Example 4.5. We can use property six in Proposition 4.3 to calculate the Fourier transform of a B-spline. To see how this can be done, let for example n = 2. Then Bn is simply the triangle function obtained in Example 4.2,
f (x) = e−2iπxω and we obtain
The last equation in (5) is due to Euler’s formula. Repeating the same procedure for arbitrary n it is easy to see that we would obtain
(F Bn)(ω) = sin(πω) πω n .
4.2
Gabor Frames
With the basic results about frames presented in section 3 in mind we are now ready to study the notion of Gabor frame, a concept originating from a paper of Gabor in 1946 on the theory of communications in which was examined sequences of functions on the form {MmbTnag}m,n∈Z. The idea to combine the
analysis of Gabor with the theory of frames did not come about until 1986 in a publication of Daubechies, Grossman and Meyer entitled Painless nonorthogonal expansions. For a more detalied historical note, see pages 167-169 of [2]. Definition 4.6. A Gabor frame for L2
(R) is a frame on the form {MmbTnag}m,n∈Z
for some lattice parameters a, b > 0 where g is a given function in L2
(R). That is to say, there exist A, B > 0 such that
A||f ||2≤ X m,n∈Z |hf, MmbTnagi|2≤ B||f ||2 ∀f ∈ L2(R). Remark 4.7. Since hf, MmbTnagi = Z R
f (t)g(t − na)e−2πimt·mdt = Vgf (na, mb),
it follows that the frame condition for a Gabor frame corresponds to estimating the STFT in a grid of R with lattice parameters a and b.
Example 4.8. It is known that the system {MmbTnag}m,n∈Zbecomes a Gabor
frame if g(x) = e−x2 with lattice parameters a and b such that ab < 1. See Example 1.1 of [6].
The following result by Christensen relates the properties of B-splines di-rectly to those of frames via time-frequency shifts.
Theorem 4.9. Suppose that N ∈ N and that g ∈ L2
(R) is a real-valued bounded function with supp(g) ⊂ [0, N ] for which is valid
X
n∈Z
g(x − n) = 1.
Suppose further that b ∈ (0,2N −11 ]. Then the function g and the function γ defined by γ(x) = bg(x) + 2b N −1 X n=1 g(x + n)
Proof. See Theorem 2.6 of [6].
Gabor frames allow us to discretize a space such as L2. That is to
repre-sent each object in the space in terms of a discrete number of coefficients, as illustrated through the following important result about Gabor frames.
Proposition 4.10. Let g be a function in L2
(Rd) and let α, β > 0. Then if
{TαkMβng}k,n∈Zdis a frame for L2(Rd), there exists a dual window γ ∈ L2(Rd)
such that the dual frame is {TαkMβnγ}k,n∈Zd and every f ∈ L2(Rd) can be
expanded as f = X k,n∈Zd hf, TαkMβngiTαkMβnγ = X k,n∈Zd hf, TαkMβnγiTαkMβng
with unconditional convergence in L2
(Rd).
Proof. See Proposition 5.2.1 in [1].
4.3
Boundedness of Hilbert Operators on Modulation Spaces
We formally define the class of operators of interest in this section Hb,c as
follows. Let c = {cn}n∈Z be a bounded sequence of complex numbers and let
b > 0, then Hb,c= ∞ X n=−∞ cn(MbnHM−bn− Mb(n+1)HM−b(n+1))
where H is the Hilbert transformation defined formally by (Hf )(x) = 1 π→0lim Z |t|> f (x − t) t dt.
It turns out that H is bounded on Lp for 1 < p < ∞ and that H = 12Hb,c for
any b > 0 when the sequence c is such that cn = −1 when n < 0 and cn = 1
when n ≥ 0. In addition, for bounded c, it follows from Plancherel’s theorem that Hb,c maps L2 into L2. Unfortunately, Hb,c fails to be bounded for all
bounded sequences c on Lpwhen p 6= 2 which is part of the motivation to study
them on modulation spaces instead. For proof of these assertions, see [7] and the references therein.
To conclude this work we generalize Theorem 1 from [7] which treats the bound-edness properties of the introduced operator Hb,c. The proof will be based on
Theorem 4.11. Let g ∈ M1 be such that {MβnTαkg}k,n∈Z is a Gabor frame
for L2 for given parameters α, β > 0 and let 1 ≤ p, q ≤ ∞. Then there is a dual γ ∈ M1such that every tempered distribution f in Mp,q has a Gabor expansion
f = X
k,n∈Z
hf, MβnTαkγiMβnTαkg ∀f ∈ Mp,q (6)
that converges uncondtionally in Mp,q when p, q < ∞ and in the weak∗topology
of M∞ when p = ∞ or q = ∞. Furthermore,
||f ||Mp,q ||hf, MβnTαkgi||`p,q ||hf, MβnTαkγi||`p,q, f ∈ Mp,q. (7)
Proof. See for example [8].
As a consequence of Theorem 3.7 in [9], we have the following extension of the previous result.
Theorem 4.12. Let 0 < p, q, r ≤ ∞ be such that r ≤ min(1, p, q) and let g and γ be the same as in Theorem 4.11. If in addition g, γ ∈ Mr, then every
tempered distribution f in Mp,q has the Gabor expansion (6) that converges
uncondtionally when p, q < ∞ and in the weak∗ topology of M∞ when p = ∞ or q = ∞. Furthermore, (7) holds.
Remark 4.13. The essential consequence of this characterization that will serve us in this work is that for a tempered distribution f ,
f ∈ Mp,q⇔ {hf, MβnTαkgi}k,n∈Z∈ `p,q.
This allows us to prove (or disprove) the membership of f to the modulation class Mp,q by considering its coefficients with respect to a Gabor frame as in
Theorem 4.11.
Proposition 4.14. Bn belongs to Mq for all q > 1n when n ≥ 2.
Proof. Let g(t) = Bn. By Proposition 2.29 and Parseval’s formula,
VgBn(x, ω) = hg, MωTxgi = hˆg, TωM−xˆgi = Z R sin(πτ ) πτ n sin(π(τ − ω)) π(τ − ω) n e2πix(τ −ω)dτ. (8)
It follows from (8) that |VgBn(x, ω)| .
Z
R
(1 + |τ |)−n(1 + |τ − ω|)−ndτ, (9) where the right-hand side is even in ω due to symmetry. Hence it suffices to consider the case when ω ≥ 0. Then
where I1(ω) = Z 0 −∞ (1 + |τ |)−n(1 + |τ − ω|)−ndτ, I2(ω) = Z ω2 0 (1 + |τ |)−n(1 + |τ − ω|)−ndτ, I3(ω) = Z 2ω ω 2 (1 + |τ |)−n(1 + |τ − ω|)−ndτ and I4(ω) = Z ∞ 2ω (1 + |τ |)−n(1 + |τ − ω|)−ndτ.
The integrals I1(ω) − I4(ω) may be bounded in slightly different ways. We have
giving that
I3(ω) . (1 + ω)−n. (13)
Evidently, the latter estimate also holds for 0 ≤ ω ≤ 2 and hence for all ω ≥ 0. For I4(ω) we have I4(ω) = Z ∞ 2ω (1 + |τ |)−n(1 + |τ − ω|)−ndτ . Z ∞ 2ω (1 + |τ |)−n(1 + ω)−ndτ . (1 + ω)−n. (14) It follows from bounds (11) − (14) that
Z ∞
−∞
(1 + |τ |)−n(1 + |τ − ω|)−ndτ . (1 + ω)−n
whenever ω ≥ 2. By (10) and the evenness with respect to ω of the expression on the right hand side of (9) we obtain for all ω ∈ R with |ω| ≥ 2
|VgBn(x, ω)|q ≤ Z ∞ −∞ (1 + |τ |)−n(1 + |τ − ω|)−ndτ q . (1 + |ω|)−qn. (15) It now follows from (15) and the compact support of VgBn with respect to x
when g = Bn that Z R Z R |VgBn(x, ω)|qdxdω = Z R Z n −n |VgBn(x, ω)|qdxdω ≤ Z R Z n −n C(1 + |ω|)−qndxdω = Z R 2nC(1 + |ω|)−qndω (16) for some constant C > 0. The integral on the right hand side of (16) is finite if q > 1
n, which concludes the proof.
Lemma 4.15. The B-spline B2 belongs to the modulation class Mq for q >12
and the sequence
{MnTk
2g(x)}k,n∈Z
is a Gabor frame for L2(R) when g(x) is the Fourier transform of B2.
We now present the main theorem of this work, which is a generalization of the one originating from a paper by B´enyi, Grafakos, Gr¨ochenig and Okoudjou (see Theorem 1 in [7]).
Theorem 4.16. For any b > 0 and c ∈ `∞, the operators Hb,c are bounded
from Mp,q into Mp,q for 1 < p < ∞, 1
2 < q ≤ ∞ with a norm estimate
||Hb,cf ||Mp,q≤ C||c||∞||f ||Mp,q
for some constant C that depends only on b, p and q.
Proof. Assume that b = 1 as we may conjugate Hb,c with an appropriate
dila-tion. Let
g(x) = sin(πx) πx
2 .
Lemma 4.15 tells us that g ∈ Mr when r = min(1, q) and that we obtain a Gabor frame for L2(R) through {MnTk
2g(x)}k,n∈Z. Let γ be the dual of g as in
Theorem 4.9, then γ belongs to Mras well since it is a finite linear combination
of g.
By Proposition 4.10 we may expand f as f = X k,n∈Z hf, MnTk 2γiMnT k 2g. (17)
It follows from (17) that
(CgH1,cf )(k, n) = hH1,cf, MnTk 2gi = * H1,c X k0,n0∈Z hf, Mn0Tk0 2 γiMn0Tk0 2 g , MnTk 2g + = * X k0,n0∈Z hf, Mn0Tk0 2 γiH1,c Mn0Tk0 2 g , MnTk 2g + = X k0,n0∈Z D hf, Mn0Tk0 2 γiH1,c Mn0Tk0 2g, Mn Tk 2g E = X k0,n0∈Z hf, Mn0Tk0 2 γihH1,c Mn0Tk0 2g, Mn Tk 2gi. (18)
In the equivalent condition provided by Theorem 4.12 we saw that H1,cf ∈
Mp,q if and only if CγH1,cf = {hH1,cf, MnTk
2gi}k,n∈Z∈ `
p,q
(Z2). We need to show the boundedness of the matrix
γ(k,n)(k0,n0)= hH1,cMn0Tk0 2
g, MnTk
If χ = χ[0,1], then Parseval’s formula and Proposition 2.29 give γ(k,n)(k0,n0)= hF (H1,cMn0Tk0 2 g),F (MnTk 2g)i = hm1,cTn0F (Tk0 2 g), TnF (Tk 2g)i = hm1,cTn0M −k0 2 ˆ g, TnM−k 2giˆ = X m∈Z cmχ[m,m+1]Tn0M −k0 2 ˆ g, TnM−k 2gˆ (20) We can evaluate the inner product (20) directly from the definition so that
γ(k,n)(k0,n0)= Z R X m∈Z cmχ(t − m)Tn0M −k0 2 ˆ g(t) · TnM−k 2ˆg(t) dt = Z R X m∈Z cmχ(t − m)Tn0M −k0 2 ˆ g(t)e−πiknMk 2Tng(t)ˆ dt = Z R X m∈Z cmχ(t − m)Tn0e−πik 0t ˆ
g(t)e−πikneπiktTnˆg(t)
dt = X m∈Z cme−πiknH(m, k, k0, n, n0), where H(m, k, k0, n, n0) = Z R
χ(t − m)e−πik0(t−n0)g(t − nˆ 0)eπiktˆg(t − n) dt. (21) By taking s = t − n as new variable of integration in (21) we get
H(m, k, k0, n, n0) = eπik0n0
Z
R
χ(t − m)e−πik0tg(t − nˆ 0)eπiktˆg(t − n) dt = eπik0n0
Z
R
χ(s − m + n)e−πik0(s+n)g(s + n − nˆ 0)eπik(s+n)ˆg(s) ds = eπik0n0e−πik0neπikn
Z
R
Due to the support of B2being [−1, 1] the series is convergent and if the supports
of the involved functions do not overlap in a non-degenerate interval, the inner product in the series equals zero so that all terms but the ones with m = n − 1 or m = n disappear. Therefore, γ(k,n)(k0,n0)= cn−1(−1)k 0(n0−n) hMk−k0 2 χ[−1,0]Tn0−nˆg, ˆgi + cn(−1)k 0(n0−n) hMk−k0 2 χ[0,1]Tn0−ng, ˆˆ gi.
Label these terms
t1= cn−1(−1)k 0(n0−n) hMk−k0 2 χ[−1,0]Tn0−ng, ˆˆ gi t2= cn(−1)k 0(n0−n) hMk−k0 2 χ[0,1]Tn0−nˆg, ˆgi.
The support of Tn0−nˆg is [n0− n − 1, n0− n + 1] and so t1 can only be non-zero
when n0− n − 1 = −1 or n0− n − 1 = −2 that is when n0 = n or n0= n − 1 and
likewise, t2 can only be non-zero when n0= n + 1 or n0= n.
With h = k − k06= 0 we have for n0 = n,
By similar calculations we get that when n0= n and k 6= k0, t2= cn − 1 πi(k − k0)+ 2 π2(k − k0)2+ 2((−1)k− 1) (iπ(k − k0))3 and when k = k0, t2= cn 3. Let δk denote the Kronecker delta,
ρk= ( − 1 πik+ 2 π2k2 + 2((−1)k−1) (iπk)3 if k 6= 0 0 if k = 0 and k= ( −(−1)π2kk2+1+ 2((−1)k−1) (iπk)3 if k 6= 0 0 if k = 0. Then straight forward calculations yield:
Case 1: n0= n γ(k,n)(k0,n0)= cn−1 ρk0−k+ 1 3δk0−k + cn ρk−k0+ 1 3δk−k0 . Case 2: n0= n + 1 γ(k,n)(k0,n0)= cn(−1)k 0 k−k0+ 1 6δk−k0 . Case 3: n0= n − 1 γ(k,n)(k0,n0)= cn−1(−1)k 0 k0−k+ 1 6δk0−k . Let a(k, n) = hf, MnTk
2γi. Then (18) becomes
The series in (23) are all expressible as convolutions with respect to the first variable of a(k, n). Lettingρ(k) = ρ(−k),˜ (k) = (−k) and (−1)˜ ·a(·, n) ∗ = {P
k0∈Z(−1)k 0
a(k0, n)k−k0}k∈Zwe can rewrite (18) as
(CgH1,cf )(k, n) = cn a(·, n) ∗ ρ(k) +1 3a(k, n) + cn−1 a(·, n) ∗ ˜ρ(k) +1 3a(k, n) + cn (−1)·(a(·, n + 1) ∗ (k) +1 6a(k, n + 1) + cn−1 (−1)·a(·, n − 1) ∗ ˜(k) +1 6a(k, n − 1) . By the triangle inequality with respect to the variable k and the `p-norm,
||(CgH1,cf )(·, n)||p (24) ≤ |cn| · ||a(·, n) ∗ ρ(k)|| + |cn| 3 ||a(·, n)|| (25) + |cn−1| · ||a(·, n) ∗ ˜ρ(k)|| + |cn−1| 3 ||a(·, n)|| + |cn| · ||(−1)·(a(·, n + 1) ∗ (k)|| + |cn| 6 ||a(·, n + 1)|| + |cn−1| · ||(−1)·a(·, n − 1) ∗ ˜(k)|| + |cn−1| 6 ||a(·, n − 1)||.
Since f ∈ Mp,q by assumption, a(k, n) ∈ `p,q and since the sequence {cm}m∈Z
was assumed bounded, |cn| 3 ||a(·, n)|| + |cn−1| 3 ||a(·, n)|| + |cn| 6 ||a(·, n + 1)|| (26) +|cn−1| 6 ||a(·, n − 1)||
≤ ||c||∞(||a(·, n)|| + ||a(·, n + 1)|| + ||a(·, n − 1)||)
The sequence (k) is in `1, so by letting r = p and q = 1 in Proposition 2.27 we
get
|cn| · ||(−1)·a(·, n + 1) ∗ (k)|| + |cn−1| · ||(−1)·a(·, n − 1) ∗ ˜(k)|| ≤ (27)
||c||∞(||a(·, n + 1)||p· ||(k)||1+ ||a(·, n − 1)||p· ||˜(k)||1) ≤
K||c||∞(||a(·, n + 1)||p+ ||a(·, n − 1)||p)
for some K > 0. The convolution of a(·, n) and 1
πik is bounded for 1 < p < ∞
(see section 8.12 of [10]) so that we obtain
|cn| · ||a(·, n) ∗ ρ(k)|| + |cn−1| · ||a(·, n) ∗ ˜ρ(k)|| ≤ (28)
for some K0> 0. (25) − (28) imply that
||(CgH1,cf )(·, n)||p≤ C||c||∞||a(·, n)||p (29)
for some C > 0 and taking the q-norm with respect to the variable n in (29) we obtain
||CgH1,c(f )||`p,q ≤ C||c||∞||a||`p,q.
Therefore, H1,cf ∈ Mp,qand due to the norm-equivalence provided by Theorem
4.11,
||Hb,cf ||Mp,q ≤ C||c||∞||f ||Mp,q
for some other C > 0 than the one in (29) which proves the theorem.
5
Discussion
5.1
Summary
Frames were introduced by Duffin and Schaeffer in the context of sequences of complex exponential functions in a paper from 1952, although their main con-cern was nonharmonic Fourier series. Since the publication of their fundamental paper, frames have found applications in applied signal analysis and in various other parts of mathematics, like for example in time-frequency analysis as we have seen, which is a part of mathematics that has benefited greatly from its introduction (see [2] for more context about the theory of frames). We have studied the most basic facts about frames and given a simple example to illus-trate the fundamental ideas about decomposition and reconstruction that the properties of frames allow for. One could ask initially why one would be in-terested in the frame category in a given space when in fact orthonormal bases allow for similar decomposition and reconstruction of the elements in that space. Some reasons for why such interest is justified include that the condition for a sequence to qualify as an orthonormal basis is quite strong compared to that of the frame condition so that it may be harder to find a suitable orthonormal basis in practice. It may also be that one is interested in finding a sequence allowing for decomposition and reconstruction satisfying additional constraints, for which orthonormal bases may not exist.
Having introduced and studied the basic objects of interest in time-frequency analysis and frame theory, we were able to use the properties of the special class of piece-wise polynomial functions known as B-splines as well as some fairly advanced results in Theorems 4.9 − 4.11 to prove the main theorem of the work in Theorem 4.16, with a proof based on the one given in [7].
Crucial to the success of the proof of Theorem 4.16 was the compact support of the B-spline B2 that together with Proposition 4.10 allowed us to calculate
explicitly the Gabor coefficients of the Hilbert transformation applied to the tempered distribution f . Coefficients that we were later able to bound using Young’s inequality, so that the equivalence condition provided by Theorem 4.11 could be applied for us to obtain the boundedness of this class of Hilbert trans-formations.
5.2
Continuation of Work
Due to Theorem 3.7 in [9] by Yevgeniy V. Galperin and Salti Samarah Theorem 4.11 is valid also in the case when q < 1. Therefore, since the decay of the Fourier transforms of B-splines increases as the order of the B-spline increases (Proposition 4.3), we should expect these transforms to belong to Mqfor smaller
and smaller values of q, which is exactly what we found in Proposition 4.14. Therefore we may proceed in approximately the same manner as in the proof of Theorem 4.16 using a B-spline of higher order. We could potentially generalize Theorem 4.16 to the case of q > 13 using B3 as the window of the short-time
Since B2(x) = max(0, 1 − |x|) we obtain for all x ∈ [−32,32] B3(x) = Z 12 −1 2 max(0, 1 − |x − t|) dt
and from Example 4.5 we know that the corresponding Fourier transform will be
(F B3)(ω) =
sin(πω) πω
3
Which belongs to Mq for all q > 1
3 due to Proposition 4.14. Applying the same
argument as in the provided proof of Theorem 4.16 we would be interested again bounding the sequence (19). Then
γ(k,n)(k0,n0)= X m∈Z cm(−1)k 0(n0−n) hMk−k0 2 χ[m−n,m−n+1]Tn0−ng, ˆˆ gi (30)
References
[1] Karlheinz Gr¨ochenig. Foundations of time-frequency analysis. Springer Science & Business Media, 2013.
[2] Ole Christensen. An introduction to frames and Riesz bases, volume 7. Springer, 2003.
[3] Walter Rudin. Real and Complex Analysis. International series in pure and applied mathematics. McGraw-Hill, 1987.
[4] Elias M. Stein and Rami Shakarchi. Fourier analysis: an introduction, volume 1. Princeton University Press, 2003.
[5] Walter Rudin. Functional Analysis. International series in pure and applied mathematics. McGraw-Hill, 2006.
[6] Ole Christensen. Pairs of dual gabor frame generators with compact sup-port and desired frequency localization. Applied and Computational Har-monic Analysis, 20(3):403–410, 2006.
[7] ´Arp´ad B´enyi, Loukas Grafakos, Karlheinz Gr¨ochenig, and Kasso Okoudjou. A class of fourier multipliers for modulation spaces. Applied and Compu-tational Harmonic Analysis, 19(1):131–139, 2005.
[8] Hans G. Feichtinger and Karlheinz Gr¨ochenig. Banach spaces related to in-tegrable group representations and their atomic decompositions, i. Journal of Functional analysis, 86(2):307–340, 1989.
[9] Yevgeniy V. Galperin and Salti Samarah. Time-frequency analysis on modulation spaces mmp. Applied and Computational Harmonic Analysis, 16(1):1–18, 2004.
