J. Math. Anal. Appl. 399 (2013) 100–107

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Directional short-time Fourier transformHossein Hosseini GivDepartment of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

a r t i c l e i n f o

Article history:Received 3 September 2011Available online 6 October 2012Submitted by Pierre Lemarie-Rieusset

Keywords:Short-time Fourier transformRidgeletContinuous ridgelet transformGabor ridge functionDirectional short-time Fourier transformInversion formula

a b s t r a c t

A directionally sensitive variant of the short-time Fourier transform is introduced whichsends functions on Rn to those on the parameter space Sn−1

× R × Rn. This transform,which is named directional short-time Fourier transform (DSTFT), uses functions in L∞(R) aswindow and is related to the celebrated Radon transform. We establish an orthogonalityrelation for the DSTFT and explore some operator-theoretic aspects of the transform,mostly in terms of proving a variant of the Hausdorff–Young inequality. The paper isconcluded by some reconstruction formulas.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries

The continuous wavelet and short-time Fourier transforms are important mathematical tools in areas like signal andimage processing. Almost one decade after the first appearance of wavelets, E.J. Candés introduced a directionally sensitivevariant of continuous wavelet transforms, known as the continuous ridgelet transform, [1–3]. Candés’ idea was to constructfrom a Schwarz function ψ : R → R a family of building blocks

ψa,b,ξ (x) := a−1/2 ψ

ξ · x − b

a

; x ∈ Rn, (1.1)

where a ∈ (0,∞), b ∈ R and ξ ∈ Sn−1, and then use them to define the continuous ridgelet transform of appropriatef : Rn

→ C as a function on (0,∞)× R × Sn−1 by

Rψ f (a, b, ξ) = ⟨f , ψa,b,ξ ⟩.

According to (1.1), the function ψa,b,ξ behaves like a one-dimensional wavelet in the direction of ξ and is constant in itsorthogonal complement.

The short-time Fourier transform (STFT) of f ∈ L2(Rn)with respect to a window g ∈ L2(Rn), [4,5], is the function definedon Rn

× Rn via

Vg f (x, w) =

Rn

f (t)g(t − x)e−2π iw·tdt. (1.2)

Assuming that Tx and Mw are the operators of translation and modulation on L2(Rn) given by

(Txg)(y) = g(y − x), (Mwg)(y) = e2π iw·yg(y), (1.3)

E-mail address: hossein.giv@gmail.com.

0022-247X/$ – see front matter© 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2012.09.053

H.H. Giv / J. Math. Anal. Appl. 399 (2013) 100–107 101

we may interpret the STFT in each of the forms

Vg f (x, w) = ⟨f ,MwTxg⟩ (1.4)

= f · Txg(w), (1.5)

where the last function is the Fourier transform of f · Txg .There is a transform, intimately related to the STFT, that is obtained from changing the order of translation and

modulation operators in (1.4). As a matter of notation, let us denote this transform by Ug :

Ug f (x, w) = ⟨f , TxMwg⟩ = e2π ix·w Vg f (x, w). (1.6)

In [6], Grafakos and Sansing introduced a directionally sensitive variant of Ug , which in view of (1.6) provides such avariant of the STFT. The basic idea in [6] was to transform an appropriate function f defined on Rn, with respect to someSchwarz function g defined on R, to a function on Sn−1

× R × R:

(ξ , x, w) →

Rn

f (t)g(ξ · t − x)e−2π iw(ξ ·t−x)dt. (1.7)

As in the case of ridgelets, a so-called ‘‘Gabor ridge function’’

t → g(ξ · t − x)e2π iw(ξ ·t−x)

behaves like a one-dimensional ‘‘Gabor function’’ in the direction of ξ and is constant in its orthogonal complement. (Itshould be mentioned that the transform sending f to the function given by (1.7) does not provide a full reconstruction off , and this led the above mentioned authors to use ‘‘weighted Gabor ridge functions’’ for the analysis and synthesis of thefunctions. See [6] for details.) The directionally sensitive variant of the STFT mentioned above can be obtained from (1.7) bypulling e2π ixw out of the integral, i.e., it is the transform that sends f to a function defined on Sn−1

× R × R by

(ξ , x, w) →

Rn

f (t)g(ξ · t − x)e−2π iw(ξ ·t)dt. (1.8)

Noticing (1.8), we find that this transform is a directionally sensitive variant of the STFT that corresponds to its interpretation(1.4); if we interpret the STFT as in (1.4), we have to enter the directional parameter ξ in both the window g and theexponential function defining the modulation operator.

The aim of the present paper is to introduce a directionally sensitive variant of the STFT that is based on the interpretation(1.5). In fact, if we think of the STFT as in (1.5), we are led to enter the directional parameter ξ only in the window g . This isbecause of the scope of the STFT, best represented in (1.5): to localize the Fourier transform by multiplying the function tobe analyzed by translations of an appropriate window and then taking Fourier transform.

In summary, we are interested in an integral transform that sends a function f defined on Rn to one on Sn−1× R × Rn

given by

DSg f (ξ , x, w) :=

Rn

f (t)g(ξ · t − x)e−2π it·wdt, (1.9)

where g is an appropriate window defined onR. It follows from (1.9) that, under certain assumptions on f and g, f → DSg fcan still be a certain Fourier transform, andwe can therefore legitimately name it the directional short-time Fourier transform(DSTFT). Here, the parameters ξ and x appear in the ‘‘directional window’’

gξ,x : t ∈ Rn

→ g(ξ · t − x), (1.10)

which behaves like a one-dimensional window in the direction of ξ and is constant in its orthogonal complement, and as in(1.2), the parameterw is used only for taking the Fourier transform.

To introduce the DSTFT more completely, we organized this paper as follows. In the remainder of the present section,we fix our notation, make our conventions and present the required preliminaries. In Section 2, the DSTFT is formallyintroduced and conditions on f and g are imposed to ensure that the integral in (1.9) is absolutely convergent. Relation tothe Radon transform and an orthogonality relation are also established in this section. The section is concluded by provinga Hausdorff–Young inequality for the DSTFT and one interesting corollary of it. The inequality allows better understandingof the operator theoretic aspects of the transform. Finally, in Section 3, weak-sense and pointwise inversion formulas andan approximate reconstruction procedure are obtained for the DSTFT.

Suppose f and h are measurable functions defined on the samemeasure space (X,N, ν). IfX f (x)h(x)dν(x) is absolutely

convergent, we denote it by ⟨f , h⟩.For an arbitrary function g defined on Rm and x, w ∈ Rm,m ≥ 1, we define the functions Txg andMwg on Rm as in (1.3).

The resulting operators Tx and Mw are called translation and modulation operators, respectively. If g ∈ L1(Rm), the Fouriertransform of g will be denoted byg .

In the remainder of this paper, n is a fixed integer greater than 1. If g : R → C is any function and (ξ , x) ∈ Sn−1× R, we

define gξ,x : Rn

→ C by (1.10). We also set gξ,x,w = Mwgξ,x for everyw ∈ Rn.

102 H.H. Giv / J. Math. Anal. Appl. 399 (2013) 100–107

The unit sphere Sn−1⊂ Rn is equippedwith the normalized surface areameasure, andwe denote integrating against this

measure by dξ . Also, we will consistently use the notation∆ = Sn−1× R × Rn. When integrating functions on∆, we write

∆forSn−1

R

Rn , δ for the triple (ξ , x, w) and for the subscript of functions like g

ξ,x,w , and we set dδ = dwdxdξ , so thatSn−1

R

Rn f (ξ , x, w) dwdxdξ becomes

∆f (δ) dδ,

∆

DSg f (δ) gδ dδ is written forSn−1

R

Rn DSg f (ξ , x, w) gξ,x,wdwdxdξ ,

and so on.The Radon transform of f ∈ L1(Rn), [7–9], is defined by

R(f )(ξ , s) = Rξ (f )(s) =

ξ ·t=s

f (t) dm(t); ξ ∈ Sn−1, s ∈ R,

where m is the Lebesgue measure on the hyperplane ξ · t = s. Then, the Fourier slice theorem asserts that for all ξ ∈ Sn−1

and f ∈ L1(Rn), Rξ (f ) ∈ L1(R), and Rξ (f )(s) =f (sξ) for every s ∈ R. In particular,f ∈ L1(Rn) implies Rξ (f ) ∈ L1(R), forevery ξ ∈ Sn−1.

We now recall some measure-theoretic inequalities which will be used in Theorem 2.5. They all can be found in [10].

Theorem 1.1 (Hausdorff–Young Inequality). Let 1 ≤ q ≤ 2 and p be the conjugate exponent of q. Then the Fourier transformmaps Lq(Rn) into Lp(Rn), and

∥f ∥p ≤ ∥f ∥q. (1.11)

Theorem 1.2 (Minkowski’s Inequality for Integrals). Let (X,M, µ) and (Y ,N , ν) be σ -finite measure spaces, and f ≥ 0 be an(M

N )-measurable function defined on X × Y . Then for every 1 ≤ p < ∞

X

Yf (x, y) dν(y)

p

dµ(x)1/p

≤

Y

Xf (x, y)p dµ(x)

1/pdν(y).

Proposition 1.3. Let (X,M, µ) be a measure space. If 0 < p < q < r ≤ ∞, then Lp(µ) ∩ Lr(µ) ⊂ Lq(µ) and

∥f ∥q ≤ ∥f ∥λp∥f ∥1−λr ,

where λ ∈ (0, 1) is given by

λ =q−1

− r−1

p−1 − r−1.

2. Directional short-time Fourier transform and its properties

We begin with a formal definition of the DSTFT.

Definition 2.1. Let g ∈ L∞(R) be a non-zero function. For every f ∈ L1(Rn), we denote the directional short-time Fouriertransform (DSTFT) of f with respect to g by DSg f and define it as a function on Sn−1

× R × Rn via

DSg f (ξ , x, w) :=

Rn

f (t)g(ξ · t − x)e−2π it·wdt. (2.1)

The assumption g ∈ L∞(R) ensures that for every (ξ , x) ∈ Sn−1× R, the function g

ξ,x defined by (1.10) is essentiallybounded with

∥gξ,x∥∞ ≤ ∥g∥∞, (2.2)

and this shows that the integral in (2.1) is absolutely convergent.Now, suppose g has a compact support K ⊂ R. For f ∈ L1(Rn), ξ ∈ Sn−1 and x ∈ R,DSg f (ξ , x, ·) is the Fourier transform

of the function f gξ,x concentrated on the set {t ∈ Rn

: ξ · t ∈ x + K}, which is a ‘‘restricted’’ collection of hyperplanes inRn that are perpendicular to ξ . If, in particular, g = χ

[0,1] and we choose to work in R2, then the last set is nothing but thestrip {t ∈ R2

: x ≤ ξ · t ≤ x + 1} perpendicular to ξ ∈ S1, and DSg f (ξ , x, ·) gives the Fourier transform of the restrictionof f itself to that strip. Thus, the DSTFT allows us to take the Fourier transform on sets which are in some sense of ‘‘limitedwidth’’ and have favored direction.

The name ‘‘directional short-time Fourier transform’’ is perhapsmore suitable for the operatorDSg that sends f ∈ L1(Rn)to the function DSg f defined on ∆. The range of this operator heavily depends on its domain of definition and propertiesof the window g . For example, we will see in Theorem 2.4 that if g ∈ L1(R) ∩ L∞(R), then DSg maps L1(Rn) ∩ L2(Rn) intoL2(∆). For now,we can show that if g is only assumed to be in L∞(R), thenDSg maps L1(Rn), i.e., its full domain, into L∞(∆).

H.H. Giv / J. Math. Anal. Appl. 399 (2013) 100–107 103

Proposition 2.2. If g ∈ L∞(R), then DSg is a bounded operator from L1(R) into L∞(∆) with the operator norm ∥DSg∥ ≤

∥g∥∞.

Proof. For every (ξ , x, w) ∈ ∆

|DSg f (ξ , x, w)| ≤

Rn

|f (t) gξ,x(t)| dt ≤ ∥g

ξ,x∥∞ ∥f ∥1,

and the result now follows from (2.2). �

The following proposition presents some useful facts about the DSTFT. The first item determines the relation of the DSTFTto the Radon transform, and the second one shows that under some stronger conditions on the involved functions, the DSTFTcan be regarded as a one-dimensional STFT.

Proposition 2.3. 1. If g ∈ L∞(R) and f ∈ L1(Rn), then for every (ξ , x, w) ∈ ∆

DSg f (ξ , x, w) = f gξ,x(w) = ⟨Rξ (M−wf ), Txg⟩.

2. If f ,f ∈ L1(Rn) and g ∈ L1(R)∩L∞(R), thenDSg f (ξ , x, w) is the STFT of the function Rξ (M−wf )with respect to the windowg, evaluated at (0,−x).

Proof. 1. The first equality is obvious from the definition. In view of (2.1), for (ξ , x, w) ∈ ∆we can write

DSg f (ξ , x, w) =

R

ξ ·t=s

f (t)g(ξ · t − x)e−2π it·wdm(t)ds

=

RRξ (M−wf )(s)Txg(s)ds,

giving the second equality.2. Since f ∈ L1(Rn), Rξ (M−wf ) ∈ L1(R). Also,f ∈ L1(Rn) implies that Rξ (M−wf ) is in L1(R). This shows that Rξ (M−wf ) ∈

L2(R). Since g ∈ L2(R), item (1) and Plancherel’s theorem tell us that DSg f (ξ , x, w) is the same as ⟨ Rξ (M−wf ),Txg⟩,which is in turn equal to ⟨ Rξ (M−wf ),M−xT0g⟩. �

As in the case of the STFT, we can prove an orthogonality relation for the DSTFT.

Theorem 2.4. Suppose g1, g2 ∈ L∞(R) and f1, f2 ∈ L1(Rn) ∩ L2(Rn). If at least one of the gi’s is in L1(R), then the followingorthogonality relation holds.

∆

DSg1 f1(δ)DSg2 f2(δ) dδ = ⟨f1, f2⟩⟨g2, g1⟩. (2.3)

In particular, if g ∈ L1(R) ∩ L∞(R) and f ∈ L1(Rn) ∩ L2(Rn), then DSg f ∈ L2(∆) and

∥DSg f ∥2 = ∥g∥2∥f ∥2.

Proof. Denote the integral on the left-hand of (2.3) by I . For i = 1, 2, fi ∈ L2(Rn), and hence fig iξ,x ∈ L2(Rn), so that the firstequality in (1) of Proposition 2.3 and Plancherel’s theorem yield

I =

Sn−1

R

Rn

f1(t)f2(t) g1ξ,x(t)g2ξ,x(t)dtdxdξ ·

By our assumptions on the gi’s, the function x → g1ξ,x(t)g2ξ,x(t) lies in L1(R) for all (ξ , t) ∈ Sn−1× Rn. This together with

the fact that f1f2 ∈ L1(Rn) allows us to use Fubini’s theorem for interchanging the order of integration and obtain:

I =

Rn

f1(t)f2(t)

Sn−1

Rg1ξ,x(t)g2ξ,x(t)dxdξ

dt

= ⟨f1, f2⟩⟨g2, g1⟩,

because, by the properties of the Lebesgue measureRg1ξ,x(t)g2ξ,x(t)dx =

Rg1(x)g2(x)dx,

and the last integral is absolutely convergent, again by our assumptions on the gi’s. �

104 H.H. Giv / J. Math. Anal. Appl. 399 (2013) 100–107

Comparing Proposition 2.2 and Theorem 2.4 one finds that if g ∈ L1(Rn) ∩ L∞(Rn), then for every f ∈ L1(Rn) ∩ L2(Rn),DSg f ∈ L2(∆) ∩ L∞(∆). Using the known properties of Lp-spaces, this shows that for all such f and g,DSg f ∈ Lp(∆) forevery 2 < p < ∞. In the following result, we estimate the value ∥DSg f ∥p.

Theorem 2.5. Assume f ∈ L1(Rn) ∩ L2(Rn), g ∈ L1(R) ∩ L∞(R) and 2 < p < ∞. Then DSg f ∈ Lp(∆) and

∥DSg f ∥p ≤ ∥g∥p ∥f ∥q, (2.4)

where q is the conjugate exponent of p. Writing ∥g∥p in terms of ∥g∥2 and ∥g∥∞, and ∥f ∥q in terms of ∥f ∥1 and ∥f ∥2 we obtain

∥DSg f ∥p ≤ (∥f ∥2 ∥g∥2)2/p (∥f ∥1 ∥g∥∞)

1−2/p . (2.5)

Proof. Since f ∈ L1(Rn) ∩ L2(Rn) and g ∈ L∞(R), for every (ξ , x) ∈ Sn−1× R, f g

ξ,x ∈ L1(Rn) ∩ L2(Rn). Also, p > 2 impliesq < 2, and this in turn gives f g

ξ,x ∈ Lq(Rn) every (ξ , x) ∈ Sn−1× R. Since p > 0, we have q > 1 and the Hausdorff–Young

inequality of Theorem 1.1 therefore gives:Rn

|DSg f (ξ , x, w)|pdw1/p

=

Rn

|f gξ,x(w)|

pdw1/p

≤

Rn

|f gξ,x(v)|

qdv1/q

.

Hence

∥DSg f ∥qp =

Sn−1

R

Rn

|DSg f (ξ , x, w)|pdwdxdξq/p

≤

Sn−1

R

Rn

|f gξ,x(v)|

qdvp/q

dxdξ

q/p

.

Since p/q > 1, we may now apply Minkowski’s inequality (Theorem 1.2) to the last integrals to obtain

∥DSg f ∥qp ≤

Rn

Sn−1

R

|f gξ,x(v)|

pdxdξq/p

dv

=

Rn

|f (v)|q

Sn−1

R

|g(ξ · v − x)|pdxdξq/p

dv.

But Sn−1

R

|g(ξ · v − x)|pdxdξ =

Sn−1

R

|g(x)|pdxdξ = ∥g∥pp.

Hence

∥DSg f ∥qp ≤

∥g∥p ∥f ∥q

q,

giving (2.4). Now, (2.5) follows from (2.4), Proposition 1.3 and a simple calculation. �

In analogy with the Hausdorff–Young inequality (1.11), inequality (2.4) can be called the Hausdorff–Young inequality forDSTFT. This is because, as we mentioned in the proof of the above theorem, p > 2 implies 1 < q < 2. The inequality alsoshows that DSg is a bounded operator of norm less than or equal to ∥g∥p from (L1(Rn) ∩ L2(Rn), ∥ · ∥q) into Lp(∆).

As our final result in this section, we use Theorem 2.5 to prove the following interesting assertion which finds upperbounds for

U |DSg f (δ)|2 dδ in terms of the measure |U| of U ⊂ ∆ and the relevant norms of f and g .

Proposition 2.6. If U is a measurable subset of ∆, f ∈ L1(Rn) ∩ L2(Rn) and g ∈ L1(R) ∩ L∞(R), then for every 2 < p < ∞U

|DSg f (δ)|2 dδ ≤ (∥f ∥2∥g∥2)4/p (∥f ∥1∥g∥∞)

2p−2p

|U|

p−2p . (2.6)

Proof. Fix 2 < p < ∞. Apply Hölder’s inequality with the exponents p2 and p

p−2 to obtainU

|DSg f (δ)|2 dδ ≤

∆

|DSg f (δ)|2·p2 dδ

2p

∆

χU (δ)p

p−2

p−2p

= ∥DSg f ∥2p |U|

p−2p .

Now, using (2.5) we get (2.6). �

H.H. Giv / J. Math. Anal. Appl. 399 (2013) 100–107 105

3. Some inversion formulas

We begin with establishing a weak-sense inversion formula for the DSTFT which is an immediate corollary ofTheorem 2.4.

Proposition 3.1. Suppose g1, g2 ∈ L∞(R) and f ∈ L1(Rn) ∩ L2(Rn). If at least one of the gi’s is in L1(R) and ⟨g1, g2⟩ = 0, then

f =1

⟨g2, g1⟩

∆

DSg1 f (δ) g2δ dδ, (3.1)

meaning that f is the unique element of L2(Rn) such that for every function u in L1(Rn) ∩ L2(Rn)

⟨f , u⟩ =1

⟨g2, g1⟩

∆

DSg1 f (δ) ⟨g2δ , u⟩dδ. (3.2)

In particular, if g ∈ L1(R) ∩ L∞(R) is a non-zero function, then

f =1

∥g∥22

∆

DSg f (δ) gδ dδ.

Proof. If u ∈ L1(Rn) ∩ L2(Rn), ⟨g2δ , u⟩ = DSg2u(δ). Hence (3.2) follows from (2.3) with f1 and f2 replaced by f and u. Theuniqueness is an outgrowth of the density of L1(Rn) ∩ L2(Rn) in L2(Rn). �

In the case of the STFT, there is a method of reproducing the function f from its STFT coefficients by an approximationprocedure. This is done in terms of a nested exhausting sequence of subsets of Rn

× Rn, [5]. In what follows, we develop asimilar reproduction procedure in our context.

Definition 3.2. A nested exhausting sequence in ∆ is a sequence (Km)m≥1 of subsets of ∆ such that ∪m≥1 Km = ∆ and foreverym ≥ 1, Km ⊂ Km+1.

Theorem 3.3. Suppose g1, g2 ∈ L1(R)∩L∞(R), f ∈ L1(Rn)∩L2(Rn), and ⟨g2, g1⟩ = 0. Also, let (Km)m≥1 be a nested exhaustingsequence in∆. If we define

fm =1

⟨g2, g1⟩

Km

DSg1 f (δ) g2δ dδ, (3.3)

then ∥f − fm∥2 → 0 as m → ∞.

Although Theorem 3.3 holds for every nested exhausting sequence in∆, it is more beneficial to consider those sequenceswhose elements are ‘‘small’’ in an appropriate sense. Of course, compactness of a set in a topological space is a good instanceof such smallness. For this reason, we equip∆with the product topology in which Sn−1 has the topology inherited from Rn,and R and Rn have their Euclidean topologies. Then, letting Om = {ξ = (ξ1, . . . , ξn) ∈ Sn−1

: ξn = 0 or |ξn| ≥ 1/m}, Cm =

[−m,m] × [−m,m]n and Bm = [−m,m] × {w ∈ Rn

: |w| ≤ m} for m ≥ 1, we find that (Om × Cm)m≥1 and (Om × Bm)m≥1are nested exhausting sequences in∆ that consist of compact sets.

In the proof of Theorem 3.3, we will need the following lemma.

Lemma 3.4. If φ ∈ L2(∆) and (Km)m≥1 is a nested exhausting sequence in∆, thenK cm

|φ(δ)|2 dδ → 0

as m → ∞.

Proof. For m ≥ 1 and (ξ , x, w) ∈ ∆ define

hm(ξ , x, w) = |φ(ξ, x, w)|2 χKcm(ξ , x, w).

Since for everym, Km ⊂ Km+1, K cm+1 ⊂ K c

m and so χKcm+1

(ξ , x, w) ≤ χKcm(ξ , x, w) for all (ξ , x, w) ∈ ∆. This gives

hm(ξ , x, w) ≥ hm+1(ξ , x, w) ≥ 0,

for every (ξ , x, w) ∈ ∆ andm ≥ 1. Also, if (ξ , x, w) ∈ ∆, then∆ = ∪m≥1 Km implies that there existsm0 such that (ξ , x, w)∈ Km0 . This shows (ξ , x, w) ∈ K c

m for allm ≥ m0, giving hm(ξ , x, w) → 0 asm → ∞. Since∆

|h1(δ)| dδ ≤∆

|φ(δ)|2 dδ <∞, it follows from a variant of Lebesgue’s monotone convergence theorem (see page 32 of [11]) that

∆

hm(δ) dδ → 0

106 H.H. Giv / J. Math. Anal. Appl. 399 (2013) 100–107

asm → ∞. The result now follows by noticing that∆

hm(δ) dδ =

K cm

|φ(δ)|2 dδ. �

Proof of Theorem 3.3. In view of (3.1), (3.3), the Schwarz inequality, and Theorem 2.4, for any u ∈ L1(Rn) ∩ L2(Rn) with∥u∥2 = 1 we can write:

|⟨f − fm, u⟩| =1

|⟨g2, g1⟩|

K cm

DSg1 f (δ)DSg2u(δ) dδ

≤1

|⟨g2, g1⟩|

∆

|DSg2u(δ)|2 dδ

1/2 K cm

|DSg1 f (δ)|2 dδ

1/2

=∥g2∥2

|⟨g2, g1⟩|

K cm

|DSg1 f (δ)|2 dδ

1/2

.

Since L1(Rn) ∩ L2(Rn) is dense in L2(Rn), the first term above is less than or equal to the last one, for every u ∈ L2(Rn)with∥u∥2 = 1. Therefore

∥f − fm∥2 ≤∥g2∥2

|⟨g2, g1⟩|

K cm

|DSg1 f (δ)|2dδ1/2

,

and the result follows from Lemma 3.4 by lettingm → ∞, because DSg1 f ∈ L2(∆) by Theorem 2.4. �

We conclude this paper with a pointwise reconstruction theorem whose proof needs the following lemma.

Lemma 3.5. If f ,f ∈ L1(Rn) and g,g ∈ L1(R), then for all (ξ , x) ∈ Sn−1× R the functions f g

ξ,x andf g

ξ,x lie in L1(Rn).

Proof. That f gξ,x ∈ L1(Rn) follows from f ∈ L1(Rn) and g ∈ L∞(R). Aswe saw in theproof of (2) of Proposition 2.3,Rξ (M−wf )

∈ L2(R) for everyw ∈ Rn. Hence, we can use (1) of Proposition 2.3 and Plancherel’s theorem to write

∥f gξ,x∥1 =

Rn

|⟨Rξ (M−wf ), Txg⟩|dw =

Rn

|⟨ Rξ (M−wf ),Txg⟩|dw.But for every s ∈ R, Rξ (M−wf )(s) = M−wf (sξ) = T−w

f (sξ) =f (sξ + w), and Txg(s) = M−xg(s) = e−2π ixsg(s). Therefore∥f g

ξ,x∥1 ≤

Rn

R

|f (sξ + w)g(s)|dsdw = ∥g∥1 ∥f ∥1 < ∞. �

Theorem 3.6. Assume f ,f ∈ L1(Rn), g1,g1 ∈ L1(R), and f and g1 are continuous. Then for each window g2 ∈ L∞(R) with⟨g2, g1⟩ = 0, the following reconstruction formula holds for every t ∈ Rn.

f (t) =1

⟨g2, g1⟩

∆

DSg1 f (δ) g2δ (t) dδ. (3.4)

If, in particular, f ,f ∈ L1(Rn) and 0 = g,g ∈ L1(R) are continuous, then for every t ∈ Rn

f (t) =1

∥g∥22

∆

DSg f (δ) gδ (t) dδ.

Proof. For every (ξ , x, t) ∈ ∆

Jξxt :=

Rn

DSg1 f (ξ , x, w)g2ξ,x,w (t) dw = g2ξ,x(t)

Rn

f g1ξ,x(w) e2π it·wdw.

Since fg1ξ,x is continuous, the Fourier inversion theorem, that is applicable because of Lemma 3.5, implies

Jξxt = g2ξ,x(t)f (t)g1ξ,x(t),

for every t ∈ Rn. Hence, for all such t , the right-hand side of (3.4) becomes

f (t)⟨g2, g1⟩

Sn−1

Rg2ξ,x(t)g1ξ,x(t)dxdξ =

f (t)⟨g2, g1⟩

⟨g2, g1⟩ = f (t). �

Although the properties of f and g1 in the above theorem seem to be restrictive, these functions can still be chosen fromdense subspaces of their relevant L2 spaces, the Schwarz class for example.

H.H. Giv / J. Math. Anal. Appl. 399 (2013) 100–107 107

Acknowledgment

The author is grateful to Professor Mehdi Radjabalipour for his helpful suggestions and arguments.

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