FR´ ECHET FRAMES FOR SHIFT INVARIANT WEIGHTED SPACES

Vol. 40, No. 1, 2010, 19-28

FR´ ECHET FRAMES FOR SHIFT INVARIANT WEIGHTED SPACES

Suzana Simi´c²

Abstract. In the present paper we analyze Fr´echet frame of the form {ϕ(· −j)|j∈Z^d}. With a known condition onϕ, we show that the given sequence constitutes a frame for a test space isomorphic to the space of periodic smooth functions so that its dual is the multiple of the space of periodic distributions byϕ.b

AMS Mathematics Subject Classification (2000): 53C25

Key words and phrases: Banach frame, Fr´echet frame, pre-F-frame,F- Bessel sequence, weightedL^ps-spaces, weighted sequence spaces.

1. Introduction

Frame theory was introduced in [9] and up to now it has developed very much in connection to wavelet theory, time frequency analysis, and sampling theory (see [1], [2], [5], [7], [11], [14], [15], [16],. . .). Shift invariant spaces are generated by frames of the form {ϕ(x−na)}_n∈Zd and this field is, in Banach spaces, especiallyL^p spaces, very much dealt with by Aldroubi, Sun and Tang [4], who studied the frames of the form {ϕi(· −j)| j ∈ Z^d,1 6i 6r} in L^p spaces. On the other hand, in [17] and [18] authors introduced Fr´echet frames and thus unabled the analysis of various test function spaces and their duals spaces of distributions.

In Section 2 we recall from [17] and [18] definitions concerning Fr´echet frames. Section 3 contains preliminary results on shift invariant weighted spaces, extensions of the corresponding results given in [4]. Our main result is given in Section 4. We prove that {ϕ(· −j)| j ∈R^d} is a frame for weighted shift invariant spaces through several equivalent conditions. In the end we conclude that {ϕ(· −j) | j ∈ R^d} forms a Fr´echet frame for a space of test functions XF =F⁻¹(ϕb· P(−π, π)), whereP is the space of periodic test functions.

2. Notation and notions

We will recall basic notions following [6], [12], [17].

We denote by (X,k·k) a Banach space, by (X^∗,k·k^∗) its dual space, (Θ,|||·|||) is a Banach sequence space. If the coordinate functionals on Θ are continuous,

1Supported by the Project No. 144016.

2Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovi´ca 12, Kragujevac 34000, Serbia, e-mail: [email protected]

or, equivalently, if the convergence in Θ implies the convergence of the corre- sponding coordinates, then Θ is called aBK-space.

We refer to [12] for the basic definitions for frames. p-frames in shift-invariant spaces ofL^pwere considered in [4], whilep-frames in general Banach spaces were studied in [8].

Let{(Ys,| · |s)}s∈N0,N0=N∪ {0}, be a family of separable Banach spaces such that

(1) {0} 6= \

s∈N0

Ys⊆ · · · ⊆Y2⊆Y1⊆Y0,

(2) | · |06| · |16| · |26· · · ,

(3) YF := \

s∈N0

Ys is dense inYs, s∈N0.

ThenYF is a Fr´echet space with the sequence of norms| · |s,s∈N0.

We will always assume that {(Xs,k · ks)}s∈N0 and {(Θs,||| · |||s)}s∈N0 are the sequences of Banach spaces which satisfy (1)-(3). For a fixed s∈ N0, an operator V : ΘF → XF will be called s-bounded if there exists a constant Ks>0 such thatkV({ci}i∈N)ks 6Ks|||{ci}i∈N|||s for all{ci}i∈N ∈ΘF. If V iss-bounded for everys∈N0, thenV will be calledF-bounded.

Let{(Θs,||| · |||s)}s∈N0 be a sequence ofBK-spaces, as well. Then a sequence {gi}i∈N∈(X_F^∗)^Nis called a pre-F-frame forXF with respect to ΘF, if for every s∈N0, there exist constants 0< As6Bs<+∞such that

(4) {gi(f)}i∈N∈ΘF, f ∈XF,

(5) Askfks6|||{gi(f)}i∈N|||s6Bskfks, f ∈XF.

The constants Bs and As, s ∈ N0, are called resp. upper and lower bounds for{gi}i∈N. IfAs=Bs, s∈N0, then the pre-F-frame is called tight. If there exists an F-bounded operatorV : ΘF →XF such thatV({gi(f)}i∈N) =f for allf ∈XF, then a pre-F-frame{gi}i∈N is called anF-frame (Fr´echet frame) for XF with respect to ΘF andV is called anF-frame operator for{gi}i∈N. When (4) holds and at least the upper inequality in (5) holds, then {gi}i∈N is called anF-Bessel sequence forXF with respect to ΘF with boundsBs,s∈N0.

WhenX=XF =Xsand Θ = ΘF = Θs, then one obtains the definitions of Θ-frame, Banach frame and Θ-Bessel sequence, respectively.

If {gi}i∈N is a pre-F-frame for XF with respect to ΘF with lower bounds As

and upper boundsBs ,s∈N0, then for everys∈N0 we have A_skfk_s6|||{g_i^s(f)}_i∈N|||_s6λ_sB_skfk_s, f ∈X_s,

whereg_i^sis the continuous extension ofgionXs. We will consider the following operators

(6) Us:Xs→Θs, Usf ={g_i^s(f)}i∈N, s∈N0,

(7) U :XF →ΘF, U f ={gi(f)}i∈N, and

(8) U_s⁻¹:R(Us)→Xs, U⁻¹:R(U)→XF. The shift invariant spaces of the form

V(ϕ) =n X

j∈Z^d

cjϕ(· −j)o ,

wherec={cj}_j∈Zd is taken from some sequence space, are considered in [4]. ϕ is called generator of V(ϕ). The spaceVp(ϕ) is the shift invariant space of the form Vp(ϕ) = { P

k∈Z^d

ckϕ(· −k)| c={ck}_k∈Zd ∈`<sup>p</sup>. Let V0(ϕ) be the space of finite linear combination of integer translates ofϕandV0,p(ϕ) be theL<sup>p</sup>closure of V0(ϕ). Obviously, we have V0(ϕ)⊂Vp(ϕ)⊂V0,p(ϕ). A function inV0,p(ϕ) is not necessarily generated by `^p coefficients. If Vp(ϕ) itself is closed, i.e, a Banach space, then V_p(ϕ) =V_0,p(ϕ).

3. Preliminary result

Considering p-frames for shift invariant subspaces of L^p space, Aldroubi, Sun and Tung [4] proved that when a sequence of translations of a finite set of appropriate functionsϕ1, . . . , ϕr forms an`<sup>p</sup>-frame for the shift-invariant space Vp(ϕ1, . . . , ϕr)⊆L<sup>p</sup>, for somep >1, then this sequence is also an`^r-frame for Vr(ϕ1, . . . , ϕr) for all values ofr >1.

In this paper we will consider weighted L^p_s, s > 0, spaces. A function f belongs to L^p_s with weight functionωs(x) = (1 +|x|)^s, x∈ R^d, s >0, if ωsf belongs to L^p. Equipped with the norm kfk_L^p_s =kωsfkL^p, the spaceL^p_s is a Banach space. Lets>0, 16p <+∞and

L^p_s:=n f ¯

¯kfk_L^p_s :=³ Z

[0,1]^d

³ X

j∈Z^d

|f(x+j)|(1 +|x+j|)^s´_p dx´_1/p

<+∞o ,

L^∞_s :=n f ¯

¯kfk_L^∞_s := sup

x∈[0,1]^d

X

j∈Z^d

|f(x+j)|(1 +|x+j|)^s<+∞o

;

W_s^p:=

n f ¯

¯kfk_Ws^p:=³ X

j∈Z^d

sup

x∈[0,1]^d

|f(x+j)|^p(1 +|j|)^ps

´_1/p

<+∞

o

;

`^p_s:=

n

c={ci}i∈N

¯¯kck_`^p_s =³ X

i∈N

|ci|^p(1 +|i|)^sp

´_1/p

<+∞

o .

Obviously, we haveW_s^p⊂W_s^q ⊂ L^∞_s ⊂ L^q_s⊂ L^p_s⊂L^p_s, where 16p6q6+∞.

Forp= 1 ands= 0 we also haveL¹=L¹. Next we recall the inequalities from [3].

Lemma 1. a)Let f ∈L^p_s,g∈L¹_s and16p6+∞. Then (9) kf∗gk_L^p_s 6kfk_L^p_skgk_L¹_s.

b)If f ∈L^p_s,16p6+∞, andg∈W_s¹, thenf∗g∈W_s^p and (10) kf∗gk_Ws^p6kfk_L^p_skgkW_s¹.

c)If c∈`<sup>p</sup><sub>s</sub> andd∈`¹_s, thenc∗d∈`<sup>p</sup><sub>s</sub> and (11) kc∗dk<sub>`</sub>^p_s 6kck<sub>`</sub><sup>p</sup><sub>s</sub>kdk`¹_s.

For any sequence c={ci}i∈N ∈`^p_s and f ∈ L^p_s, 16p6+∞, define, as in [4], their semi-convolutionf∗⁰c by

(f ∗⁰c)(x) = X

j∈Z^d

cjf(x−j), x∈R^d.

Lemma 2. a) If f ∈W_s^p, 1 6p6+∞, and c ∈`¹_s, then the function f ∗⁰c belongs toW_s^p and

(12) kf∗⁰ck_Ws^p6kck_`¹_skfk_Ws^p,

and also iff ∈W_s¹ andc∈`^p_s,16p6+∞, then the functionf∗⁰c belongs to W_s^p and

(13) kf∗⁰ck_Ws^p6kck<sub>`</sub><sup>p</sup><sub>s</sub>kfkW<sub>s</sub><sup>1</sup>. b) Iff ∈ L<sup>p</sup><sub>s</sub> andc∈`¹_s, thenf∗⁰c belongs tof ∈ L^p_s and (14) kf∗⁰ck_L^p_s 6kck`¹_skfk_L^p_s.

c) f ∗⁰·is a continuous map from`<sup>p</sup><sub>s</sub> toL<sup>p</sup><sub>s</sub>, and also from`¹_s toL^p_s iff ∈ L^p_s, 16p6+∞.

We will give the proof of the next lemma since it is differently posed in [4].

Lemma 3. Let f ∈L^p_s andg ∈W_s¹,1 6p6+∞, s>0. Then the sequence n R

R^d

f(x)g(x−j)dxo

j∈Z^d belongs to`^p_s and we have (15)

°°

° n Z

R^d

f(x)g(x−j)dx o

j∈Z^d

°°

°`^ps

6kfk_L^p_skgkW_s¹.

Proof. Using inequality (11) for fixedx∈R^d, we obtain

°°

° n Z

R^d

f(x)g(x−j)dxo

j∈Z^d

°°

°`^ps

=³ X

j∈Z^d

¯¯

¯ Z

R^d

f(x)g(x−j)dx

¯¯

¯^p(1 +|j|)^sp´_1/p

6³ X

j∈Z^d

³ Z

R^d

|f(x)||g(x−j)|dx

´_p

(1 +|j|)^sp

´_1/p

=³ X

j∈Z^d

³ Z

[0,1]^d

X

k∈Z^d

|f(x+k)||g(x+k−j)|dx´p

(1 +|j|)^sp´1/p

6³ X

j∈Z^d

Z

[0,1]^d

³ X

k∈Z^d

|f(x+k)||g(x+k−j)|dx´p

(1 +|j|)^spdx´1/p

=³ X

j∈Z^d

Z

[0,1]^d

³ X

k∈Z^d

|f(x+k)||g(x+k−j)|(1 +|k|)^s

´_p dx

´_1/p

6

³ Z

[0,1]^d

X

j∈Z^d

³ X

k∈Z^d

|f(x+k)||g(x+k−j)|(1 +|k|)^s

´_p dx

´_1/p

6

³ Z

[0,1]^d

X

j∈Z^d

|f(x+j)|^p(1 +|j|)^sp³ X

k∈Z^d

|g(x−k)|(1 +|k|)^s

´_p dx

´_1/p

6kfk_L^p_s

³ sup

x∈[0,1]^d

³ X

k∈Z^d

|g(x−k)|(1 +|k|)^s

´_p´_1/p

6kfk_L^p_skgkW_s¹.

2

4. Main result

Our main result is related to Theorem 1 in [4].

Letϕ∈ L^p_s, 16p6∞. We consider shift-invariant spaces of the form

(16) V_s^p(ϕ) =n X

j∈Z^d

cjϕ(· −j)

¯¯

¯c∈`^p_s o

.

Note, ifs= 0, then we have the spaceV^p(ϕ) considered in [4].

Theorem 1. Let ϕ∈ T

s>0

W_s¹. Then the following statements are equivalent to each other.

i) V_s^p(ϕ)is closed inL^p_s for alls>0 and for all16p6+∞.

ii) For alls>0 and16p6+∞, the family{ϕ(· −j)|j∈Z^d} is ap-frame forV_s^p(ϕ), i.e. there exist positive constantsAs,Bs(depending onϕand

s) such that (17)

Askfk_L^p_s 6

°°

° n Z

R^d

f(x)ϕ(x−j)dxo

j∈Z^d

°°

°`^p_s 6Bskfk_L^p_s, ∀f ∈V_s^p(ϕ).

iii) There exist positive constants C1 andC2 such that (18) 0< C16 X

j∈Z^d

|ϕ(xb +j)|²6C2<+∞, a.e. x∈R^d.

iv) There exist positive constants K_s¹ and K_s² (depending on ϕ and s) such that for all 16p6+∞we have

(19) K_s¹kfk_L^p_s 6 inf

c∈Mkck_`^p_s 6K_s²kfk_L^p_s, ∀f ∈V_s^p(ϕ), s>0, where

(20) M =n

c={ck}_k∈Zd∈`^p_s|f(·) = X

k∈Z^d

ckϕ(· −k)o .

v) There existsψ∈ T

s>0

W_s¹ such that (21)

f = X

j∈Z^d

hf, ψ(· −j)iϕ(· −j) = X

j∈Z^d

hf, ϕ(· −j)iψ(· −j), ∀f ∈V_s^p(ϕ).

Proof.

v)⇒iv) Let f = P

j∈Z^d

hf, ψ(· −j)iϕ(· −j) and let M be given by (20). Using (15) we have

c∈Minf kck`^p_s 6

°°

° n Z

R^d

f(x)ψ(x−j)dx o

j∈Z^d

°°

°`^ps

6kfkL^p_skψkW_s¹.

ForK_s²=kψkW_s¹ we have the right-hand side of the inequality.

Using (13), we have

kfkL^p_s 6kfkW_s^p=kϕ∗⁰ckW_s^p6kϕkW_s¹kck`^p_s, and forK_s¹=_kϕk¹

W1 s

we prove the left-hand side of inequality (19).

Assertionsv)⇒ii),ii)⇔iv), andiv)⇒i) are simple and their proofs will be omitted.

iii)⇒iv)

We have already seen that forϕ∈W_s¹ andc∈`<sup>p</sup><sub>s</sub>, 16p6+∞, the inequality kϕ∗<sup>0</sup>ck<sub>Ws</sub><sup>p</sup>6kck<sub>`</sub>^p_skϕkW_s¹,

holds. Withkϕ∗⁰ck_L^p_s 6kϕ∗⁰ck_Ws^p for all 16p6+∞, andK_s¹=kϕk⁻¹_W1 s, we have that the left-hand side of the inequality (17).

The family {ϕ(· −k) |k ∈ Z^d} with the condition (18) is a Riesz basis of V²(ϕ) (see [3]), so there exists a unique functionψ ∈V²(ϕ) such that {ψ(· − k) | k ∈ Z^d} is also a Riesz basis for V²(ϕ), and such that it satisfies the biorthogonality relations

hψ(x), ϕ(x)i= 1, hψ(x), ϕ(x−k)i= 0, k6= 0.

Theorem 2.3 in [3] says that if ϕ∈W_s¹and the family{ϕ(· −k)|k∈Z^d}is a Riesz basis for V²(ϕ), then the dual generator ψis inW_s¹. Since we have that ϕ∈W_s¹for alls>0, then we have thatψ∈ T

s>0

W_s¹. Since

(ϕ∗⁰c)(x) = X

k∈Z^d

ckϕ(x−k)∈V_s^p(ϕ),

thenck,k∈Z^d, can be expressed in the form ck =

Z

R^d

(ϕ∗⁰c)(x)ψ(x−k)dx.

For 16p6+∞(with usual changes forp=∞), we have

|ck(1 +|k|)^s|^p=

¯¯

¯ Z

R^d

(ϕ∗⁰c)(x)ψ(x−k)(1 +|k|)^sdx

¯¯

¯^p

6³ Z

[0,1]^d

X

j∈Z^d

|ϕ∗⁰c|(x+j)|ψ(x+j−k)|(1 +|k|)^sdx´_p

6 Z

[0,1]^d

³ X

j∈Z^d

|ϕ∗⁰c|(|ψ(x+j−k)|(1 +|k|)^s´_p dx.

We sum over k∈Z^d and obtain X

k∈Z^d

|ck|^p(1 +|k|)^sp

6 Z

[0,1]^d

X

j∈Z^d

³ X

k∈Z^d

|ϕ∗⁰c|(x+j)|ψ(x+j−k)|(1 +|k|)^s

´_p dx

6 Z

[0,1]^d

X

k∈Z^d

|ϕ∗⁰c|^p(x+k)|(1 +|k|)^sp³ X

k∈Z^d

|ψ(x+k)|(1 +|k|)^s

´_p dx 6 kψk^p_W1

skϕ∗⁰ck^p_Lp s. It follows

kck_`^p_s 6kψkW_s¹kϕ∗⁰ck_L^p_s.

For the lower bound in the inequality (19) one may chooseK_s²=kψkW_s¹. Finally, kck_`^p_s 6K_s²kfk_L^p_s.

i)⇒iii)

Since V_s^p(ϕ) is closed inL^p_s for all 16p6+∞,s>0, then for p= 2 and s= 1 we have the standard assumption on the generatorϕ, i.e. there exist two constantsC1 andC2 such that

0< C16 X

j∈Z^d

|ϕ(xb +j)|²6C2<+∞, a.e. x∈R^d.

2 Corollary 1. Let ϕ∈ T

s>0

W_s¹. Then V_s^p(ϕ)⊂V_s^q(ϕ), for all 16p6q6+∞

ands>0.

Proof. Letf(x) = P

k∈Z^d

ckϕ(x−k), for somec={ck}_k∈Z^d ∈`^p_s, 16p6+∞.

Since`<sup>p</sup><sub>s</sub>⊂`^q_s, 16p6q6+∞, Theorem 1 implies the inequalities kfk_L^q_s 6Bskck_{`</sub><sup>q</sup><sub>s</sub> 6B<sub>s</sub><sup>0</sup>kck<sub>`}^p_s 6kfk_L^p_s, ∀s>0, 16p6q6+∞.

2 Remark 1. From the inequalities(19)and(17)we can conclude that`^p_s andV_s^p are isomorphic Banach spaces for alls>0and16p6+∞, and forf ∈V_s^p(ϕ) we have the equivalence between inf

c∈M{kck_`^p_s} and the L^p_s-norm of f.

As a consequence of Theorem 1 and from [3, Theorem 1], and since`<sup>p</sup><sub>s1</sub> ⊂`^p_s2, for 06s26s1, we have the following corollary.

Corollary 2. Let ϕ ∈ T

s>0

W_s¹. Then V_s^p₁(ϕ) ⊂ V_s^p₂(ϕ) for 0 6 s2 6 s1 and every 16p6+∞.

We construct Fr´echet spaces XF,p, p>1, as the intersection of translator invariant spacesV_s^p(ϕ),s∈N. Note that, for 16p6+∞,

{0} 6= \

s∈N0

V_s^p(ϕ)⊆ · · · ⊆V₂^p(ϕ)⊆V₁^p(ϕ)⊆V₀^p(ϕ) =V^p(ϕ).

Also, we have thatX_F,p = T

s∈N0

V_s^p(ϕ) is dense inV_s^p(ϕ) for all s ∈N₀. The corresponding sequence space QF,p, p >1, is the intersection of the weighted sequence space `^p_s, s ∈ N0. Note that T

s∈N0

`^p_s, for every p > 1, is actually the space of rapidly decreasing sequences s. We proved that ifϕ∈W_s¹, then a sequence{ϕ(·−k)|k∈Z^d}is ap-frame forV_s^p(ϕ) as well as{ϕ(·−k)|k∈Z^d}

is an r-frame for V_s^r(ϕ), for all 1 6 r 6+∞. So we have that the definition of XF,p does not depend on p > 1, so {ϕ(· −k) | k ∈ Z^d} is a pre-F-frame for X_F,p as well as that {ϕ(· −k) |k ∈Z^d} is a pre-F-frame forX_F,r, for all 16r6+∞.

Since the corresponding function space forsis the space of rapidly increasing functions

S={f | kfkm= sup

n6m(1 +|x|²)^m/2|f⁽ⁿ⁾(x)|<+∞},

and its dual is S⁰- the space of slowly decreasing distributions, we obtain that dual space of Fr´echet spaceXF =XF,p, for anyp, is isomorphic to (a comple- mented subspace of) the spaceS⁰.

Denote by P(−π, π) the space of smooth 2π-periodic functions with the family of norms|θ|k = sup{|θ^(k)(t)|;t∈(−π, π)},k∈N0. It is a Fr´echet space and its dual is the space of 2π-periodic tempered distributions. Denote by F andF⁻¹the Fourier transformation and its inverse transformation, respectively.

We have

Theorem 2. Let ϕ ∈ T

s>0

W_s¹ and XF = T

s∈N0

V_s^p(ϕ) for some1 6p6 +∞.

Then

XF =F⁻¹( ˆϕ· P(−π, π)), in the topological sense.

Proof. For f ∈ XF we have f = P

j∈Z^d

cjϕ(· −j), for some sequence c = {cj}_j∈Zd∈s. Then

fb= X\

j∈Z^d

cjϕ(· −j) =³ X

j∈Z^d

cje^ij·´ b ϕ.

This implies the assertion. 2

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Received by the editors June 5, 2009