Tight wavelet frames in Lebesgue and Sobolev spaces

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Lasse Borup, Rémi Gribonval, Morten Nielsen

To cite this version:

Lasse Borup, Rémi Gribonval, Morten Nielsen. Tight wavelet frames in Lebesgue and Sobolev spaces.

Journal of function spaces, Hindawi, 2004, 2 (3), pp.227–252. �inria-00567343�

AALBORG UNIVERSITY

!

"

#

$

Tight wavelet frames in Lebesgue and Sobolev spaces

by

Lasse Borup, R´emi Gribonval and Morten Nielsen

March 2003 R-2003-05

D EPARTMENT OF M ATHEMATICAL S CIENCES A ALBORG U NIVERSITY

Fredrik Bajers Vej 7 G DK - 9220 Aalborg Øst Denmark Phone: +45 96 35 80 80 Telefax: +45 98 15 81 29 URL: www.math.auc.dk/research/reports/reports.htm

ISSN 1399–2503 On-line version ISSN 1601–7811

L. BORUP, R. GRIBONVAL, AND M. NIELSEN

ABSTRACT. We study tight wavelet frame systems inL_p(R^d), and prove that such systems (under mild hypotheses) give atomic decompositions ofL_p(R^d) for 1<p<F. We also characterizeL_p(R^d)and Sobolev space norms by the analysis coefficients for the frame. We consider Jackson inequalities for bestm- term approximation with the systems inL_p(R^d)and prove that such inequalities exist. Moreover, it is proved that the approximation rate given by the Jackson in- equality can be realized by thresholding the frame coefficients. Finally, we show that in certain restricted cases, the approximation spaces, for best m-term ap- proximation, associated with tight wavelet frames can be characterized in terms of (essentially) Besov spaces.

1. I

NTRODUCTION

A tight wavelet frame (TWF) for L

₂

( R

^d

) is a fi nite collection of functions } = {

^!

}

!∈E

in L

₂

( R

^d

), E = {1, 2, . . ., L}, for which the system X (}) := {2

^{jd/2 !}

(2

^j

·

−k)| j ∈ Z , k ∈ Z

^d

, ! ∈ E} is a tight frame for L

₂

( R

^d

), i.e., there exists a constant A > 0 such that L

_g∈X(})

|# f , g$|

²

= A% f %

²_L

2

for any f ∈ L

₂

( R

^d

). The functions } are called the generators of the TWF. The construction and properties of TWFs in L

₂

( R

^d

) have been studied extensively by many authors (see e.g. [16, 17]). The purpose of this paper is to study such frames in spaces different from L

₂

( R

^d

).

In particular, we will study TWFs in L

_p

( R

^d

) and L

_p

-based Sobolev spaces. We prove that most reasonable TWFs give atomic decompositions of L

p

( R

^d

), 1 < p <

F , and it is proved that we can characterize the L

_p

( R

^d

) and Sobolev norm by the analysis coef fi cients associated with the frame. An important consequence of the characterization is that there is a Jackson inequality for nonlinear approximation with TWFs, and moreover we will show that the rate of convergence given by the Jackson inequality can be reached simply by thresholding the analysis coef fi cients.

The structure of the paper is as follows. In Section 2 we review the most common method to construct TWFs, the so-called extension principles of Ron and Shen.

The TWFs generated through an extension principle are based on a multiresolution analysis, and the generators are often called framelets. They have been studied ex- tensively, see e.g. [2, 4, 15, 16, 17]. Section 3 contains the analysis of the properties of TWF expansions in L

_p

( R

^d

) and L

_p

-based Sobolev spaces. We give a complete characterization of the L

_p

-norm, 1 < p < F , in terms of analysis coef fi cients as- sociated with the frame, and prove that a TWF gives an atomic decomposition for

2000Mathematics Subject Classification. Primary 41A17; Secondary 41A15, 41A46.

Key words and phrases. wavelet frames, nonlinear approximation.

1

L

_p

( R

^d

). The characterization has the same form as the classical characterization of the L

_p

-norm by wavelet coef fi cients, see e.g. [13]. In Section 3.3 the analysis is extended to L

_p

-based Sobolev spaces. In Section 4, we consider Jackson inequal- ities for best m-term approximation with TWFs in L

p

( R

^d

), and we discuss some cases where a complete characterization of the approximation spaces – associated with best m-term approximation in L

p

( R

^d

) with TWFs – in terms of (essentially) Besov spaces is possible. Two of the present authors have studied approximation with spline based framelets, de fi ned on R , in [11].

2. T

IGHT WAVELET FRAMES

The most common methods to construct TWFs are the extension principles of Ron and Shen. Tight wavelet frames build through the extension principle are based on a multiresolution analysis and we will brie fl y touch upon some of the main ideas in the construction, see [4, 17, 16]. There is also the (signi fi cant) advantage with the MRA based constructions that there are fast associated algorithms.

For historical notes on this construction, we refer the reader to [4]. MRA based TWFs are called framelets. We begin by introducing some basic notation and general assumptions.

Let › = (›

⁰

, ›

¹

, . . . , ›

^L

) be a vector of 2N Z

^d

-periodic measurable functions with ›

⁰

the mask of a re fi nable scaling function fl of a MRA {V

j

}

j∈Z

. We assume that fl satis fi es lim

_→0

! fl( ) = 1 and there exist 0 < c ≤ C < F such that c ≤ L

k∈Z^d

|! fl(· − 2 N k)|

²

≤ C, i.e., fl generates a Riesz basis of the scaling space V

₀

of the MRA. We associate the “wavelets” } = {

^!

}

_!∈E

to › by letting "

^!

(2 ) = ›

^!

( )! fl( ).

The following is the fundamental tool to construct framelets:

Theorem 2.1 (The Oblique Extension Principle (OEP) [4]). Suppose there exists a 2N Z

^d

-periodic function n that is non-negative, essentially bounded, continuous at the origin with n(0) = 1. If for every ∈ [−N, N]

^d

and ƒ ∈ {0, N}

^d

,

(2.1) n(2 )›

⁰

( )›

⁰

( + ƒ) +

L

L

!=1

›

^!

( )›

^!

( + ƒ) =

# n( ), ƒ = 0, 0, otherwise, then the wavelet system {2

^{jd/2 !}

(2

^j

· −k)| j ∈ Z , k ∈ Z

^d

, ! ∈ E }, defined by › is a tight wavelet frame.

The system X (}) is usually called the framelet system generated by }.

Remark 2.2. Theorem 2.1 can be stated in slightly more generality by introducing the notion of a spectrum for the scaling space V

₀

and dropping the requirement that fl generates a Riesz basis, see [4].

Remark 2.3. For n ≡ 1, Theorem 2.1 reduces to the Unitary Extension Principle

(UEP) of Ron and Shen [17]. The advantage of the OEP compared to the UEP is

that one can construct framelets with a high number of vanishing moments using

the OEP. This is not possible with the UEP, where at least one of the generators

has only one vanishing moment.

3. T

IGHT WAVELET FRAMES IN

L

_p AND

S

OBOLEV

S

PACES

In this section we study TWFs in L

_p

( R

^d

), 1 < p < F , (Section 3.1) and L

_p

- based Sobolev spaces (Section 3.3). In Section 3.2 we show that thresholding the analysis coef fi cients associated with a TWF is a bounded operation in L

_p

( R

^d

), 1 < p < F . For notational convenience we let D denote the set of dyadic cubes I = 2

^−j

([0, 1]

^d

+ k), j ∈ Z , k ∈ Z

^d

, and write

_I

(x) := 2

^jd/2

(2

^j

x − k).

3.1. TWFs in L

_p

( R

^d

). Theorem 3.1 below will show that we can characterize the L

_p

-norms by the analysis coef fi cients associated with the TWF. Theorem 3.3 will show that there is a stable way to reconstruct L

p

-functions using the TWF, and this leads to two results: TWFs form atomic decompositions of L

p

( R

^d

) (see Corollary 3.6), and thresholding (or shrinkage of) the frame coef fi cients are stable operations in L

p

( R

^d

) (see Section 3.2).

Theorem 3.1. Let {

^!

}

!∈E

be the generators of a tight wavelet frame for L

₂

( R

^d

).

Suppose for all ! ∈ E, some ʼ > 0 and some ¼ > 0,

^!

∈ C

^ʼ

( R

^d

),

$ !

(x) dx = 0 and |

^!

(x)| ≤ C(1 + |x|)

^−d−¼

.

Then

(3.1) % f %

p

)

% %

% %

&

I∈D,!∈E

L

|# f ,

^!_I

$|

²

|I|

⁻¹

‒

I

(x) '

1/2

%

% %

%

p

,

for 1 < p < F , where ‒

I

denotes the indicator function for the subset I ⊂ R

^d

.

¹

Proof. Let {

^m,s

}

²_s=1^d−1

be the orthonormal Meyer wavelet(s) de fi ned on R

^d

. For each ! ∈ E we consider the integral kernel

K

^!

(x, y) := L

I∈D m,1

I

(x)

^!_I

(y).

Notice that the corresponding operator T

^!

: f +→

$

R^d

K

^!

(x, y) f (y) dy

is bounded on L

₂

( R

^d

) due to the fact that {

^!_I

}

I∈D

is a subset of a frame. Also, standard estimates show that (see e.g. [3])

|K

^!

(x, y)| ≤ C|x − y|

^−d

,

|K

^!

(x

^,

, y) − K

^!

(x, y)| ≤ C|x − x

^,

|

^˜

|x − y|

^−d−˜

, and

|K

^!

(x, y

^,

) − K

^!

(x, y)| ≤ C|y − y

^,

|

^˜

|x − y|

^−d−˜

,

because of the smoothness and decay of

^!

. Thus T

^!

is a Calder´on-Zygmund operator and therefore bounded on L

p

( R

^d

), 1 < p < F. However T

^!

f has a nice

1ByF)Gwe mean that there exist two constants 0<C₁≤C₂<Fsuch thatC₁F≤G≤C₂F.

expansion in the orthonormal Meyer wavelet, so using the L

_p

-characterization of such expansions we get

% %

% %

&

I∈D

L

|# f ,

^!_I

$|

²

|I|

⁻¹

‒

I

(x) '

1/2

% %

% %

p

) %T

^!

f %

p

≤ C% f %

p

. Using this estimate for ! = 1, 2, . . ., L, and the fact that !

1

" → !

2

we get

²

% %

% %

&

I∈D,!∈E

L

|# f ,

^!_I

$|

²

|I|

⁻¹

‒

I

(x) '

1/2

%

% %

%

_p

=

% %

% %

&

!∈E

L

&(

I

L

∈D

|# f ,

^!_I

$|

²

|I|

⁻¹

‒

I

(x)

)

1/2

'

2

'

1/2

%

% %

%

p

≤

% %

% % L

!∈E

(

I∈D

L

|# f ,

^!_I

$|

²

|I|

⁻¹

‒

I

(x) )

1/2

%

% %

%

p

≤ L · C% f %

p

.

Now we turn to the converse estimate. Notice that since we have a tight wavelet frame we have the identity

# f , g$ = A L

I∈D,!∈E

# f ,

^!_I

$#g,

^!_I

$, f , g ∈ L

₂

( R

^d

), where A > 0 is a constant depending only on the frame. Write

W f (x) = {|I|

^−1/2

# f ,

^!_I

$‒

I

(x)}

_I,!

and notice that for f ∈ L

₂

( R

^d

) ∩ L

_p

( R

^d

) and g ∈ L

₂

( R

^d

) ∩ L

_p^,

( R

^d

), with p

⁻¹

+ (p

^,

)

⁻¹

= 1,

|# f , g$| = A

* *

* *

$

#W f (x),W g(x)$

_!2

dx

* *

* *

≤ A

* *

* *

$

#W f (x),W f (x)$

!₂

#W g(x),W g(x)$

!₂

dx

* *

* *

≤ A%#W f (x),W f (x)$

_!2

%

p

%#W g(x),W g(x)$

_!2

%

_p^,

≤ AC%#W f (x),W f (x)$

_!2

%

p

%g%

_p^,

.

Taking the supremum of this estimate for {g ∈ L

₂

( R

^d

) ∩ L

_p^,

( R

^d

) : %g%

_p^,

≤ 1} we obtain

% f %

p

≤ C%#W f ˜ (x),W f (x)$

!₂

%

p

.

This proves the result for f ∈ L

₂

( R

^d

) ∩ L

_p

( R

^d

). To complete the proof we just notice that from the fi rst part of the proof it follows that f +→ #W f (x),W f (x)$

!₂

is

continuous on L

_p

( R

^d

). !

2The notationV "→W means that the two (quasi)normed spacesV andW satisfyV⊂W and there is a constantC<Fsuch that% · %W≤C% · %V.

From Theorem 3.1 we see that the following sequence space plays an important role.

De fi nition 3.2. Let d

_p

denote the sequences {c

^!_I

}

I∈D,!∈E

for which

|||{c

^!_I

}|||

p

:=

% %

% %

&

I∈D,!∈E

L

|c

^!_I

|

²

|I|

⁻¹

‒

I

(x) '

1/2

%

% %

%

p

< F.

In fact, let us show that there is a stable reconstruction operator de fi ned on d

_p

. Theorem 3.3. Let {

^!

}

!∈E

be the generators of a tight wavelet frame for L

₂

( R

^d

).

Suppose for all ! ∈ E, some ʼ > 0 and some ¼ > 0,

^!

∈ C

^ʼ

( R

^d

),

^{+ !}

(x) dx = 0 and |

^!

(x)| ≤ C(1 + |x|)

^−d−¼

. Then the map T : d

_p

+→ L

_p

( R

^d

) defined by

T {c

^!_I

} = L

I∈D,!∈E

c

^!_I ^!_I

is a bounded linear map. Moreover, the sum defining T converges unconditionally.

Proof. We consider the dual (T

^!

)

^,

of the operator T

^!

used in Theorem 3.1, i.e., the operator with kernel

K ˜

^!

(x, y) := L

I∈D

!

I

(x)

^m,1_I

(y).

By exactly the same arguments as given in the fi rst part of the proof of Theo- rem 3.1, it can be shown that (T

^!

)

^,

is bounded on L

_p

( R

^d

). Take {c

^!_I

}

I∈D,!∈E

∈ d

_p

and consider f

^!

:= L

I∈D

c

^!_I ^m,1_I

. This is a well-de fi ned function in L

_p

( R

^d

) with

% f

^!

%

p

)

% %

% %

&

I∈D

L

|c

^!_I

|

²

|I|

⁻¹

‒

I

(x) '

1/2

%

% %

%

p

,

where we used the characterization of L

_p

( R

^d

) using wavelets. Thus,

% % L

I∈D,!∈E

c

^!_I ^!_I

%

%

p

≤ L

!∈E

% % L

I∈D

c

^!_I ^!_I

%

%

p

= L

!∈E

% % (T

^!

)

^,

f

^!

%

%

p

≤ C L

!∈E

% % f

^!

% %

p

≤ C ˜ L

!∈E

% %

% %

&

I∈D

L

|c

^!_I

|

²

|I|

⁻¹

‒

I

(x) '

1/2

%

% %

%

p

≤ L C ˜

% %

% %

&

I∈D,!∈E

L

|c

^!_I

|

²

|I|

⁻¹

‒

I

(x) '

1/2

% %

% %

p

,

and it follows that T : d

_p

+→ L

_p

( R

^d

) is bounded. Unconditionality follows easily from the observation that none of the above estimates depend on the sign of each

c

^!_I

. !

Recall the Lorentz space !

p,q

(q), 1 ≤ p < F, 0 < q ≤ F, for some countable set q, as the set of sequences {a

m

}

_m∈q

satisfying %{a

m

}%

_!p,q

< F, where

%{a

m

}%

!_p,q

=

 

 / L

^Fj=0

0 2

^j/›

a

^∗₂j

1

q

2

1/q

, 0 < q < F,

sup

_j≥0

2

^j/›

a

^∗_2j

, q = F,

with {a

^∗_j

}

^F_j=0

a decreasing rearrangement of {a

m

}

_m∈q

.

It is known from the orthonormal wavelet case [10, 11], that there exist constants c,C > 0 such that

c%{c

I

}%

_!p,F(D)

≤

% %

% %

&

I

L

∈D

|c

I

|

²

|I|

^−2/p

‒

I

(x) '

1/2

%

% %

%

p

≤ C%{c

I

}%

_!p,1(D)

, for any {c

I

} ∈ !

p,1

(D). Notice that for any {c

^!_I

} ∈ !

p,1

(D × E),

|{I, ! : |c

^!_I

| > ¼}| =

L

L

!=1

|{I : |c

^!_I

| > ¼}| ≤ L%S({c

^!_I

}

I,!

)%

^p_p

¼

^−p

, where S({c

^!_I

}

I,!

) := 0

L

I∈D,!∈E

|c

^!_I

|

²

|I|

^−2/p

‒

I

(x) 1

1/2

. Furthermore, since !

1

" → !

2

we have

%S({c

^!_I

}

I,!

)%

p

≤

L

L

!=1

% %

% %

&

I∈D

L

|c

^!_I

|

²

|I|

^−2/p

‒

I

(x) '

1/2

% %

% %

p

≤ C

L

L

!=1

%{c

^!_I

}%

_!p,1(D)

≤ CL%{c

^!_I

}%

_!p,1(D×E)

.

Combining these two estimates, we get that there exist constants c,C > 0 such that (3.2)

c%{c

^!_I

}%

_!p,F(D×E)

≤

% %

% %

&

I∈D,!∈E

L

|c

^!_I

|

²

|I|

^−2/p

‒

I

(x) '

1/2

%

% %

%

p

≤ C%{c

^!_I

}%

_!p,1(D×E)

, for any {c

^!_I

} ∈ !

p,1

(D × E).

Remark 3.4. We denote by

^!,p_I

the function

^!_I

normalized in L

_p

( R

^d

), i.e. %

^!,_I ^p

%

p

)

|I|

^1/2−1/p

%

^!_I

%

p

. Notice that by Theorem 3.3 and (3.2) the TWF system is !

_p,1

- hilbertian in L

_p

( R

^d

), 1 < p < F , that is to say we have

% %

% L

I∈D,!∈E

c

^!_I ^!,_I ^p

% %

%

p

≤ C

_p

%{c

^!_I

}%

_!p,1(D×E)

, for any sequence {c

^!_I

} ∈ !

p,1

(D × E).

We have in fact proved that any reasonable (in the sense of theorem 3.1) TWF system induces an atomic decomposition of L

p

( R

^d

), 1 < p < F. Let us recall the de fi nition of an atomic decomposition:

De fi nition 3.5. Let X be a Banach space and X

_d

a Banach sequence space indexed by N . Let { f

_k

} ⊂ X , {g

k

} ⊂ X

^∗

. Then ({ f

_k

}, {g

k

}) is an atomic decomposition of X with respect to X

_d

if

• {g

k

( f )} ∈ X

_d

for all f ∈ X .

• For any f ∈ X we have

% f %

X

) %{g

k

( f )}%

Xd

.

• f = L

^F_k=1

g

_k

( f ) f

_k

, ∀ f ∈ X .

From this de fi nition we read off the following:

Corollary 3.6. Let {

^!

}

!∈E

be the generators of a tight wavelet frame for L

₂

( R

^d

), with frame constant A. Suppose for all ! ∈ E, some ʼ > 0 and some ¼ > 0,

!

∈ C

^ʼ

( R

^d

),

^{+ !}

(x) dx = 0, and |

^!

(x)| ≤ C(1 + |x|)

^−d−¼

. Then the system ({A

^{−1 !}_I

}

I,!

, {

^!_I

}

I,!

) is an atomic decomposition of L

_p

( R

^d

), 1 < p < F, with re- spect to the sequence space d

p

.

3.2. Thresholding the TWF analysis coef fi cients. From a practical point of view, it is interesting to study different types of thresholding (or shrinkage) operators for the framelet system in L

_p

. Let ʻ : C × R

⁺

+→ C be a function for which there exists a constant C such that

(3.3) |x − ʻ(x, ⁄)| ≤ C min(|x|, ⁄).

We call such a function ʻ a shrinkage rule, see e.g. [18]. The well-known notions of hard and soft thresholding are two of the prime examples of shrinkage rules. The expressions are given by ʻ(x, ⁄) = x1(|x| > ⁄) and ʻ(x, ⁄) = x(1 −⁄/|x|)1(|x| > ⁄), respectively.

We de fi ne the associated shrinkage operator T

_⁄^ʻ

as T

_⁄^ʻ

f = L

I∈D,!∈E

ʻ(# f ,

^!_I

$, ⁄)

^!_I

.

We claim that T

_⁄^ʻ

f → f in L

_p

( R

^d

), 1 < p < F , as ⁄ → 0. To see this, we use the estimates given by Theorem 3.3,

% f − T

_⁄^ʻ

f %

p

≤ C

% %

% %

&

I∈D,!∈E

L

|# f ,

^!_I

$ − ʻ(# f ,

^!_I

$, ⁄)|

²

|I|

⁻¹

‒

I

'

1/2

%

% %

%

p

.

By (3.3) and the dominated convergence theorem we see that % f − T

_⁄^ʻ

f %

p

→ 0 as

⁄ → 0.

3.3. Sobolev Spaces. We now turn our attention to L

_p

-based Sobolev spaces. For 1 ≤ p ≤ F and r ≥ 0 denote by W

^r

(L

p

( R )) the Sobolev space consisting of func- tions f ∈ L

_p

( R

^d

) satisfying

% f %

_Wr(Lp(R^d))

:= %(I − U)

^r/2

f %

p

< F,

with U the Laplace operator. We prove in this section that for TWFs with some smoothness and vanishing moments, we can actually characterize the Sobolev norm using the frame coef fi cients.

For a nonnegative integer N, we say that a function f belongs to the set S

^N

( R

^d

) if there exist constants C,C

_˜

< F and ¼ > 0, such that

(3.4)

 



+

x

^˜

f (x)dx = 0 for ˜ ∈ N

^d

, |˜| ≤ N,

| f (x)| ≤ C(1 + |x|)

^{−d−1−N−¼}

for x ∈ R

^d

,

|K

^˜

f (x)| ≤ C

_˜

(1 + |x|)

^−d−¼

for x ∈ R

^d

, ˜ ∈ N

^d

, |˜| ≤ N + 1.

Here, |˜| := L

^d_k=1

˜

k

.

Remark 3.7. Given N ∈ N , it is possible, using the oblique extension principle, to construct a generator } of a framelet system such that ⊂ S

^N

( R

^d

) (see e.g. [11]).

Theorem 3.8. Given 1 < p < F and r ≥ 0. Let {

^!

}

_!∈E

be the generators of a TWF for L

₂

( R

^d

) such that

^!

∈ S

^2r3

( R

^d

) for all ! ∈ E. Then,

(3.5) % f %

_Wr(Lp(R^d))

)

% %

% %

&

I∈D,!∈E

L

|# f ,

^!_I

$|

²

(1 + |I|

^−2r/d

)|I|

⁻¹

‒

I

(x) '

1/2

%

% %

%

p

,

for all f ∈ W

^r

(L

p

( R

^d

)), with equivalence depending only on p and r.

Proof. For notational convenience we write

W

^r

_{f (x) :=} ^/ L

I∈D,!∈E

|# f ,

^!_I

$|

²

(1 + |I|

^−2r/d

)|I|

⁻¹

‒

I

(x) 2

1/2

,

for any f ∈ W

^r

(L

p

( R

^d

)). Let us fi rst consider the case r ∈ N . According to The- orem 6.6.21 in [12], and the characterization of Sobolev functions using wavelet expansions, there exist constants C and C

^,

depending only on r and p such that

% W

^r

_{f %}

_p

_{≤ CL} ^{% %} _% ^/ L

I∈D

|# f ,

^m_I

$|

²

(1 + |I |

^−2r/d

)|I|

⁻¹

‒

I

(x) 2

1/2

% %

%

p

≤ C

^,

L% f %

_W^r_(Lp(R^d))

,

for all f ∈ W

^r

(L

p

( R

^d

)). This gives us the lower bound in (3.5) for r ∈ N .

To get the upper bound we recall that % f %

_W^r_(Lp(R^d))

) % f %

p

+ %(−U)

^r/2

f %

p

, and since % f %

p

≤ C% W

⁰

_{f %}

_p

_{≤ C}

^,

_% W

^r

_{f %}

_p

by Theorem 3.1, it suf fi ces to show that

%(−U)

^r/2

f %

p

≤ C% W

^r

_{f %}

_p

. Fix two functions f ∈ W

^r

(L

p

( R

^d

)) ∩ L

₂

( R

^d

) and g ∈ L

_p^,

( R

^d

) ∩W

^r

(L

2

( R

^d

)), where 1 = 1/p + 1/p

^,

. Since {

^!

}

!∈E

are generators of a TWF, we have

# f ,(−U)

^r/2

g$

= A L

I∈D,!∈E

# f ,

^!_I

$#(−U)

^r/2

g,

^!_I

$

= A

$

L

I∈D,!∈E

# f ,

^!_I

$|I|

^−r/d−1/2

#(−U)

^r/2

g,

^!_I

$|I|

^r/d−1/2

‒

I

(x) dx.

Thus, by the Cauchy-Schwartz inequality

|#(−U)

^r/2

f , g$| = |# f , (−U)

^r/2

g$|

≤ A

$

W

^r

_{f (x)} ^/ L

I∈D,!∈E

|#(−U)

^r/2

g,

^!_I

$|

²

|I|

^2r/d

|I|

⁻¹

‒

I

(x) 2

1/2

dx

= A

$

W

^r

_{f (x)} ^/ L

I∈D,!∈E

|#g, ˜

^!_I

$|

²

|I|

⁻¹

‒

I

(x) 2

1/2

dx

where ˜

^!

:= (−U)

^{r/2 !}

and we have used that #(−U)

^r/2

g,

^!_I

$|I|

^r/d

= #g, ˜

^!_I

$ in the

last inequality. According to [13, p. 170], (−U)

^r/2

: S

^r

+→ S

⁰

. Thus, Theorem 6.4.9

in [12], and the characterization of Lebesgue functions using wavelet expansions gives,

|#(−U)

^r/2

f , g$| ≤ A% W

^r

_{f %}

_p

^{% %} _% ^/ L

I∈D,!∈E

|#g, ˜

^!_I

$|

²

|I|

⁻¹

‒

I

(x) 2

1/2

%

% %

p^,

≤ CL% W

^r

_{f %}

_p

^{% %} _% ^/ L

I∈D

|#g,

^m_I

$|

²

|I|

⁻¹

‒

I

(x) 2

1/2

%

% %

p^,

≤ C

^,

% W

^r

_{f %}

_p

_%g%

_p,

.

Now, taking the supremum over all g ∈ L

_p^,

( R

^d

) ∩L

^r₂

( R

^d

) with %g%

p^,

≤ 1, we obtain

%(−U)

^r/2

f %

p

≤ C

^,

% W

^r

_{f %}

_p

_,

for f ∈ W

^r

(L

p

( R

^d

)) ∩ L

₂

( R

^d

) and thus for f ∈ W

^r

(L

p

( R

^d

)) since f +→ W

^r

_{f is}

continuous from W

^r

(L

p

( R

^d

)) to L

_p

( R

^d

). In order to conclude the theorem we need to prove (3.5) for a general r > 0. De fi ne for each I ∈ D, ! ∈ E and x ∈ R

^d

the discrete weight function w

_r

:= w

_r

(I, !) := (1 + |I|

^−2r/d

)|I |

⁻¹

‒

I

(x). Notice that (3.6) % W

^r

_{f %}

_L

p

=

&

_$

R^d

% % {# f ,

^!_I

$} %

%

_!^p

2(wr)

dx '

1/p

, r > 0.

De fi ne

J _{: W}

^N

_(L

_p

₍ R

^d

)) +→ L

_p

(!

2

(w

N

)) by f +→ {# f ,

^!_I

$}

I∈D,!∈E

and de fi ne

P _{: L}

_p

_(!

₂

_(w

_N

_{)) +→ W}

^N

_(L

_p

₍ R

^d

)) by {c

^!_I

}

_I∈D,!∈E

+→ L

I∈D,!∈E

c

^!_I ^!_I

. Then the arguments above show that P _◦ J _{= Id}

W^N(Lp(R^d))

for all N ∈ N

₀

, in other words, W

^N

(L

p

( R

^d

)) is a retract of L

p

(!

₂

(w

N

)).

W

^N

(L

p

( R

^d

))

J

W

^N

(L

p

( R

^d

))

P

L

p

(!

₂

(w

N

)) Id

_WN(Lp(R^d))

For a given r > 0, r 5∈ N , take N ∈ N

₀

such that r = (1 − ¡)N + ¡(N + 1) for some

¡ ∈ (0, 1). Notice that w

^r

) w

^(1−¡)_N

w

^¡_N+1

. Now, according to Theorem 5.5.3 in [1]

we have

!

2

(w

r

) = 0

!

2

(w

N

), !

2

(w

N+1

) 1

[¡]

,

with equivalent norms. Here, (X ,Y )

_[¡]

denotes complex interpolation between X and Y .

³

Furthermore, by Theorem 5.1.2 in [1],

0 L

_p

(!

2

(w

N

)), L

_p

(!

2

(w

N+1

)) 1

[¡]

= L

_p

00

!

2

(w

N

), !

2

(w

N+1

) 1

[¡]

1 = L

_p

(!

2

(w

r

)),

3We have adopted the notation for complex interpolation as used by Bergh and L¨ofstr¨om in [1]

with equivalent norms. Thus, W

^r

(L

p

( R

^d

)) is a retract of L

_p

(!

2

(w

r

)) for a general

r ≥ 0. !

Remark 3.9. The reader will notice that the spaces studied so far, L

_p

( R

^d

) and L

_p

- based Sobolev spaces, belong to the Triebel-Lizorkin scale of function spaces. It can be veri fi ed (at the expense of “messy” estimates) that suf fi ciently nice TWFs also can be used to characterize the Triebel-Lizorkin norms. We leave the details for the reader.

4. J

ACKSON INEQUALITIES FOR TIGHT WAVELET FRAMES

We will now look at some of the implications that can be derived from the various characterizations given in the previous section. The main result will be a Jackson inequality that will give a certain rate for m-term approximation for “nice” func- tions. We consider two interpretations of the word “nice”. When we do not assume any smoothness or vanishing moments for the TWF, we get the Jackson estimates for functions in a sparsity class de fi ned in terms of the TWF. If we assume the generators for the TWF has some smoothness and vanishing moments (the OEP tells us that such nice generators do exist), then we can state the Jackson inequality in terms of smoothness measured on the Besov scale.

First we introduce some notions that will be used later. A dictionary D _{= {g}

_k

_}

_k∈N

in L

_p

( R

^d

) is a countable collection of quasi-normalized elements from L

_p

( R

^d

).

For D we consider the collection of all possible m-term expansions with elements from D _:

x

m

( D _{) :=} ³ L

i∈q

c

_i

g

_i

* *

* c

_i

∈ C , card q ≤ m 4

.

The error of the best m-term approximation to an element f ∈ L

_p

( R

^d

) is then

«

m

( f , D ₎

_p

_{:= inf}

fm∈xm(D₎

% f − f

_m

%

_Lp(R^d)

.

De fi nition 4.1 (Approximation spaces). The approximation space A

_q^\

_(L

_p

₍ R

^d

), D ₎

is defined by

| f |

_Aq^\_(Lp(Rd),D₎

:=

&

_F

m=1

L

0 m

^\

«

m

( f , D ₎

_p

¹

^q

¹

m '

1/q

< F,

and (quasi)normed by % f %

_Aq^\_(Lp(Rd),D₎

= % f %

p

+ | f |

_Aq^\_(Lp(Rd),D₎

, for 0 < q, \ < F , with the !

q

norm replaced by the sup-norm when q = F .

It is well known that the main tool in the characterization of A

_q^\

_(L

_p

₍ R

^d

), D _{) comes}

from the link between approximation theory and interpolation theory (see e.g. [7, Theorem 9.1, Chapter 7]). Let X

_p

( R

^d

) be a Banach space with semi-(quasi)norm

| · |

Xp

continuously embedded in L

_p

( R

^d

). Given ˜ > 0, the Jackson inequality

«

m

( f , D ₎

_p

_{≤ Cm}

^−˜

_{| f |}

Xp(R^d)

, ∀ f ∈ X

_p

( R

^d

), ∀m ∈ N

(4.1)

and the Bernstein inequality

|S|

_Xp(R^d)

≤ C

^,

m

^˜

%S%

p

, ∀S ∈ x

m

( D ₎

(4.2)

(with constants C and C

^,

independent of f , S and m) imply, respectively, the con- tinuous embedding

/

L

_p

( R

^d

), X

_p

( R

^d

) 2

\/˜,q

" → A

^\

q

(L

p

( R

^d

), D ₎

and the converse embedding /

L

_p

( R

^d

), X

_p

( R

^d

) 2

\/˜,q

← # A

^\

q

(L

p

( R

^d

), D ₎

for all 0 < \ < ˜ and q ∈ (0, F].

We want to obtain a Jackson estimate for «

m

when D is any (reasonable) TWF.

For this we need to de fi ne a class of “nice” and “smooth” functions. This will be the following class as introduced in [8] for Hilbert spaces.

De fi nition 4.2 (Sparsity class). Let X (}) be a TWF. For p ∈ (1, F), › ∈ (0, F) and q ∈ (0, F], we let K

_›,q

_(L

_p

₍ R

^d

), X (}), M) denote the closure (in L

p

( R

^d

)) of the set

(

f ∈ L

p

( R

^d

)

* *

* * ∃q ⊂ N , card q < F, f = L

(I,!)∈q

c

^!_I ^!,p_I

, %{c

_k

}%

_!›,q

≤ M )

.

Then we define

K

_›,q

_(L

_p

₍ R

^d

), X (})) :=

⁵

M>0

K

_›,q

_(L

_p

₍ R

^d

), X(}), M), with

| f |

K_›,q(Lp(R^d),X(}))

= inf{M : f ∈ K

_›,q

_(L

_p

₍ R

^d

), X (}), M)}.

Remark 4.3. Since the TWF system is !

p,1

-hilbertian (cf. Remark 3.4), Proposi- tion 3 in [10] gives an equivalent de fi nition of K

_›,q

_(L

_p

₍ R

^d

), X(})) for p ∈ (1, F),

› < p and q ∈ [1, F]:

K

_›,q

⁰ _L

_p

₍ R

^d

), X (}) 1

= (

f ∈ L

_p

( R

^d

)

* *

* * ∃{c

^!_I

}

I,!

∈ !

_›,q

, f = L

I∈D,!∈E

c

^!_I ^!,p_I

)

,

and | f |

K_›,q(Lp(R^d),X(}))

equals the smallest norm %{c

^!_I

}

_I,!

%

_!›,q

such that f = L

I,!

c

^!_I ^!,p_I

. 4.1. A general Jackson inequality. For the sparsity class K

_›,q

_(L

_p

₍ R

^d

), X (})) we have the following rather general Jackson inequality.

Proposition 4.4. Let X (}) be a TWF that satisfies the hypothesis of Theorem 3.3.

Then for 1 < p < F, › < p, and ˜ = 1/› − 1/p, we have the Jackson inequality (4.3) «

m

( f , X (}))

p

≤ Cm

^−˜

| f |

K_›,1(Lp(R^d),X(}))

, ∀m ∈ N ,

for all f ∈ K

_›,1

_(L

_p

₍ R

^d

), X (})).

Proof. Given f ∈ K

_›,1

_(L

_p

₍ R

^d

), X(})), let {c

^!_I

}

I,!

∈ !

_›,1

be a sequence satisfying f = L

I,!

c

^!_I ^!,_I ^p

and | f |

K_›,1(Lp(R^d),X(}))

= %{c

^!_I

}

_I,!

%

_!›,1

. Let q ⊂ D × E be a fi nite set, card q = m < F, such that {c

^!_I

}

_(I,!)∈q

is the m largest coef fi cients from the sequence {c

^!_I

}

I,!

. Then,

% %

% f − L

(I,!)∈q

c

^!_I ^!,_I ^p

% %

%

p

= %

% % L

(I,!)∈q^c

c

^!_I ^!,_I ^p

% %

%

p

≤ C

p

%{c

^!_I

}

_(I,!)∈q^c

%

_!p,1

≤ C

^,

L

F k=log₂(m)+1

2

^j/p

|c

^∗₂j

|, where {c

^∗_j

}

j∈N

is a decreasing rearrangement of {c

^!_I

}

I,!

. Since ˜ = 1/› − 1/p we get

L

F k=log₂(m)+1

2

^j/p

|c

^∗₂j

| ≤ m

^−˜

L

F k=log₂(m)+1

2

^j/›

|c

^∗₂j

|

≤ m

^−˜

%{c

^!_I

}

I,!

%

!_›,1

= m

^−˜

| f |

K_›,1(Lp(R^d),X(}))

.

! Remark 4.5. By Proposition 4.4, K

_›,1

_(L

_p

₍ R

^d

), X (})) " → A

_F^˜

_(L

_p

₍ R

^d

), D _{), ˜ =}

1/› + 1/p. It is easy to verify that (see e.g. [11])

K

_›,q

_(L

_p

₍ R

^d

), X(})) " → 0 K

_›

1,1

(L

p

( R

^d

), X (})), K

_›

2,1

(L

p

( R

^d

), X (})) 1

¡,q

, for

¹_›

=

_›^¡

1

+

^1−¡_›

2

, and thus,

K

_›,q

_(L

_p

₍ R

^d

), X (})) " → A

^˜

q

(L

p

( R

^d

), D _{), ˜ = 1/› − 1/ p,}

for a general q ∈ [1, F].

It may not always be easy to check whether a function f ∈ K

_›,q

_(L

_p

₍ R

^d

), X(})).

Therefore, it is interesting to study the set of functions in L

_p

( R

^d

) depending only on the behavior of the coef fi cients # f ,

^!,p_I ^,

$. For p ∈ (1, F), › ∈ (0, F) and q ∈ (0, F], we let ˜ K

_›,q

_(L

_p

₍ R

^d

), X (})) denote the set

(

f ∈ L

p

( R

^d

)

* *

* * | f |

K˜_›,q(Lp(R^d),X(}))

:= %{# f ,

^!,_I ^p,

$}%

!_›,q

< F, 1 = 1/p + 1/p

^,

)

. Lemma 4.6. Given p ∈ (1, F) and › ∈ [1, p), let N ∈ N

₀

be a nonnegative integer such that 1 − ›/p < (N + 1)/d. Suppose

^!

∈ S

^N

( R

^d

) for ! = 1, 2, . . .,L. Then

K ˜

_›,›

_(L

_p

₍ R

^d

), X (})) = K

_›,›

_(L

_p

₍ R

^d

), X (})), with equivalent norms.

Proof. By Remark 4.3 we have ˜ K

_›,q

_(L

_p

₍ R

^d

), X (})) " → K

_›,q

_(L

_p

₍ R

^d

), X (})). Thus,

given f ∈ L

_p

( R

^d

), we only need to show that if f = L c

^!_I ^!,p_I

with {c

^!_I

} ∈ !

_›

then

{# f ,

^!,p_I ^,

$} ∈ !

_›

, with %{# f ,

^!,p_I ^,

$}%

!_›

≤ C%{c

^!_I

}%

!_›

. First, notice that

# f ,

^!,_I ^p,

$ = L

I^,,!^,

c

^!_I,^,

#

^!_I^,,^,p

,

^!,_I ^p,

$.

Hence, it suf fi ces to show that the double in fi nite matrix (#

^!_I^,,^,p

,

^!,p_I ^,

$)

_I,I^,_,!,!^,

is bounded on !

_›

. Let us introduce the notation

^!_j,k

:=

^!_I

and c

^!_j,k

:= c

^!_I

for I = [2

^−j

k, 2

^−j

(k + 1)], j ∈ Z , k ∈ Z

^d

. By Proposition 6.6.20 in [12] we have for j ≤ j

^,

|#

^!_j^,,,k^,

,

^!_j,k

$| ≤ C2

^{(j−j,)(d/2+N+1)}

(1 + |k − 2

^j−j,

k

^,

|)

^d+\

,

for some \ > 0. Since

^!,p_j,k^,

) 2

^{jd(1/p,−1/2) !}_j,k

, this gives the bound (4.4) |#

^!_j^,,^,p,k^,

,

^!,_j,k^p,

$| ≤ C2

^{−|j,−j|(N+1)}

a

_j^,_−j;k,k^,

, where

a

_m;k,k^,

:=

# 2

^−md/p,

(1 + |k − 2

^−m

k

^,

|)

^−d−\

for m ≥ 0, 2

^md/p

(1 + |2

^m

k − k

^,

|)

^−d−\

for m < 0,

For notational convenience we suppress the index ! in the following. For fi xed j ∈ Z and k ∈ Z

^d

we have

|# f ,

^p_j,k^,

$| ≤ L

j^,∈Z

L

k^,∈Z^d

|#

^p_j,,k^,

,

^p_j,k^,

$||c

j^,,k^,

|

Using the bound (4.4) and H¨olders inequality for the sum over j

^,

, with 1 = 1/› + 1/›

^,

, we get

|# f ,

^p_j,k^,

$| ≤ C L

j^,∈Z

2

^{−|j,−j|(N+1)}

/

L

k^,∈Z^d

a

_j^,_−j;k,k^,

|c

_j^,_,k^,

| 2

= C L

j^,∈Z

2

^−|j,−j|(N+1)(1/›^,+1/›)

/

L

k^,∈Z^d

a

_j^,_−j;k,k^,

|c

_j^,_,k^,

| 2

≤ C / L

j^,∈Z

2

^{−|j,−j|(N+1)}

2

_1/›^,

(4.5)

&

j

L

^,∈Z

2

^{−|j,−j|(N+1)}

/

L

k^,∈Z^d

a

_j^,_−j;k,k^,

|c

_j^,_,k^,

| 2

›

'

_1/›

≤ C

^,

&

m∈

L

Z

2

^−|m|(N+1)

/

L

k^,∈Z^d

a

_m;k,k^,

|c

m+j,k^,

| 2

›

'

1/›

.

Lemma 8.10 in [14] implies for any {d

k^,

}

k^,

∈ !

_›

, 1 ≤ › < F,

k∈

L

Z^d

/ L

k^,∈Z^d

a

_m;k,k^,

|d

k^,

| 2

›

≤ C2

^{md(›/p−1)}

L

k^,∈Z^d

|d

k^,

|

^›

, for m ∈ Z .