L
~
Constr. Approx. (1999) 00: 1-15
CONSTRUCTIVE
APPROXIMATION
© 1999 Springer-Verlag New York Inc.
Nonlinear Approximation with Local Fourier Bases
K. Grochenig and S. Samarah
Abstract.
It is shown that local Fourier bases are unconditional
bases for the mod-
ulation spaces on R, including the Bessel potential spaces and the Segal algebra So.
As a consequence, the abstract function spaces, that are defined by the approximation
properties with respect to a local Fourier basis, are precisely the modulation spaces.
1. Introduction
A guiding principle of approximation theory states that good approximation properties
of a function are equivalent to smoothness. Recently many classical questions of approximation theory have gained new momentum through the discovery of new types of
orthonormal bases for L 2(R). Moreover, a number of results on nonlinear approximation
[5], [6], [9] have had a direct impact on applications in signal and image processing and
in statistics.
This paper is devoted to approximation theory with local Fourier bases [1], [2], [3],
[18], [19] and some consequences in time-frequency analysis. Such orthonormal bases
can be constructed for any partition of R and have been invented for the flexible segmentation of signals [19]. This construction leads to orthonormal bases as diverse as wavelet
bases and Wilson bases [4]. To stay within the realm of time-frequency analysis, we
restrict ourselves to local Fourier bases which are close to orthonormal Wilson bases.
Traditional approximation theory either starts from a known function space and tries
to understand it in terms of approximation properties by a basis; or it defines abstract
function spaces by certain approximation properties [7], [8]. In the latter case, the main
problem is to find a concrete realization of the abstract space.
In the case of an orthonormal basis, such abstract spaces are easily defined. Let
B = {cpk, k E I} be an orthonormal basis of a separable Hilbert space 1i, where I
is a countable index set, and let ~n be the subset of all linear combinations of n elements
ofB:
(1)
~n
=
If
E
1i:
f
kEF
= LCkf/!k,
F ~
I,
cardF:S
nJ.
Date received: April 22, 1998. Date accepted: May 18, 1999. Communicated by Vladimir N. Temlyakov.
AMS classification: •••
Key words and phrases: Nonlinear approximation, Local Fourier bases, Wilson basis, Modulation spaces,
Schur's test.
I
I
L
~
2
K. Grochenig and S. Samaroo
Note that Ln is not a linear subspace. The approximation error of an element
:En is
(2)
Un
(f) = seI;.
inf III -
f E H in
sll1t·
By imposing conditions on the asymptotic behavior of Un (f), one defines subsets of H.
If done suitably, the approximation behavior leads to conditions on the coefficients of
in the orthogonal expansion
= Lkel(f, CPk)CPk.Stechkin's fundamental result and its
generalization by DeVore and Temlyakov can be stated as follows:
I
f
00
(3)
.L: I {f, CPkW < 00
if and only if
kel
.L:n-PI2un{f)P
n=1
< 00,
where a < p < 2. See [8], and [22] and Lemma 5. If p = I, the approximation
property (3) characterizes exactly all absolutely convergent orthogonal expansions. In
particular, for the trigonometric basis {e21rikX, k E Z} for L2(0, 1), (3) characterizes the
absolutely convergent Fourier series by their nonlinear approximation properties.
In the context of a given orthonormal basis the abstract result often requires further
elaboration. Of particular interest are characterizations of the resulting spaces directly
in terms of
rather than through the coefficients {f, CPk)' For an orthonormal wavelet
basis, deVore, Jawerth, and Popov [5] have shown that the concrete spaces that arise
from the characterization (3) are certain Besov spaces. The proof of this result rests on
the fundamental fact that wavelet bases are unconditional bases for Besov spaces [20]
and that these spaces can be characterized by size estimates for the coefficients in the
orthogonal expansion.
Our goal is the identification of the correct function spaces which occur when a local
Fourier basis is used in the approximation process.
We first describe the idea of local Fourier bases. On any interval [a, /'3] there are several
types of orthonormal bases for L 2(a, fJ) given by trigonometric functions, for instance,
the collection {(2/(/'3 - a»1/2 sin TCk(x - a)/(/'3 - a) . X[a.,Bj(x), kEN}. If a partition
of R is given through a sequence {ak, k E Z}, ak < aHI, limk--+±ooak = ±oo, then
such bases can be patched together to yield an orthonormal basis of L2(R). However,
the brutal cut-off to the intervals [ak, ak+d introduces artificial discontinuities and is
extremely undesirable. The construction of local Fourier bases avoids this problem by
replacing the characteristic functions X[akoak+d by Coo functions with compact support.
At the same time, orthonormality and the trigonometric structure are preserved [1], [2].
Precisely, given a partition {ab k E Z} with interval lengths 11k = ak+1 - ab and a
sequence Ck > 0, such that
f
(4)
ak
+ Ck < aHI
- cHI,
then for any smoothness N EN U roo} there exist so-called bell functions bk E eN (R)
with supp bk S; [ak - Cb ak+1 + ck+d such that
(5)
I
1/!kl(X)
= -11k
if bk(x) sin -(x
11k
TCI
- ad,
lEN,
k E Z,
I
L
~
Nonlinear Approximation
3
with Local Fourier Bases
is an orthonormal basis for L2(R). Depending on the choice of a polarity, there are several
other bases of this type, where sin(nl I Llk)(x - ak) is replaced by cos(n II Llk)(x - ak)
or sin[n(2/1)/2Llk](x - ak) or cos[n(2/-1)/2~k](x
- ak), where I EN. In all cases
the index set is := Z x N. To be explicit, we will work with the basis functions in (5) in
the proofs. Using the other trigonometric functions requires typographical modifications
at best, for instance, writing (21 - 1) /2 instead of l.
In the case of the uniform partition ak = k /2 + ~ the bell functions can be chosen
[4] as
independent of k and one obtains the Wilson bases of Daubechies-Jaffard-Journe
a special case of the local Fourier bases [I].
The bell functions can be written explicitly in the following form: choose an auxiliary
function ~ E CN-1(R)
with supp ~ ~ [-1,1] and fR ~(x)dx = n12. Now define
eSk(x) = (lICk) f~oo ~(t/ck)dt. Then
I
(6)
bk(x)
=
sin eSk (x - ak) cos eSk+Jx - ak+l),
k E
Z,
has the desired properties [1], [2].
It cannot be expected that the approximation spaces resulting from the use of a particular local Fourier basis are all independent of the partition. However, if we impose the
restriction
1
(7)
o
<A
::s
ak+l - ak
= Llk ::s A < 00
for all
k E Z,
for some A > 1 it turns out that the corresponding approximation properties are independent of the precise details of the basis. Furthermore, the corresponding approximation
spaces can be identified with function spaces that are already known and well studied.
These function spaces are defined by the decay properties of the short-time Fourier
transform. For this, let
(8)
Txf(t)
= f(t
- x)
and
Myf(t)
= e2rriytf(t)
be the operators of translation and modulation by x, Y E R. For a fixed nontrivial window
function g E S(R) the short-time Fourier transform (STFT) on the time-frequency plane
R2 is defined as
(9)
Sgf(x,
y)
= (t, My Txg) =
i
f(t)g(t
- x)e-2rriyt dt.
Any norm on Sgf induces a norm on f. In this way one obtains the family of socalled modulation spaces. Those spaces were introduced in the 1980s by H. Feichtinger
[10], [11] who developed their theory in analogy to the Besov spaces. In particular,
they possess atomic decompositions. They are not as well known as their counterparts
in wavelet theory, the Besov spaces, but in recent years they have become increasingly
important in time-frequency analysis. They appear in a natural way in the study of
pseudodifferential operators, in Gabor theory, and in uncertainty principles [17], [23].
In the approximation theory with local Fourier bases they arise in the following way:
Theorem 1. Assume that the partition {ad satisfies II A ::s ak+1 - ak ::s A, that
infk Ck = C > 0, and that {'hz, (k, I) E Z x N} is an associated local Fourier basis in
I
I
L
~
4
CN(R), N
K. Grochenig and S. Samarah
>
1. Thenfor 1:::; p
<2
00
(10)
:L)n-1/2an(f))P
n=1
< 00,
if and only if
(11)
ff
ISg/(x,
yW dx dy
< 00.
R2
f
Condition (11) is usually expressed by saying that belongs to the modulation space
M p' Thus a function
can be approximated well by a local Fourier basis, if and only if
it belongs to a certain modulation space.
We shall present several theorems of this type. For this we prove first that local Fourier
bases are unconditional bases for modulation spaces and then deduce approximation
theorems from that fact.
f
As is to be expected from the paradigm of approximation theory, condition (11)
expresses a form of smoothness. If p = 1, then (11) is equivalent to a decomposition
f = LnEZ Tnfn, where in ELI, supp fn S; [-1, 1], and L~ll1inll1
< 00 [11]. Such
functions are locally in the Fourier algebra and thus "smooth."
The paper is organized as follows. In the next section the modulation spaces and some
of their properties are discussed along with a technical lemma about the STFT. Section 3
is devoted to the proof that the local Fourier bases are unconditional bases for the
modulation spaces. Section 4 provides a proof of Theorem 1 and several generalizations.
2. The Modulation Spaces
In this section we formally introduce the modulation spaces and discuss a few properties
needed later.
To quantify decay properties in the time-frequency plane, we consider nonnegative
continuous weight functions w on R2 that satisfy the condition
(12)
w(x
+ y)
:::;C(l
+ Ixl)aw(y)
for
x,y
E R2,
for some constants C > 0, a ::: 0. Condition (12) implies that the weighted LP-space
L~(R2) = {f: IIf11p,w= IIf,wllp < oo} is invariant under translations. Typical weights
are of the form w(x) = (1 + IxlY for S E R.
The modulation spaces are a mathematical tool to measure the joint time-frequency
distribution of a tempered distribution
E 5' (R). They are defined by decay properties
f
oftheSTFT.Giyen
I:::; p,q:::: oo,andaweightwonR2,themodulationspaceM;q(R)
E 5' (R) for which the norm
is the space of all distributions
f
IlfIIM;, ~
(I. (f.IS,[(X,
y)IPw(x, y)P dx fP
dy
r'
< OC
is finite, with obvious modifications, if p = 00 or q = 00. M;q is a Banach space
whose definition is independent of the window g [13, Lemma 24J. This means that
I
I
L
~
Nonlinear Approximation
5
with Local Fourier Bases
different windows define the same space and yield equivalent norms. Furthermore, M;'q
is invariant under the operators Tx My, x, Y E R. For nonlinear approximation we only
need the diagonal case p = q, and we write M;' p = M';.
We shall make use of the fact that the dual space of M'; is M~!W,where pi is the
conjugate index, 1/ p + 1/ pi = 1.
Examples of modulation spaces:
1. Feichtinger's Segal algebra So(R) = MI,I (R) [11).
2. M2,2(R) = L2(R).
3. The Bessel potential space HS = {f E $I: II/IIH'
Mf, where w(x, y) = (1 + IYI2r/2.
=
11(/ -
fj,y/2
f112
<
oo}
=
For more information, see [10], [11], [12], [13], and [16].
Similar to the qJ-expansion of Besov-Triebel-Lizorkin
spaces [15] the modulation
o
spaces have nice atomic decompositions. Given cp E S, there exist f3 > 0 and y > 0
small enough and a dual window qJ0 E S, independent of 1 ~ p, q ~ 00 and w, such
that every f E $I has an expansion
f =
(13)
L (f,
TfJmMyncpO)TfJmMynCP.
m,nEZ
f
Moreover,
E
M;, if and
only if
L l(f,
(14)
TfJmMynqJ°)IPw(f3m,
< 00
yn)P
( m,nEZ
)I/P
and the sequence space norm in (14) is equivalent to the M';-norm. If 1 ~ p, q < 00,
then the so-called Gabor expansion (13) converges unconditionally in the norm of M;'q'
See [10], [12], and [16].
In the sequel we need pointwise estimates for the STFr. For this define a set of nice
windows by
(15) C =
C(B, K, N)
Lemma 1.
(16)
Let
sup
cp
=
{g E CN (R): suppg ~ [-K,
E COO(R), suppcp S; [-L,
ISlOg(x,
y)1
~
1
Co /1 , L."'"
K], k=~:e,N
L], and set C
=K
forall
X[-c,C](x)
IIg(k)III ~ B}.
+ L. Then
x, y E R,
gEe
with a constant Co
> 0 depending
only on B, K, and N.
Proof. If Ixl > C, then S<pg(x, y) = 0 for all y E R.
If Ixl ~ C and Iyl ~ 1, then IS<pg(x, y)1 = I(g, My TxqJ)I ~ IIgll1IIcplioo~ BllqJlloo.
If Ixl ~ C and Iyl 2: 1, then integration by parts N times yields
S<pg(x, y)
I
= jK (g.
-K
Txcp)(t)e-27rlyt
.
dt
=.
1
(2my)N
jK
-K
(g.
Txcp)(N) (t)e-2rrlyt dt.
.
I
L
~
6
K. Grochenig and S. Samarah
Since g E C, we can estimate
ISq>g(x, y)1 ::: 12rr~IN
<
II~(~)
g(k) (TxW)(N-kt
1 B
12rrYIN
These estimates are independent of g
E
f-
£:0 (N)
k
IIcp(k) 1100'
•
C and thus imply (16) .
The explicit construction of the bell functions (6) of a local Fourier basis leads to the
following consequence:
Lemma 2. If 11 A ::: C¥k+1- C¥k ::: A and if Ck 2: C
Lakbk,
k E Z} ~ C(B, K, N)for some B, K, and N.
Proof.
By construction
choose K = 2A. Since
sin eek (x - C¥k)cos e.k+!
eW and e~{l! and sines
is bounded for j = 1, ...
supp gk ~ [-ck,
C¥k+l -
>
0 for all
+ ck+d
C¥k
k E
[-A,
~
Z, then {gk
=
2A], and we can
=
8ek(x)
= (lick) J~co ~(tlck) dt, the derivatives of bk(x)
(x - C¥k+I) are sums of products containing terms of the form
and cosines thereof. Since Ck 2: c, lIe~{llloo
, n. This is enough to verify that {gb k E Z}
= c;j
1I1;(j-l) 1100
£ C(B,
K, N) .•
Note that the basis functions (5) can be written as
(17)
Vrkl = '21
( 11k
2 )
1/2 Tak(MI/26.k
3. Unconditional
± M-l/2b.k)gk.
Bases for Modulation
Spaces
In this section we prove that the local Fourier bases subject to the geometric condition
(7) are unconditional bases for the modulation spaces M;;'- Recall that a set
of vectors in a Banach space B forms an unconditional basis, if:
{e;,
i
E
I}
(1) the finite linear combinations of the e; 's are dense in B; and if
2: 1, such that
(2) there exists a constant
e
II~ciAieill
holds for any finite subset F
Theorem
2.
Suppose
partition
a weightfunction
on R2 satisfying
(18)
I and any sequence (Ai)
£ C.
that {Vrkl, (k, I) E Z x N} ~ eN (R) is a local Fourier
whose underlying
is an unconditional
£
::: c s~PIAilll~cieill
satisfies
11A ::: C¥k+1- C¥k::: A and infk
w(x
+ y)
:::
(1+ Ixl)aw(y)for
basis for M; . Every distribution
f
=
L (f,
f E M;
a
Ck
basis
= C > O. Ifw
< N -1,
is
then {Vrkz}
has a unique expansion
Vrkl)Vrkl
(k.I)El
I
I
L
~
Nonlinear Approximation
with unconditional
(19)
convergence
-lIfIlM;:S
C
1
for some constant
(18) converges
7
with Local Fourier Bases
in the norm of M;.
L l(f,
( (k,l)EI
C 2: 1. If
p
=
VrklWW
Moreover,
ctk, 21::1k)p)I/P
(1
:S CIlfIlMp"
00, then {Vrkd is a weak basis, that is, the expansion
only in the weak *-topology
with respect to the predual
Mt/w.
Remark.
A similar statement holds for M';.q, 1 :S p, q < 00, p i= q [21]. The proof
requires a little more effort because of the mixed norms. However, for the nonlinear
approximation problem it suffices to consider only the case p = q.
i
Orthonormal Wilson bases are a special case ofIocal Fourier bases which are associated
to a uniform partition ctk = k/2 + and the choice of a certain polarity [1]. Then Vrkl =
V2( MI + (-1 )k+1M -I) Tk/2b, I = 0, 1,2, ... , k E Z (and a different normalization, if
I = 0) requires only a single bell function b. In this case, the bell function need not be
compactly supported, but the technical details of the proofs are simpler than for general
local Fourier bases. As a consequence we recover some of the results of [14]:
Corollary 1.
unconditional
Corollary 2.
N> 1, andfor
The Wilson basis of exponential
basis for M; for 1 ::::p
decay constructed
in [1] and [4] is an
< 00.
Any local Fourier basis in eN (R) is an unconditional bases for So, if
HS, ifs < N - 1.
To prove Theorem 2 we extend the orthonormal expansion (18) from L2(R) to the
modulation spaces. For this purpose we need to understand the action of certain associated
operators.
The analysis operator r is defined by
(20)
r(f)
= «(f, Vrkl))(k,l)EI.
Since {Vrkl, (k, 1) E I} is an orthonormal basis, r is well defined and maps L2(R)
onto £2(1). The formal adjoint is the synthesis operator r* which acts on "sequences"
c = (Ckl)(k.I)EI as
(21)
r*«cklh,l)
=
L
CklVkl.
(k,l)EI
We will show that both operators extend to other function or sequence spaces. We write
=
TIki
(ctk, 1/2I::1k), (k,l) E Z2, for the points in the time-frequency plane associated to
Vkl. For a given weight function w, Wi denotes its restriction w'(k, 1)
W(Tlkl) to the
=
discrete set {TIki}. Then £~, consists of all sequences on I for which
IIcllp,w'
=
(
L
(k,l)EI
< 00.
IcklIPw(1]k/)P
)l/P
We first prove an estimate for the STFT of a local Fourier basis.
I
I
L
~
8
K. Grochenig and S. Samarah
Lemma 3. Using the notation of Lemma 1, set G(x, y) = X[-c.C](x)[I/(1
+ lyI)N].
If {1/1kl, (k, I) E I} ~ eN (R) is a local Fourier basis satisfying the assumptions
of
Theorem 2, then there exists CI > 0, such that
(22)
Proof.
IS<p1/1kl(x, y)1 ::;
CI(T~'IG(x,
y)
+ T~,._IG(x,
foral!
y»
x, y E
R.
We start with the well-known covariance property of the STFf
IS<p(TxMyf)(u,
v)1
= IS<pf(u
- x, v - y)1
= IT(x.y)S<p(u,
v)1 ,
where T(x,y) is the two-dimensional translation. Using the time-frequency
the basis functions (17), we estimate
=
ISrp1/1kil
(
2!:!.k
1 )
I/Z
IS<p(Ta,(Ml/ZL1,
< ( 2!:!.k
1 )
I/Z
(IT~klSrpgkl
structure of
± M-l/ZL1k)gk)I
+ IT~k._ISrpgkl) .
The assumptions made on the local Fourier basis imply that (1/(2fik»I/Z ::: (A/2)I/z
and Lemmas 1 and 2 state that S<pgk(X, y) ::: CoG(x, y), These inequalities combined
yield (22) .•
To prove that r is bounded, we need a weighted version of Schur's test.
Lemma 4 (Schur). Suppose
L
(23)
that WI (i), i E I, and wz(j),
J,
tions on index sets I and
that
respectively,
laj;lwI (i)-1
=
Let A
CoWZ(j)-1
j E
= (aji )jEJ.iEI
< 00
J,
are two weightfunc-
be an infinite matrix such
foral!
j E J
iEI
and
L
(24)
laj;lwz(j)
jEJ
for some constants
Co,
C1
= Clwl(i) < 00
> 0. Then
for-al!
the map A is boundedfrom
iEI
e~l
(l)
into e~2 (J) for
1 ::: p ::: 00.
Proof.
First assume 1
IIAcll:~=
2
<
p
I:
<
jEJ II:ajiCilP
iEI
< ~
Wz(j)P
00 and let c
= (Ci)iEI
Wz(j)P
(~laj;ll/p+I/P'
IctiWI (i)I/P'-I/P)
pIp'
<
I
~wz(j)P
E e~l (I). Then
(~lajilwl(i)-)
P
'
(~lajillctlPWI(i)P/P)
I
L
~
Nonlinear Approximation
9
with Local Fourier Bases
by HOlder's inequality
<
cg/P'
LL
IIc; IPwl
W2(J)P-p/p'laj;
(i)p/p'
j
cg/pl
<
cg/pl
I;(~
laj;!W2(J))
CI 2;:
ICiIPWI
I
IC;!PWI
(i)p/p'+1
=
(i)p/p'
cg/pl C] IIcllf~, .
The cases p = 1 and p = 00 are easily verified directly. Each of them requires only one
of conditions (23) and (24) .•
Proposition 1. Under the hypotheses of Theorem 2, r is a bounded operator from M;
into e~,(I)for 1 ~ p ~ 00.
Proof. Given qJ E Coo with compact support and fJ, y
E
there exists a dual window qJ0 E S, such that every
f
L (j,
j =
> 0 small
enough, we know that
S' has an atomic decomposition
TfJmMynqnTfJmMynrp,
m,nEZ
with
f E M;,
if and only if
(
L
l(j, Tf3mMynqJ°Ww(fJm,
<
yn)P
m,nEZ
00.
)]/P
Then
(rfhz
=
L(f,
Tf3mMynqJ°)(Tf3mMynrp, 1/Jk/).
m,n
Therefore the proposition is proved, if we can show that the operator defined by the
matrix A(k,l),(m.n) = {TfJmMynrp,1/fkil maps the sequence Cmn = {j, Tf3mMynqJ°) E
e~,(Z2), WI (m, n) = w(fJm, yn), into e~,(I), w'Ck, 1) = w(/7kI)' For this it is enough
to verify the conditions of Schur's test.
With the help of Lemma 3, condition (23) becomes
L
I (Tf3mMynqJ, 1/Jkz)lw(fJm,
yn)-I
m,nEZ
:S clL
, Z (G
m,nE
(fJm - ak, yn __ 2Ak
1_)
+G (fJm
- ak, yn
+ 2~J )
w(fJm, yn)-I.
Using (12) in the form
w (ak,
I
2~J :Sw(fJm,
yn) (1 + lak
- fJml
+ I:k
-
ynlr,
I
L
~
10
K. Grochcnig and S. Samarah
each of the two tenus can be estimated by
I:
m,nEZ
G (f3m - ctk, yn - 2~ k ) w(f3m, yn)-I
S
±l
W ( ctk, 211-;
)-1 m~z
X[-C,c](f3m
- ctk)
( 1 + I 211k
±l -
(1 + Ictk - f3ml + 12~k - ynlr = (*).
f3m I S C and since the sum over m contains
yn
I)-N
x
Since Ictk continue
(*) S
±l
w ( ctk, 211k
)-1
( 1 + I 211k
±l - yn /)-N
2C ~
7'"
at most 2C / f3 terms, we
( 1+ C
+
I 211k
±l - yn I)a
As a < N -1, this sum is clearly finite with a bound independent of k and t. Condition (24)
is derived in a similar fashion. Since by (7) there are at most 2C A tenus ctk in every
interval of length 2C, we have
1:
k,lEf
I (T,BmMyn<P,
S
1fikl)lw(1Jkl)
L
C1 w(f3m, yn) (k,l)Ef
. (1 + Ictk -
f3ml
G (f3m - ctk, yn _ 211k
±l)
+ 12~k
- ynlr
L
S C1w(f3m, yn)2CA sup
±l l)a(1 + Iyn _ 2±l
11k /)-N
kEZ lEZ (1 + C + Iyn _ 211k
Since 1/ A S 11k S A and a < N - 1, the sum is finite with a bound independent of m
and n .•
Proposition 2.
Proof.
r* is a bounded map from .e~,(l) into M; for 1 S p S 00.
First assume that p
dual space M~!W of
k,l
1(1:
ckl1fikl, h)1
< 00,
and let (Ckl)(k,l)Ef be finitely supported. For h in the
M;, Proposition
=
II:
k,l Ckl(rhhll
I implies
S
I
IIclle~, IIrhlle'l/w
, S IIrllopllhllMl:wl/clle~,.
P
Thus
IIr*cIIM;
=
IIhIlMl/w::;1
sup
p'
I(I:
k,l ckl1fikl, h)1
S IIrliop l/Cllep,
w
shows that r* is bounded on .e~,.
I
I
L
~
Nonlinear Approximation
Furthermore, for any
II
11
with Local Fourier Bases
L
(k,l)',tF
£
> 0 there
C",""
II
M;
exists a finite subset Fe ~ I, such that
:'
11,11"
(
L
(k,l)¢F
ICklIPw(",aJP
q
for all finite subsets F :2 Fe. This means that the sums in TC converge unconditionally.
If p = 00, then taking the supremum over MJ/w shows that r* is bounded on M~
and that the sum is w* -convergent .•
We now prove Theorem 2.
Proof of Theorem 2.
Since rand r* are bounded on M;;' and
(25)
f
=
L(f,
1/fkl}1/fk/
=
£~"
the identity
r*rf
k,l
extends from L2(R) to M;;', I :S P :S .00. For p < 00 the series converges unconditionally and thus finite linear combinations are dense in M;;'. The norm equivalence (19)
follows from
:S IIr*lIopll«(f,
IIf11M;
1/fkl})(k,I)Ellle~
:S IIrllopllr*lIopllfIlM;.
Furthermore, since in a (finite) linear combination f = Lk I ckl1fikl the coefficients are
uniquely determined as Ckl = (f, 1fik/) = (rfhl,
we can esti~ate
II
L
k,l
'Aklckl1fikl
<
IIr* lIop II('AkICkl)(k,l) lie:,
<
IIrllopll'Alioollclle:,
11M;
II'A II00 IIf11Mff"
:S IIrllopllr*lIop
•
This shows that {1/fkl, (k, l) E /} is an unconditional basis for M;'
If P = 00, then (25) still holds with w* -convergence .
4. Characterization
of Modulation
Spaces
In this section we consider the nonlinear approximation with local Fourier bases under
the uniformity assumption (7). Theorem 2 identifies the functions
E L2(R) with
coefficients (f,1/fkd in lP(Z x N) as members of the modulation spaces Mp(R). The
Approximation Theorem I and its generalizations can deduced from a simple lemma
about decreasing sequences.
Let {'Pb k E /} be an orthonormal basis of a Hilbert space and let f = Lk Ck'Pb Ck =
(f, 'Pk). By rearranging the coefficients ick I by magnitude there is a bijection 1l': N -+ I,
such that ICrr(1) I 2': ICrr(2)I 2': .... For the approximation error we have obviously
f
(26)
I
an(f)
=
inf
SEEn
IIf-Sllrt=llf-tcrr(i)'Prr(i)11 i=1
rt
=
(f:
i=n+1 \crr(i)12)1/2
I
L
~
12
K. Grochenig and S. Samarah
Lemma 5 ([8]). Leta
=
an,q
(L,~n
we have
(27)
N} beadecreasingsequenceojpositivenumbers.
p, q > 0 set a = lip - 1/q· Then for 0 < p
k E
a%)I/q andfor
Tllp
( ~a~
00
with a constant
c
if and
(an) E f,P
= {ak:
S ( ~00
) lip
> 0 depending
only
if an, 00
~ ) lip
Sc
only on p. In particular,for
q
( ~a~
00
<q
S 00
) lip
= 00, an,OO = Ian I, and
E f,p.
Proof. Since (ak) is nonincreasing
equality
aZm
[m"am,qY
Set
and positive, we can estimate with Holder's in-
S aZm-l S m1 Zm-[
2:= ak S m mllq
2:= a%
k=m
k=m
1
,(Zm-l)
Ilq
S m-llqam,q
.
Therefore
00
,,(
L
m=1
Writing
a
2l/p(.L~=1
00
2"
+ aZm
P ) <
Lm
m=1
P
aZm-l
=
lip
- l/q,
(m"am,q)p(l/m»IIP.
this yields the left-hand inequality
For the converse we use the inequalities
00
00
L
(28)
-plq am,q'
P
"2k
k=m
q
aZk+!
and the embedding f,P ~ f,q for p
q
S aZm,q
=
lIalip
<
00
L
'" akq
k=zm
SL
"'2k aZkq
k=m
S q in the form
(29)
azm,q:::
Since k"p-l
ak
(00~
(2klq aZk)q )llq
:::
~
(00
(2klq azk)P )IIP
is also decreasing, by (28) it suffices to show that
00
"k"p-laP
L
k=l
00
L
1=1
k,q <
- "i"p-laP
00
ZI,q 21
=L
"i"PaP
1=1
00
L1=1
ZI.q -< c "iaP
ZI
for some constant c. Using (29), this follows from
00
"2m"PaPzm,q
L
m=1
<
~2m"p
= ~oo(k)f,;
Since ap
I
+ plq =
(~2Pklqaf,)
2m"p
2Pklq afk ::: c ~00 2k("P+Plq)afk'
1, we get the right-hand inequality, and the lemma is proved .
•
I
L
~
..
o
NonIlnear Approximation
13
with Local Fourier Bases
Theorem 1 follows now from (26) and Lemma 5 with p
< q = 2.
As a generalization we may measure the approximation error in an M;-norm and we
define for 1 :S q :S 00
(30)
= SEbn
inf III-
an(f)MW
q
sliMw.
q
In practice, the case q = 2, W == I, is the most interesting, since the
concrete interpretation as the "energy."
Theorem
3.
< q < 00,
If 1 :S p
and a
=
1/ p - 1/ q, then
1I/11Mq"
IU,
;:::: (L(k,l)EI
n
q
< 00.
by (19), the error
VrkIWW(1]kl)P)I/p
2:: IU, 1fik/)
inf
c:#suppc:::::n
E M;;', if and only if
1
00
~(n"an(f)Mw)PL-n=1
Proof.
Since
equivalent to
f
L 2-norm has a
-
ckzlPw(1]kl)P
( (k,I)EI
)I~
To minimize this expression, we rearrange the index set I into a sequence (kj, lj), j
such that aj = I (f, VrkJ,IJ) I W(1]kJ ,IJ) is decreasing. Then
an (f)M;'
;::::
,2::
( =n+1
00
f E M;, if and only if (aj )jEN
L::t (n"an(f)M;')P I/n < 00 .•
Consequently,
and only if
With the choice p = 2 and w(x, y)
of the Bessel potential spaces HS (R).
Corollary
3.
q
For
IE
=
aJ
E £P.
+ lylY,
(I
N,
By Lemma 5 this is the case, if
Theorem 3 yields a characterization
=4
- 1/ q, we have
if and
only
if
1
00
HS
E
)I/q
a
> 2, and
is
an (f)Mq"
~(naan(f)M"')1/2-q
~
n=1
n
< 00.
The case q = 00 can also be treated and may be of some practical relevance, because
pointwise comparisonof STFrs is as easy as comparing energies. The approximation
error is explicitly
an,oo(f)
= sup
sup
ISg(f
- s)(x,
y)lw(x,
y).
SEbn (x,Y)ER2
Lemma 5 yields immediately
Corollary
I
4.
I
E
M;
if and
only
if
(an,oo)nEN
E £P.
I
L
~
14
K. Grochenig and S. Samarah
In the above proof the basis property of {1/tkl} was used in an essential way to rewrite
the approximation error in terms of a sequence space norm. For linearly dependent sets
it is not clear how much of Theorem 3 still holds. We next prove one-half of Theorem 3
under the weaker assumption that the set {Tf3mMynf/J, m, n E Z} is a Banach frame for
M;;'. This means that there exists a dual window f/Jo, such that every
E M;;' has an
atomic decomposition (13) with an equivalent norm (14). Such Banach frames exist in
abundance and have become a useful tool in time-frequency analysis [10], [16].
The following result sheds some light on a question about nonlinear approximation
with linearly dependent sets, which was raised by DeVore: and Temlyakov [8]. :En and
O"nare taken with respect to {Tf3mMynf/J, m, n E Z}.
f
Proposition 3.
p,
Let {TemMynf/J: m, n E
I~p ~ 00. I~p <
If
Proof.
q and f EM;,
Z}
be a Banach frame for
M;;'(R)
then L:I(naO"n(f)M~,)P(1/n)
for all
< 00.
The argument is similar to the proof of Theorem 3. We find a permutation of
Z2, such that aj = IU, Tf3mjMynjf/JO)lw(fJmj, ynj) is decreasing, and by (14) Iiallp
IIf11Mp' Then clearly
ffo(f)M;
"
11f:
(f, T,m,M,o,~OIT,m,M,o,~~
J=n+1
M:;'
::=::
"C (f:=n+1 a;)"',
where the last inequality is a property of Banach frames. With Lemma 5 we obtain
(00~(naO"n(f)M:Y~
Remarks.
)I/P
::S
C'II(aj)lIp
::S
C"lIf11Mp'
•
With the usual technical problems which have to be overcome in the case of
quasi-Banach spaces, it is possible to establish a theory of the modulation spaces M; (R)
for 0 < p < I and to show that the local Fourier bases are unconditional bases for these
quasi-Banach spaces [21]. :Since Lemma ~ is valid for p < I, Theorem:3 remains true
for p < 1. In this way, quasi-Banach spaces appear naturally in problems of nonlinear
approximation.
References
[1]
[2]
[3]
I
P. AUSCHER(1994): Remarks on the local Fourier bases. In: Wavelets: Mathematics and Applications
(J. Benedetto, M. Frazier, eds). CRC Press, Boca Raton, FL, pp. 203-218.
P. AUSCHER, G. WEISS, M. WICKERHAUSER(1992): Local sine and cosine bases of Coifman and
Meyer and the construction of smooth wavelets. In: Wavelets: A Tutorial in Theory and Applications
(c. K. Chui, ed.). San Diego: Academic Press, pp. 237-256.
R. COIFMAN, Y. MEYER (1991): Remarques sur l'analyse de Fourier afenerre, c. R. Acad. Sci. Paris
ser. 1,312: 259-261.
[4]
I. DAUBECHIES,S. JAFFARD,J. JOURNE (1991): A simple Wilson orthonormal
decay. SIAM J. Math. Anal., 22: 554-572.
basis with exponential
[5]
R. A. DEVORE, B. JAWERTII,V. POPOV(1992): Compression of wavelet decompositions.
114: 737-785.
Amer. J. Math.,
I
L
~
Nonlinear Approximation
[6]
[7]
15
with Local Fourier Bases
R A. DEVORE, B. LUCIER (1992): Fast wavelet techniques for near-optimal image processing. In: IEEE
Military Comunications Conference Record, San Diego, October 11-14, 1992. Piscataway, NJ: IEEE
Press, pp. 1I29-1135.
R. A. DEVORE, V. A. POPov (1988): Interpolation
Lecture Notes in Math. No. 1302.
spaces and non-linear approximation.
Springer
[8]
R. A. DEVORE, V. N. TEMLYAKOV(to appear): Some remarks on greedy algorithms. Adv. in Comput.
Math.
[9]
D. DONOHO (1993): Unconditional bases are optimal bases for data compression
estimation. Appl. Comput. Harmonic Anal., 1: 100-115.
and for statistical
[10]
H. G. FEICHTINGER(1989): Atomic characterizations of modulation spaces through Gabor-type representations. Proc. Conference on Constructive Function Theory, Edmonton, July 1986. Rocky Mountain
J. Math., 19: 113-126.
[11]
[12]
H. G. FEICHTINGER(1981): On a new Segal algebra. Monatsh. Math., 92: 269-289.
H. G. FEICHTINGER,K. GROCHENIG(1992): Non-orthogonal wavelet and Gabor expansions, and group
representations. In: Wavelets and Their Applications, (M. B. Ruskai et aI., eds.). Boston: Jones and
Bartlett, pp. 353-376.
[13]
[14]
[15]
[16]
H. G. FEICHTINGER,K. GROCHENIG(1992): Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. In: Wavelets: A Tutorial
in Theory and Applications (c. K. Chui, ed.). New York: Academic Press.
H. G. FEICHTINGER,K. GROCHENIG,D. WALNUT(1992): Wilson basis and modulation spaces. Math.
Nachr., 155: 7-17.
M. FRAZIER, B. JAWERTH(1988): The cp-transform and applications to distribution spaces, injunction
spaces and applications. Springer Lecture Notes in Math., No. 1302, pp. 223-246.
K. GROCHENIG(1991): Describing Junctions: Atomic decompositions versus frames. Monatsh. Math.,
112: 1-42.
[17]
K. GROCHENIG (1996): An uncertainty principle
Math., 121: 87-104.
related to the Poisson summation formula.
Studia
[18]
E. LAENG (1990): Une base orthonormale de L2(R), dont les iliments sont bien localises dans
l'espace de phase et leurs supports adaptes a toute partition symitrique de l'espace desfrequences.
C. R. Acad. Sci. Paris, 311: 677-680.
[19]
[20]
[21]
[22]
H. MALVAR(1991): Lapped transform for efficient transform/subband
coding. IEEE Trans. Acoust.
Speech Signal Process., 83: 674-673.
Y. MEYER (1990): Ondelettes et Operateurs 1. Paris: Hermann.
S. SAMARAH(1998): Approximation Theory and Modulation Spaces. Thesis, University of Connecticut.
S. B. STECHKIN(1995): On absolute convergence of orthogonal series. Dokl. Akad. Nauk SSSR, 102:
[23]
37-40.
K. TACHIZAWA(1994): The boundedness ofpseudodif.ferential
operators on modulation spaces. Math.
Nachr., 168: 263-277.
K. Grochenig
Department of Mathematics
The University of Connecticut
Storrs
CT 06269-3009
USA
[email protected]
~
S. Samarah
Department of Mathematics
NUHAG
University of Vienna
Strudlhofg, 4
A-1090 Austria
r-