On the divergence of greedy algorithms with respect to Walsh subsystems in $L$

0.07 MB·English

by  Sergo A. Episkoposian

On the divergence of greedy algorithms
L
with respect to Walsh subsystems in
5
1
0
2 S.A.Episkoposian

n
a e-mail: [e-mail]
J

5

In this paper we prove that there exists a function which f(x) belongs to
]
A L1[0,1] such that a greedy algorithm with regard to the Walsh subsystem does
F not convergeto f(x) in L1[0,1]norm, i.e. the Walsh subsystem {Wnk} is not a
. quasi-greedy basis in its linear span in L1
h
t
a 1. INTRODUCTION
m
[ Let a BanachspaceX with a norm||·||=||·|| , and a basis Ψ={ψ }∞ ,
X k k=1
||ψ || =1, k =1,2,.. be given.
1 k X
v Denote by Σm the collectionofallfunctions inX whichcanbe expressedas
2 a linear combination of at most m- functions of Ψ. Thus each function g ∈Σ
m
3 can be written in the form
8
0
g = a ψ , #Λ≤m.
0 s s
. sX∈Λ
1
0 For a function f ∈X we define its approximation error
5
1 σ (f,Ψ)= inf ||f −g|| , m=1,2,...
v: m g∈Σm X
i
X and we consider the expansion
r ∞
a
f = a (f)ψ .
k k
kX=1
Definition 1. Let an element f ∈ X be given. Then the m-th greedy
approximant of function f with regardto the basis Ψ is given by the formula
G (f,Ψ)= a (f)ψ ,
m k k
kX∈Λ
where Λ⊂{1,2,...} is a set of cardinality m such that
|a (f)|≥|a (f)|, n∈Λ, k ∈/ Λ.
n k
1
We’ll say that the greedy approximantof f(t)∈Lp , p≥1 with regardto
[0,1]
the basis Ψ converges,if the sequence G (x,f) converges to f(t) in Lp norm.
m
This new and very important direction invaded many mathematician’s at-
tention (see [1]-[9]).
Definition 2. We calla basisΨ greedybasisif foreveryf ∈X thereexists
a subset Λ⊂{1,2,...} of cardinality m, such that
||f −G (f,Ψ)|| ≤C·σ (f,Ψ)
m X m
where a constant C =C(X,Ψ) independent of f and m.
In 1998V.N.Temlyakovprovedthat each basis Ψ which is L -equivalentto
p
the Haar basis H is Greedy basis for L (0,1), 1<p<∞ (see [4]).
p
Definition 3. We say that a basis Ψ is Quasi-Greedy basis if there exists
a constant C such that for every f ∈ X and any finite set of indices Λ, having
the property
min|a (f)|≥max|a (f)|
k k
k∈Λ n∈/Λ
we have
a (f)ψ ≤C·||f|| .
(cid:12)(cid:12) k k(cid:12)(cid:12) X
(cid:12)(cid:12)(cid:12)(cid:12)kX∈Λ (cid:12)(cid:12)(cid:12)(cid:12)X
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
In 2000 P.Wojtaszczyk [5] proved that a basis Ψ is quasi-greedy if and
only if the sequence {G (f)} converges to f, for all f ∈ X. Note that in
m
[6] S.Konyagin and V.Temlyakov constructed an example of quasi-greedy basis
that is not Greedy basis.
V.Temlyakovprovedthatthe trigonometricsystemT isnotaQuasi-Greedy
basis for Lp if p6=2 (see [7]).
In [8] it is proved that this result is true for Walsh system.
In the sequel, we’ll fix a sequence {M }∞ so that
n n=1
lim (M −M )=+∞
2k 2k−1
k→∞
and consider a subsystem of Walsh system
{W (x)}∞ ={W (x): M ≤m≤M , s=1,2,...} (1)
nk k=1 m 2s−1 2s
Inthispaperweconstructedafunctionf(x)∈L1[0,1]suchthatthesequence
{G (f)}, with respect to Walsh system, does not converge to f(x) by L1[0,1]
m
norm and we can watch for spectra of ”bad” function f(x).
Moreover the following is true.
Theorem . There exists a function f(x) belongs to L1[0,1] such that the
approximateG (f,W )withregardtotheWalshsubsystemdoesnotconverge
n nk
to f(x) by L1[0,1]norm,i.e. the Walshsubsystem{W } is nota quasi-greedy
nk
basis in its linear span in L1.
2
2. PROOF OF THEOREM
First we will give a definition of Walsh-Paly system (see [10]).
k k
W (x)=1, W (x)= r (x), n= 2ms, m >...>m , (2)
0 n ms 1 s
sY=1 Xs=1
where {r (x)}∞ is the system of Rademacher:
k k=0
1, x∈ 0,1 ;
2
r0(x)= (cid:2) (cid:1)
−1, x∈(1,1).
2

r (x+1)=r (x), r (x)=r (2kx), k =1,2,...
0 0 k 0
IntheproofoftheoremwewillusedthefollowingpropertiesofWalshsystem:
1. From (2) we have
W (x)=W (x)·W (x), if 0≤j ≤2k−1. (3)
2k+j 2k j
m−1
2. The Dirichlet-Walsh kernel D (x) = W (x) has the following prop-
m j
Xj=0
erties (see [10] p.27)
2j, x∈ 0, 1 ;
2j
D2j(x)= (cid:2) (cid:3) (4)
0, x∈ 1,1 .
2j
3. There is a sequence of natural numbers(cid:0){m (cid:3)}∞ with 2k−1 ≤ m < 2k,
k k=1 k
k =1,2,... (see [10] p.47), such that
1 1
||D || = |D (x)|dx≥ log m , k =1,2,.... (5)
mk 1 Z mk 4 2 k
0
Proofof Theorem. Takingintoaccount(1)-(3)wecantakethesequences
of natural numbers {k }∞ and {p }∞ so that the following conditions are
ν ν=1 ν ν=1
satisfied:
k >(ν−1)2+1, (6)
ν
W (x)·W (x)=W (x), 0≤i<2kν, (7)
2kν i 2kν+i
2kν +i∈[M2pν−1,M2pν), 0≤i<2kν, (8)
For any natural ν we set
f (x)= c(ν)W (x)=
ν nk nk
Nν≤nXk<Nν−1
3
2kν−1
1
= +2−(2kν+i) ·W (x)=
(cid:18)ν2 (cid:19) 2kν+i
Xi=0
2kν−1
1
=W (x)· +2−(2kν+i) ·W (x)=
2kν (cid:18)ν2 (cid:19) i
Xi=0
2kν−1 2kν−1
1 1
=W2kν(x)·ν2 Wi(x)+ 2kν 2−iWi(x)=
Xi=0 Xi=0
 
2kν−1
1 1
=W2kν(x)·ν2D2kν(x)+ 2kν 2−iWi(x), (9)
Xi=0
 
where
ν12 +2−nk, Nν ≤nk =Nν +i<Nν+1, 0≤i<Nν,
c(ν) = (10)
nk 
0, n <N , ν ≥1,
k ν

N =2kν, N =2kν+1. (11)
ν ν+1
We set
∞
f(x)= c (f)W (x)=
nk nk
kX=1
∞ ∞
= fν(x)=  cn(νk)Wnk(x), (12)
Xν=1 Xν=1Nν≤nXk<Nν−1 
where
c (f)=c(ν) for N ≤n <N −1, ν =1,2,... (13)
nk nk ν k ν
Now we will show that f(x)∈L1[0,1]
Taking into account (9)-(11)we get
∞ ∞ 2kν−1
1
f(x)= ν2W2kν(x)D2kν(x)+  2−iW2kν+i(x)=
νX=1 Xν=1 Xi=0
 
=G(x)+H(x). (14)
For function G(x) from (4)and definition of Walsh system we have
1 ∞ 1
|G(x)|dx ≤ <∞
Z ν2
0 νX=1
and we get that G(x)∈L1[0,1].
4
Analogously
1 ∞
|H(x)|dx≤ 2−ν <∞
Z
0 Xν=1
i.e. H(x)∈L1[0,1].
Hence and from (14) it follows that f(x)∈L1[0,1].
For any natural ν we choose numbers k,j so that
N ≤n <N ≤n <N .
ν k ν+1 j ν+2
Then from (10) we have
1
c (f)=c(ν+1) = +2−nj <
nj nj (ν+1)2
1
< +2−nk =c(ν) =c (f)
ν2 nk nk
i.e. c (f)<c (f).
nj nk
Analogously for any number n , N ≤n <N , ν ≥1 we have
k ν k ν+1
1 1
c(ν) = +2(−nk+1) < +2−nk =c(ν)
nk+1 ν2 ν2 nk
Thus we get
c (f)<c (f).
nk+1 nk
In other hand if k →∞ then n →∞ and ν →∞ (see (10), (11)).
k
From (13) we get lim c (f)=0 and consequently c (f)ց0.
k→∞ nk nk
For any numbers m so that
ν
2kν ≤m <2kν+1. (15)
ν
By (11) - (13) and Definition 1 we have
G (f,W )−G (f,W )=
2kν+mν nk 2kν nk
2kν+mν−1 1
= +2−(2kν+i) ·W (x)=
(cid:18)ν2 (cid:19) 2kν+i
i=X2kν
1 2kν+mν−1
= ·W (x)· W (x)+
ν2 2kν i
i=X2kν
1 2kν+mν−1 1
+ ·W (x)· W (x)=
2kν 2kν 2i i
i=X2kν
=J +J . (16)
1 2
5
By (6) we get
1 mν−1
J = ·W (x)· W (x)=
1 ν2 2kν 2kν+i
Xi=0
1
= ·W2 (x)·D (x).
ν2 2kν mν
|J |≤2kν+mν−1 1 |W (x)|≤ ∞ 1 ≤22−kν+1.
2 2i i 2i
i=X2kν i=X2kν
From this and (16)we obtain
|G (f,W )−G (f,W )|≥
2kν+mν nk 2kν nk
≥ 1 ·|D (x)|−22−kν+1. (17)
ν2 mν
Now we take the sequence of natural numbers m defined as (5) such that
ν
2kν ≤mν <2kν+1. Then from (6), (17) we have
1
|G (f,W )−G (f,W )|dx>
Z 2kν+mν nk 2kν nk
0
> 1 · 1|D (x)|dx−22−kν+1 ≥
ν2 Z mν
0
≥ 1 ·log m −22−kν+1 ≥ kν −22−kν+1 ≥
4·ν2 2 ν 4·ν2
≥ (ν−1)2+1 −22−kν+1 ≥ 1 −22−kν+1 ≥C , ν ≥2
4·ν2 8 1
Thus the sequence {G (f,W)} does not converge by L1[0,1] norm, i.e. the
n
Walsh subsystem {W }∞ is not a quasi-greedybasis in its linear span in L1.
nk k=1
The Theorem is proved.
Remark. As we well known (see [10] p.149) if the c ց 0, then the series
i
∞
c W (x) converges on (0,1). In the proof of Theorem we constructed the
n n
nX=1
series (11) so that the coefficients strongly decreasing, but the series diverges
by L1 -norm.
6
R E F E R E N C E S
[1]DeVore R.A.,TemlyakovV.N., SomeremarksonGreedyAlgorithms,Adv.
Comput. Math., 1995,v.5, p.173-187.
[2] DeVore R. A., Some remarks on greedy algorithms. Adv Comput. Math. 5,
1996, 173-187.
[3]DavisG.,MallatS.andAvalanedaM.1997,Adaptivegreedyapproximations.
Constr, Approx. 13, 57-98.
[4]TemlyakovV.N.,Thebestm-termapproximationandGreedyAlgorithms,
Advances in Comput. Math., 1998, v.8, p.249-265.
[5]WojtaszcykP.,GreedyAlgorithmforGeneralBiorthogonalSystems,Journal
of Approximation Theory, 2000, v.107, p. 293-314.
[6] Konyagin S. V., Temlyakov V. N., A remark on Greedy Approximation in
Banach spaces, East Journal on Approximation, 1999, v.5, p. 493-499.
[7] Temlyakov V. N., Greedy Algorithm and m - term Trigonometric approxi-
mation, Constructive Approx., 1998,v.14, p. 569-587.
[8] Gribonval R., Nielsen M., On the quasi-greedy property and uniformly
boundedorthonormalsystems,http://www.math.auc.dk/research/reports/R-2003-09.pdf.
[9] Grigorian M.G., ”On the convergence of Greedy algorithm”, International
Conference, Mathematics in Armenia, Advances and Perspectives, Abstract,
2003, p.44 - 45, Yerevan, Armenia.
[10]GolubovB.I.,EfimovA.V.,SkvortsovV.A.,WalshSeriesandTransforma-
tions: Theory and Applications [in Russian], NAuka, Moscow, (1987); English
transl.: Kluwer, Dordrecht (1991).
Department of Physics,
State University of Yerevan,
Alex Manukian 1, 375049 Yerevan, Armenia
e-mail: [e-mail]
7