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Georgian Mathematical Journal Volume 16 (2009), Number 3, 465–474

AN ALMOST GREEDY UNIFORMLY BOUNDED ORTHONORMAL BASIS IN Lp(·) ([0, 1]) SPACES ANA DANELIA AND EKATERINE KAPANADZE

In memory of Prof. L. Zhizhiashvili on the occasion of his 75th birthday anniversary Abstract. We construct a uniformly bounded orthonormal almost greedy basis for the variable exponent Lebesgue spaces Lp(·) ([0, 1]), 1 < p− ≤ p+ ≤ 2 (or 2 ≤ p− ≤ p+ < ∞), when the diadic Hardy–Littlewood maximal operator is bounded on these spaces. 2000 Mathematics Subject Classification: 41A17, 42C40. Key words and phrases: Greedy algorithm, non-linear approximation, variable Lebesgue spaces.

1. Introduction Let {xn }n∈N be a semi-normalized basis in a Banach space X. This means that {xn }n∈N is a Schauder basis and is semi-normalized, i.e., 0 < inf n∈N kxn k ≤ supn∈N kxn k < ∞. For an element x ∈ X we define the error of the best m-term approximation as follows ½° °¾ X ° ° σm (x) = inf °x − α n xn ° , n∈A

where the inf is taken over all subsets A ⊂ N of cardinality m at most and over all possible scalars αn . The main problem in approximation theory concerns the construction of efficient algorithms for an m-term approximation. A computationally efficient method of producing m-term approximations, which has been widely investigated in recent years, Pis the so called greedy algorithm. We define the greedy approximation of x = n an xn ∈ X as X Gm (x) = an xn , n∈A

where A ⊂ N is any set of cardinality m in such a way that |an | ≥ |al | whenever n ∈ A and l ∈A. We say that a semi-normalized basis {xn }n∈N is greedy if there exists a constant C such that for all m = 1, 2, . . . and all x ∈ X we have kx − Gm (x)k ≤ Cσm (x). This notion evolved in the non-linear approximation theory, see, e.g., [15], [16]. A result of Konyagin and Temlyakov [7] characterizes greedy bases in a Banach c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °

466

A. DANELIA AND E. KAPANADZE

spaces X as unconditional and democratic. The latter means that for some constant C > 0 ° ° ° ° ° X xα ° ° X xα ° ° ≤ C° ° ° ° ° kxα k ° kxα k ° α∈A0

α∈A

holds for all finite sets of indices A, A0 ⊂ N of the same cardinality. Wavelet systems are the well known examples of greedy bases for many function and distribution spaces. Indeed, Temlyakov showed in [15] that the Haar system (and any wavelet system Lp -equivalent to it) is greedy in the Lebesgue spaces Lp (Rn ) for 1 < p < ∞. When wavelets have sufficient smoothness and decay, they are also greedy bases for more general Sobolev and Tribel–Lizorkin classes (see, e.g., [3], [4]). A bounded Schauder basis for a Banach space X is called quasi-greedy if there exists a constant C such that for x ∈ X, kGm (x)k ≤ Ckxk for m ≥ 1. Wojtaszczyk [16] proved the following result which gives a more intuitive interpretation of quasi-greedy bases. Theorem 1.1. A bounded Schauder basis for a Banach space X is quasigreedy if and only if limm→∞ kx − Gm (x)kX = 0 for every element x ∈ X. A bounded Schauder basis for a Banach space X is almost greedyP if there exists a constant C such that for x ∈ X, kx − Gm (x)k ≤ C inf{kx − n∈A < x, xn > xn k : A ⊂ N, |A| = m}. It was proved in [2] that a basis is almost greedy if and only if it is quasigreedy and democratic. Given a measurable function p(·) : [0, 1] −→ [1, ∞), Lp(·) ([0, 1]) denotes the set of measurable functions f on [0, 1] such that for some λ > 0 Z1 µ

|f (x)| λ

¶p(x) dx < ∞.

0

This set becomes a Banach function space when equipped with the norm ( ) ¶p(x) Z1 µ |f (x)| kf kp(·) = inf λ > 0 : dx ≤ 1 . λ 0

Also, we define p+ = ess supx∈[0,1] p(x) and p− = ess inf x∈[0,1] p(x). We need two facts about variable Lp spaces. Firstly, generalize Hölder’s inequality Z |f (x)g(x)|dx ≤ 2kf kp(·) kgkp0 (·) , [0,1]

secondly, C1 kf kp− ≤ kf kp(·) ≤ C2 kf kp+ for some constants C1 , C2 (see [9]). M. Nielsen in [13] proved that there exists uniformly bounded orthonormal almost greedy basis in Lp ([0, 1]), 1 < p < ∞, that shows that it impossible to

AN ALMOST GREEDY UNIFORMLY BOUNDED ORTHONORMAL BASIS

467

extend Orlicz’s theorem stating that there are no uniformly bounded orthonormal unconditional bases for Lp ([0, 1]), p 6= 2, to the class of almost greedy bases. In variable exponent Lebesgue spaces an analogue of Orlicz’s theorem is valid. Namely, the following statement is true. Theorem 1.2. Let 1 < p− ≤ p+ < ∞ and p(·) 6= const = 2. Then there are no uniformly bounded orthonormal unconditional bases for Lp(·) ([0, 1]). There arises a question whether there exists a uniformly bounded orthonormal almost greedy basis in variable exponent Lebesgue spaces Lp(·) ([0, 1]). Note that orthogonal wavelet bases normalized in Lp(·) ([0, 1]) are not greedy bases of Lp(·) ([0, 1]) (p(·) 6= const) when the Hardy–Littlewood maximal operator is bounded on this space (see [8]). The purpose of this paper is to study such problems in the class of variable exponent Lebesgue spaces Lp(·) ([0, 1]) when Hardy–Littlewood maximal operator is bounded. Namely, the following theorem is obtained. Theorem 1.3. Let 1 < p− ≤ p+ ≤ 2 or 2 ≤ p− ≤ p+ < ∞ and let the Hardy–Littlewood maximal operator be bounded on the space Lp(·) ([0, 1]). Then there exists a uniformly bounded orthogonal almost greedy basis in Lp(·) ([0, 1]). 2. Preliminaries Let us start with the proof of the extension of Orlicz’s Theorem. First note that if 1 < p− ≤ p+ < ∞, then the space Lp(·) ([0, 1]) is a uniform convex Banach space (see [12]). Using this fact and the Maurey theorem [11, Theorem 1.d.6] we have Theorem 2.1. Let 1 < p− ≤ p+ < ∞ and let {ϕi )}∞ i=1 be an unconditional basis of Lp(·) ([0, 1]). Then there is a constant D such that for every choice of scalars {ai }ni=1 we have °µ X °X ° °µ X ¶ 21 ° ¶ 21 ° ° n ° ° n ° ° ° n −1 ° 2 2 ° ≤° ° . ° D ° |ai ϕi | ai ϕi ° |ai ϕi | ° ° ° ° ≤ D° p(·)

i=1

i=1

p(·)

i=1

p(·)

Proof of Theorem 1.2. Let A = {t : t ∈ [0, 1], 1 < p(t) ≤ 2} and A = [0, 1] \ A. Let there exist uniformly bounded orthonormal unconditional bases {ϕn }∞ n=1 for Lp(·) ([0, 1]), p(·) 6= 2. Without loss of generality let us suppose that |A| > 0. 0 1 = 1) for which the (If |A| = 0, we consider the space Lp (·) ([0, 1]) ( p01(t) + p(t) ∞ system {ϕn }n is also an unconditional basis). Let f ∈ Lp(·) ([0, 1]). For the function f1 := f χA ∈ L2 ([0, 1]) we have ½X ½X ¾ 12 ¾ 21 ∞ ∞ 2 2 P (f1 , x) := [hf1 , ϕn iϕn (x)] ≤M [hf1 , ϕn i] ≤ M kf1 k2 n=1

n=1

for a.e. x ∈ (0, 1), i.e., kP (f1 , ·)kp(·) ≤ M kf1 k2 . On the other hand, using Theorem 2.1 we obtain kP (f1 , ·)kp(·) ≥ D−1 kf1 k(·) . Thus kf1 kp(·) ≤ DM kf1 k2 , that a contradiction. ¤

468


Lemma 2.2 (Khintchine’s inequality). Let 1 ≤ p− ≤ p+ < ∞ and let rk (t), k ≥ 1, be Rademacher functions. Then there exist Ap(·) , Bp(·) such that for any sequence {ak }k≥1 , ° µX ¶ 12 ° X µX ¶ 21 ° ° 2 2 ° ° Ap(·) |ak | ≤° ak rk (t)° ≤ Bp(·) . |ak | k

p(·)

k

k

Proof. It is known (see [5]) that for 1 ≤ p < ∞ there exist Ap , Bp such that for any sequence {ak }k≥1 , ° µX ¶ 12 µX ¶ 12 ° X ° ° 2 2 |ak | ak rk (t)° Ap ≤° . |ak | ° ≤ Bp ° p

k

k

k

Taking into account the fact that Bk · kp− ≤ k · kp(·) ≤ Ck · kp+ and the above inequality we obtain Lemma 2.2. ¤ Lemma 2.3. Let 1 < p− ≤ p+ < ∞ and let rk (t), k ≥ 1, be Rademacher functions.Then for f ∈ Lp(·) ([0, 1]) we have ¶ 12 µX ∞ |hf, rk i|2 ≤ Cp(·) kf kp(·) . k=1

Proof. For any n ≥ 1, by the Hölder’s inequality and Khintchine’s inequality we obtain ¶ µX Z1 n n X 2 rk (x)hf, rk i dx |hf, rk i| = f (x) k=1

0

k=1

° n ° °X ° ° ≤ 2° hf, r ir k k ° ° k=1

This implies

µX n

kf kp(·) ≤ C

µX n

p0 (·)

|hf, rk i|

2

¶ 12

¶1/2 |hf, rk i|

2

kf kp(·) .

k=1

≤ Bp(·) kf kp(·) .

k=1

Now taking the limit as n → ∞ we obtain our result.

¤

Let us denote by Md the dyadic maximal operator defined for any f ∈ L1 [0, 1] function is defined as follows Z 1 Md f (x) = sup |f (t)|dt, x∈I |I| I

where x ∈ [0, 1] and I is a dyadic interval containing x. The following theorem is valid. Theorem 2.4. Let 1 < p− ≤ p+ < ∞. The following statements are equivalent: p(·) 1) the Haar system {χk }∞ ([0, 1]). k=1 is a basis in L ∞ 2) the Haar system {χk }k=1 is an unconditional basis in Lp(·) ([0, 1]).


469

p(·) 3) the Walsh system {Wk }∞ ([0, 1]). k=0 is basis in L 4) Paley’s inequality °µ X ¶1 ° ° ∞ 2 2 2° ° C1 kf kp(·) ≤ ° ak χk ° ° ≤ C2 kf kp(·) p(·)

k=1

is valid, where ak = hf, χk i, k ∈ N and C1 and C2 are absolute constants. 1 5) sup |I| kχI kp(·) kχI kp0 (·) ≤ C < ∞, |I|

6) the maximal operator Md is bounded in Lp(·) ([0, 1]). Proof. The equivalence 4) ⇔ 2) automatically follows from Theorem 2.1. The relation 4) ⇒ 3) can be proved in the same way as it is done in [5]. To prove of 5) ⇔ 6) we can use Lerner’s method [10]. The relation 6) ⇒ 4) is a consequence of the extrapolation theorem obtained by Cruz-Uribe, Fiorenza, Martel and Perez (see [1]) and the weight inequality for Paley’s function obtained by Kazarian in [6]. Note that the extrapolation method is considered for the common maximal operator, but it also is valid for dyadic maximal operator. The conclusions 2) ⇒ 1) and 3) ⇒ 1) are trivial. For proof of 1) ⇔ 5) we can use Lerner’s method [10]. ¤ 3. Proof of the Main Results Let us construct some system in the manner as follows. For k = 1, 2, . . . , we (k) k define the 2k × 2k Olevskii matrix Ak = (aij )2i,j=1 by k

aki1 = 2− 2 for i = 1, 2, . . . , 2k , and for j = 2s + ν, with 1 ≤ ν ≤ 2s and s = 0, 1, . . . , k − 1, we let  s−k  for (ν − 1)2k−s < i ≤ (2ν − 1)2k−s−1 , 2 2 s−k (k) aij = −2 2 for (2ν − 1)2k−s−1 < i ≤ ν2k−s ,  0 otherwise. It is known [14] that Ak are orthogonal matrices and there exists a finite constant C such that for all i, k we have k

2 X

(k)

|ai,j | ≤ C.

j=1 10k

Put Nk = 2 and define Fk such that F0 = 0, F1 = N1 − 1 and Fk − Fk−1 = Nk −1, k = 1, 2, . . . . We consider the Walsh system W = {Wn }∞ n=0 on [0, 1]. We split W into two subsystems. The first subsystem W1 = {rk }∞ k=1 is Rademacher functions with their natural ordering. The second subsystem W2 = {φk }∞ k=1 is the collection of Walsh functions not in W1 with the ordering from W. We now impose the ordering φ1 , r1 , r2 , . . . , rF1 , φ2 , rF1 +1 , . . . , rF2 , φ3 , rF2 +1 , . . . , rF3 , φ4 , . . . .

470

A. DANELIA AND E. KAPANADZE k

The block Bk := {φk , rFk−1 +1 , . . . , rFk } has length Nk and we apply A10 to Bk (k) k to obtain a new orthonormal system {ψi }N i=1 given by N

(k) ψi (1)

(1)

k X φk (10k ) √ = + aij rFk−1 +j−1 . Nk j=2

(2)

(2)

The system ψ1 , . . . , ψN1 , ψ1 , . . . , ψN2 , . . . will be denoted by B = {ψk }∞ k=1 . It k is easy to verify that B is an orthonormal basis for L2 since each matrix A10 is orthogonal and is also uniformly bounded. Lemma 3.1. Let 1 ≤ p− ≤ p+ < ∞. The system B = {ψk }∞ k=1 is democratic p(t) in L ([0, 1]) with X 1 k ψk kp(·) ³ |A| 2 . k∈A

Proof. Taking into account the fact that Bk · kp− ≤ k · kp(·) ≤ Ck · kp+ and the estimate (see [13]) ° ° °X ° 1 ° 2 ψk ° ° ° ³ |A| for any 1 < p < ∞ k∈A

p

(see [13]) we obtain our result.

¤

Lemma 3.2. Let 1 < p− ≤ p+ < ∞ and let the Hardy–Littlewood maximal operator be bounded on the space Lp(·) ([0, 1]).Then the system B = {ψk }∞ k=1 is a p(·) Schauder basis for L ([0, 1]). Proof. Notice that span(B) = span(W) by construction, so span(B) is dense in Lp(·) ([0, 1]) sincePW is a Schauder basis for Lp(t) ([0, 1]). Let Sn (f ) = nk=1 hf, ψk iψk be the partial sum operator. We need to prove p(·) that the family of operators {Sn }∞ ([0, 1]). Let n=1 is uniformly bounded on L ∞ 2 f ∈ L ([0, 1]) ⊂ L ([0, 1]). For n ∈ N we can find L ≥ 1 and 1 ≤ m ≤ NL such that Sn (f ) =

Nk n L−1 X m X X X (k) (k) (L) (L) hf, ψk iψk := T1 + T2 . hf, ψk iψk = hf, ψj iψj + k=1

k=1 j=1

k=1

Let us estimate T1 . If L = 1, then T1 = 0, so we may assume L > 1. The construction of B shows that T1 is the orthogonal projection of f onto µ L−1 ¶ Nk [[ (k) ψk span = span{{W0 , W1 , . . . , WL−2 } ∪ {rl0 , rl0 +1 , . . . , rFL−1 }}, k=1 j=1

with l0 = [log2 (L)]. It follows that we can rewrite T1 as T1 =

L−2 X k=0

hf, Wk iWk + PR (f ),


471

where PR (f ) is the orthogonal projection of f onto span{rl0 , rl0 +1 , . . . , rFL−1 }. Thus, using the fact that W is a Schauder basis for Lp(·) ([0, 1]), Khintchine’s inequality and Lemma 2.3, we will have kT1 kp(·) ≤ Ckf kp(·) . Let us now estimate T2 . T2 = =

m ¿ X k=1

m X

(L)

k=1

φL f, √ + NL

NL X

(L)

hf, ψk iψk Àµ

N

L X φL (10L ) √ φL + akt rFL−1 +t−1 NL t=2 µ ¶ NL m X X (10L ) akj hf, rFL−1 +j−1 i

(10L ) akj rFL−1 +j−1

j=2

¶

φL m hf, φL i + √ NL NL j=2 k=1 ¶ ÀX ¿ NL µ X m φL (10L ) a rFL−1 +j−1 + f, √ NL j=2 k=1 kj ¸· X ¸ NL NL m ·X X (10L ) (10L ) akt rFL−1 +t−1 = akj hf, rFL−1 +j−1 i + =

k=1

t=2

j=2

= G1 + G2 + G3 + G4 . Using the fact that 1 ≤ m ≤ NL and Hölder’s inequality we obtain kG1 kp(·) ≤ Ckf kp(·) . Using Hölder’s and Khintchine’s inequalities, the fact that the matrices Ak are orthonormal and Lemma 2.3, we obtain kGi kp(·) ≤ Ckf kp(·) , i = 2, 3, 4, for some constant C independent of f ∈ L∞ ([0, 1]). Consequently, for some constant C independent of f ∈ L∞ ([0, 1]) we have kSn f kp(·) ≤ Ckf kp(·) . Since L∞ ([0, 1]) is dense in Lp(·) ([0, 1]), we deduce that {Sn }∞ n=1 is a uniformly bounded p(·) family of linear operators on L [0, 1] and the system B is a Schauder basis for Lp(·) [0, 1]. ¤ Lemmas 3.1 and 3.2 give rise to Theorem 3.3. Let 1 < p− ≤ p+ < ∞ and let the Hardy–Littlewood maximal operator be bounded on the space Lp(·) ([0, 1]). Then there exists a uniformly bounded orthonormal democratic basis in Lp(·) ([0, 1]). Lemma 3.4. Let 1 < p− ≤ p+ ≤ 2 or 2 ≤ p− ≤ p+ < ∞ and let the Hardy–Littlewood maximal operator be bounded on the space Lp(·) ([0, 1]).Then p(·) the system B = {ψk }∞ ([0, 1]). k=1 is a quasi-greedy basis for L Proof. First we consider 2 ≤ p− ≤ p+ < ∞. Let f ∈ Lp(t) ([0, 1]) ⊂ L2 . We have f=

∞ X i=1

hf, ψi iψi ,

472


with k{hf, ψi i}kl2 ≤ kf k2 ≤ Ckf kp(·) . We must prove that Gm (f ) is convergent in Lp(·) ([0, 1]). Let us formally write f=

Nk ∞ X X

(k)

(k)

hf, ψj iψj

k=1 j=1

=

Nk ∞ X X

φk (k) hf, ψj i √

Nk

k=1 j=1

Nk Nk ∞ X X X (k) (10k ) + hf, ψi i aij rFk−1 +j−1 = S1 + S2 . k=1 i=1

j=2

Consider εki ⊂ {0, 1}. By Kchintchine’s inequality and the fact that each k A10 is orthogonal we conclude that S2 converges unconditionally in Lp(·) ([0, 1]). Indeed, ° ° ∞ N ÃN ! k k ° °X X X k) ° ° (k) (10 k εi hf, ψi iaij rFk−1 +j−1 ° ° ° ° k=1 j=2

i=1

≤C

Ã Nk XX k

p(·)

!1/2 (k) εki |hf, ψi i|2

.

i=1

The series defining S2 converges unconditionally, so it suffices to prove that the series defining S1 converges in Lp(·) ([0, 1]) when the coefficients < f, ψ> are arranged in decreasing order. Let us consider the sets ( ) 1 1 (k) 1 Λk = j : < |hf, ψj i| < 1/10 , Nk N ½ ¾ k 1 (k) Λ2k = j : |hf, ψj i| ≤ , Nk ) ( 1 (k) Λ3k = j : |hf, ψj i| ≥ 1/10 . Nk Then S1 =

∞ X X

φk (k) hf, ψj i √

Nk

k=1 j∈Λ1k

+

∞ X

+

∞ X X

φk (k) hf, ψj i √ Nk 2

k=1 j∈Λk

X

φk (k) hf, ψj i √ = T1 + T2 + T3 . Nk 3

k=1 j∈Λk

By the construction of sets Λik we can conclude that the series defining T2 and T3 converges absolutely in Lp(·) ([0, 1]). From the definition of Λ1k we get (k)

|hf, ψi i| >

1 1 (k+1) i > |, i ∈ Λ1k , j ∈ Λ1k+1 , k = 1, 2, . . . , ≥ 1/10 ≥ |hf, ψj Nk Nk+1


473

so when we arrange T1 in decreasing order the rearrangement can only take place inside the blocks. Inside the blocks we have the estimate ° µX ¶1/2 1 1\2 X° ° ° |Λk | (k) 2 °hf, ψ (k) i √φk ° ≤ √ |hf, ψj i| , k ≥ 1. j ° ° N N k

j∈Λ1k

Hence

p(·)

X° ° °hf, ψ (k) i √φk j ° N

j∈Λ1k

k

j∈Λ1k

k

° ° ° °

→ 0,

k → ∞.

p(·)

We can conclude that Gm (f ) is convergent in Lp(·) ([0, 1]). Therefore by Theorem 1.1 B is a quasi-greedy basis and thus almost greedy in Lp(·) ([0, 1]), 2 ≤ p− ≤ p+ < ∞. Let 1 < p− ≤ p+ ≤ 2. The above arguments imply that the system B is 0 almost greedy in Lp (·) ([0, 1]). From [2, Theorem 5.4] we conclude that B is a quasi-greedy basis and consequently almost greedy in Lp(·) ([0, 1]). ¤ Acknowledgements The first author was supported by grants GNSF/PRES07/3-151,GNSF/ STO7/3-171. The second author was supported by grants GNSF/PRES07/3150, GNSF/PRES07/3-151. References ´rez, The boundedness of 1. D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pe p classical operators on variable L spaces. Ann. Acad. Sci. Fenn. Math. 31(2006), No. 1, 239–264. 2. S. J. Dilworth, N. J. Kalton, D. Kutzarova, and V. N. Temlyakov, The thresholding greedy algorithm, greedy bases, and duality. Constr. Approx. 19(2003), No. 4, 575–597. ´ s and E. Herna ´ ndez, Sharp Jackson and Bernstein inequalities for N -term 3. G. Garrigo approximation in sequence spaces with applications. Indiana Univ. Math. J. 53(2004), No. 6, 1739–1762. 4. C.-C. Hsiao, B. Jawerth, B. J. Lucier, and X. M. Yu, Near optimal compression of orthonormal wavelet expansions. Wavelets: mathematics and applications, 425–446, Stud. Adv. Math., CRC, Boca Raton, FL, 1994. 5. B. S. Kashin and A. A. Saakyan, Orthogonal series (Russian) Izdatel’stvo NauchnoIssledovatel’skogo Aktuarno-Finansovogo Tsentra (AFTs), Moscow, 1999. 6. K. S. Kazarian, On bases and unconditional bases in the spaces Lp (), 1 ≤ p < ∞. Studia Math. 71(1981/82), No. 3, 227–249. 7. S. V. Konyagin and V. N. Temlyakov, A remark on greedy approximation in Banach spaces. East J. Approx. 5(1999), No. 3, 365–379. 8. T. S. Kopaliani, Greediness of the wavelet system in Lp(t) (R) spaces. East J. Approx. 14(2008), No. 1, 59–67. ć ˇik and J. Ra ´ kosnik, On spaces Lp(x) and W k,p(x) . Czechoslovak Math. J. 9. O. Kova 41(116)(1991), No. 4, 592–618. 10. A. K. Lerner, On some questions related to the maximal operator on variable Lp spaces. Trans. Amer. Math. Soc. (to appear).

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11. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II. Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 97. Springer-Verlag, Berlin–New York, 1979. 12. J. Musielak, Orlicz spaces and modular spaces. Lecture Notes in Mathematics, 1034. Springer-Verlag, Berlin, 1983. 13. M. Nielsen, An example of an almost greedy uniformly bounded orthonormal basis for Lp (0, 1). J. Approx. Theory 149(2007), No. 2, 188–192. 14. A. M. Olevski˘ı, Fourier series with respect to general orthogonal systems. (Translated from the Russian) Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 86. SpringerVerlag, New York–Heidelberg, 1975. 15. V. N. Temlyakov, The best m-term approximation and greedy algorithms. Adv. Comput. Math. 8(1998), No. 3, 249–265. 16. P. Wojtaszczyk, Greedy algorithm for general biorthogonal systems. J. Approx. Theory 107(2000), No. 2, 293–314.

(Received 21.08.2008) Authors’ address: Faculty of Exact and Natural Sciences I. Javakhshvili Tbilisi State University 2, University St., Tbilisi 0186 Georgia E-mail: [email protected] ekaterine [email protected]