Greediness of higher rank Haar wavelet bases in L p w(r) spaces

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1 Stud. Univ. Babeş-Bolyai Math. 59(2014), No. 2, Greediness of higher rank Haar avelet bases in L (R) saces Ekaterine Kaanadze Abstract. We rove that higher rank Haar avelet systems are greedy in L (R), 1 < < if and only if A N. Mathematics Subject Classification (2010): 41A17, 42C40. Keyords: Greedy algorithm, non-linear aroximation. 1. ntroduction Let X = {x n : n N} be a semi-normalized basis in a Banach sace X. This means that {x n } n N is a Schauder basis and is semi-normalized i.e. 0 < inf n N x n su n N x n <. For an element x X e define the error of the best m term aroximation as follos σ m (x, X ) = inf{ x n A α n x n }, here the inf is taken over all subset A N of cardinality at most m and all ossible scalars α n. The main question in aroximation theory concerns the construction of efficient algorithms for m-term aroximation. A comutationally efficient method to roduce m-term aroximations, hich has been idely investigated in recent years, is the so called greedy algorithm. For x X ith x = n=1 a nx n and m N, consider a subset G(m, x) N of cardinality m such that min a n max a n. n G(m,x) n N\G(m,x) There is some ambiguity in the choice of the set G(m, x), but our considerations do not deend on the articular choice. Then the m-th greedy aroximation of x ith resect to the basis X is defined as G m (x, X ) = a n x n. n G(m,x)

2 214 Ekaterine Kaanadze Clearly, σ m (x, X ) x G m (x, X ). The basis X is called greedy if there is a constant C > 0, indeendent of m, such that for each m N and x X, x G m (x, X ) Cσ m (x, X ). Wavelet systems are ell knon examles of greedy bases for many function and distribution saces. ndeed, V. N. Temlyakov shoed in [13] that the classical dyadic Haar system (and any avelet system L equivalent to it) is greedy in the Lebesgye saces L (R n ) for 1 < <. When avelets have sufficient smoothness and decay, they are also greedy bases for the more general Sobolev and Tribel-Lizorkin classes (see [3],[5]). Some examle of greedy bases are given in [13], [14]. n most cases these bases are greedy simly because they are symmetric (e.g. Riesz bases for a Hilbert sace), or because they are equivalent to the dyadic Haar basis or to a subsequence of the Haar basis (see [7]). S. V. Konyagin and V. N. Temlyakov [8] gave a very useful abstract characterization of greedy bases in a Banach saces X as those hich are unconditional and democratic, the last meaning that for some constant C > 0 x n x n C x n x n n A holds for all finite sets of indices A, A N ith the same cardinality. The urose of this aer is to study the efficiency of greedy algorithms ith resect higher rank Haar avelet system in the saces L (R). We recall that, as M. zuki roved in [6], that the dyadic Haar avelet system (N = 2) is greedy in L (R n ) for 1 < < if and only if A 2. Characterization of almost greedy uniformly bounded orthonormal bases in rearrangement invariant Banach function saces are given [2]. By an N-adic (N N, N 2) lattice D e mean the collection of all N-adic intervals, i. e. the collection of all intervals of the form [jn k, (j + 1)N k ), j, k Z. f is an interval, e denote by its length, and by χ its characteristic function. f is an N-adic interval [jn k, (j + 1)N k ) then e denote by (l) the children intervals of : [jn k + ln (k+1), jn k + (l + 1)N (k+1) ), l = 0, 1,, N 1. We denote by L 2 (R) the Hilbert sace of square integrable (ith resect to the Lebesgue measure) comlex-valued functions on R. We consider also L (R) (1 < ) saces, here L 1 loc (R) is a ositive function called a eight. The norm of a function f : R C from the sace L (R) is ( 1/ f L = f(x) (x)dx). R Given a function f, e denote by f its average over the interval, f = 1 n A f(x)dx. We are concerned ith a secial class of eights, called A N. We say that A N, 1 < < if A = su 1/( 1) 1 <. D

3 Greediness of higher rank Haar avelet bases in L (R) saces 215 We say that an N N matrix is a Haar avelet matrix of rank N if it is unitary and the elements of the first ro are all equal to 1/ N. Many classical examles of matrices used in mathematics and signal rocessing are Haar matrices of secific tyes. These include the discrete Fourier transform matrices, the discrete cosine transform matrices, Hadamard and Walsh matrices, Radmacher matrices, and Chebyshev matrices (see [12]). Let H = (g ki ) N 1 k,i=0 be a N N Haar matrix and ϕ = χ [0,1). Define the functions ψ (k) j,n ψ (k) (x) = N The collection of functions = ψ(k) N 1 l=0 g kl ϕ(nx l) k = 1,, N 1. (1.1) ψ (k) j,n = N j/2 ψ (k) (N j x n), j, n Z, k = 1,, N 1 form an orthonormal basis of L 2 (R) (see [15]. Bello e adot the shorter notation, here = [nn j, (n + 1)N j ). The system X = {ψ (k), D, k = 1,, N 1}, here the functions ψ (k), k = 1,, N 1 are defined by (1.1), is called the Haar avelet system of rank N (corresonding to Haar matrix H). An imortant examle of a higher rank Haar avelet system is the system obtained by avelet functions ψ (k) (x) = N N 1 l=0 e 2πikl/N ϕ(nx l), k = 1,, N 1, (1.2) here ϕ is characteristic function of the interval [0, 1). Note that the avelet system constructed by functions (1.2) became of interest in connection ith some roblems of -adic (non-archimedean) mathematical hysics (see [10-11]). For a Haar avelet system X and f L 1 loc (R), e define the Littleood-Paley oerator by here P f(x) = ( N 1 k=1 D < f, ψ (k) >= R < f, ψ (k) > 2 1 χ (x)) 1/2, f(x)ψ (k) (x)dx. The characterization of the saces L (R) ( A, 1 < < ) using higher rank Haar avelet system X as given in [9]. Theorem 1.1. ([9]) Let X be a Haar avelet system of rank N and 1 < <. The folloing conditions are equivalent: 1) The system X is unconditional basis of sace L (R); 2) There exist ositive constants c and C such that c f L P (f) L C f L for all f L (R); 3) A N. The urose of this aer is to rove folloing theorem.

4 216 Ekaterine Kaanadze Theorem 1.2. Let X = {ψ (k) ; D, k = 1,, N 1} be a Haar avelet system of rank N. Suose A N (1 < < ). Then the system {ψ (k) 1,, N 1} forms a greedy basis in the sace L (R). / ψ (k) ; D, k = 2. Proof of Theorem 1.2 For simlicity e shall denote the normalized characteristic function of a set of indices Γ {1, 2,, N 1} D by 1 Γ = ψ k ψ k. (k,) Γ Observe that X is democratic in L (R) if and only if there exists a function h : N R + for hich 1 C h(card(γ)) 1 Γ L C h(card(γ)), Γ {1, 2,, N 1} D (2.1) for some C 1. Observe that from Theorem 1.1 for a single element ψ k ψ k L ()1/ 1/2, here () = (x)dx and the constants involved in are indeendent of ψk. Thus, using again the exression of the norm L it follos that 1/2 1 Γ L 1/2 χ () 2/ χ, (2.2) () (k,) Γ 2/ L Γ L here Γ = { : (k, ) Γ}. Note that Card( Γ) Card(Γ). Given a finite set of intervals Γ D, e shall denote ( ) 1/2 χ (x) S Γ (x) =. () 2/ Γ We linearize the square function S Γ (x). Note that this lineralization rocedure has been used by other authors in the context of m term aroximation (see e.g. [3-5]). For every x Γ, e define x as the smallest (hence unique) interval in Γ containing x. t is clear that S Γ (x) χ x (x) ( x ) 1/ x Γ. (2.3) We no sho that the reverse inequality holds ith some universal constant. Note that if A N, then there exist C 1, C 2 > 0 and δ > 0 such that C 1 ( A / ) (A)/() C 2 ( A / ) δ (2.4) for all D and all subsets A (see [1]).

5 Greediness of higher rank Haar avelet bases in L (R) saces 217 f e enlarge the sum to include all N adic intervals containing x e have S Γ (x) 2 = χ (x) () 1 2/ (). 2/ Γ D, x f x and = N j x, then by (2.4) e have, () C 1 2 ( x)n jδ. Thus, S Γ (x) 2 C χ x (x) ( x ) 2/. This and (2.3) sho that S Γ (x) χ x (x) ( x). 1/ Observe that S Γ (x) S Γ min (x), here Γ min denotes the family of minimal intervals in Γ, that is, Γ min = { x : x Γ }. Note that the intervals in Γ min are not necessarily airise disjoint. Given a fixed Γ D, for any Γ e define the set S() as the union of all intervals from Γ strictly contained in. We define also the set L() = \S(). t is clear that Γ min if and only if L(), and moreover Γ = Γ min L(), here the sets in the last union are airise disjoint. Therefore e can rite S Γ (x) χ L() (x) (2.5) () 1/ Γ min here in the last sum there is a most one non-zero term. Denote by Γ S the collection of all intervals from Γ ith roerty: S() > (1 1/N). Denote by Γ L the collection of all intervals from Γ ith roerty: L() /N. Observe that Γ L Γ min. We have (1 1/N)Card(Γ) Card(Γ L ) Card(Γ min ) Card(Γ), Γ D. (2.6) Clearly Card(Γ L ) Card(Γ min ) Card(Γ). Thus, e need to rove only the inequality from the left hand side of (2.6). Given D, e rite (k), k = 1,, N for the N-adic intervals contained in of size N 1. For Γ S and k = 1,, N let (k) 0 be the biggest interval from Γ ith (k) 0 (k). Note that the intervals (k) 0 exist for every Γ S ; otherise, if for some k 0 {1,, N} there is no interval from Γ contained in (k0) e have (k0) L() and then S() \ (k0) = (1 1/N), contradicting the definition of Γ S. We claim that if, R Γ S and R, then e necessarily have (k) 0 R (l) 0 for all 1 k, l N. This is trivially true if R =. Without loss of generality e may assume R and also R (1). t follos that (k) 0 R (l) 0 for all k = 1,, N and all l = 2,, N. Moreover, as R (1) 0 is the biggest interval in Γ contained in R (1) and R 1 e have that R (1) 0 R (1). Hence, for all k = 1,, N e have (k) 0 R (1) 0 and thus (k) 0 R (1) 0. n short, to each Γ S e have assigned N different intervals in Γ and these are not associated to any other interval in Γ S. We conclude that NCard(Γ S ) Card(Γ). Consequently e have Card(Γ L ) = Card(Γ L ) Card(Γ S ) Card(Γ) Card(Γ)/N = (1 1/N)Card(Γ).

6 218 Ekaterine Kaanadze Note that /N L() and by (2.4) e have χ L() L χ L. Using this estimate e can rite S Γ L χ L() (x) () 1/ (Card(Γ min )) 1/ (Card(Γ)) 1/. (2.7) Γ min L From (2.2), (2.7) one obtains the estimates (2.1), ith h(n) = n 1/. References [1] Cruz-Uribe, D., Martell, J.M., Pérez, C., Weights, Extraolation and the Theory of Rubio de Francia, Birkhauser, [2] Danelia, A., Kaanadze, E., Almost greedy uniformly bounded orthonormal bases in rearrangement invariant Banach function saces, Stud. Univ. Babes-Bolyai, Math., 56(2011), no. 2, [3] Garrigos, G., Hernández, E., Shar Jackson and Bersntein inequalities for N term aroximation in sequence saces ith alications, ndiana Univ. Math. J., 53(2004), [4] Garrigós, G., Hernanández, E., Martell, J.M., Wavelets, Orlicz saces and greedy bases, Al. Com. Harmon. Anal., 24(2008), [5] Hsiao, C., Jaerth, B., Lucier, B.J., Yu, X.M., Near otimal comression of almost otimal avelet exansions. n: Wavelets, mathematics and alications. Stud. Adv. Math. CRC. Boca Raton. FL., 1994, [6] zuki, M., The Haar avelets and the Haar scaling function in eighted L saces ith A dy,m eights, Hokkaido Math. J., 36(2)(2007), [7] Kamont, A., General Haar system and greedy aroximation, Studia Math., 45(2001), [8] Konyagin, S.V., Temlyakov, V.N., A remark on greedy aroximation in Banach saces, East J. Arox., 5(1999), [9] Koaliani, T., Higher rank Haar avelet bases in L (R) saces, Georgian Math. J., 18(3)(2011), [10] Kozyrev, S.V., Wavelet theory as -adic sectral analysis, zv. Ross. Akad. Nauk Ser. Mat., 66(2)(2002), [11] Kozyrev, S.V., -adic seudodifferential oerators and -adic avelets, Teoret. Mat. Fiz., 138(3)(2004), [12] Resnikoff, H.L., Wells, R.O., Wavelet Analysis, Ne York, Sringer-Verlag, [13] Temlyakov, V.N., The best m term aroximation and greedy algorithms, Adv. Comut. Math., 8(1998), [14] Temlyakov, V., Greedy Aroximation, Cambridge Monograhs on Alied and Comutational Mathematics 20. Cambridge University Press, Cambridge, [15] Wojtaszczyk, P., A mathematical ntroduction to Wavelets, Cambridge University Press, Cambridge, 1997.

7 Greediness of higher rank Haar avelet bases in L (R) saces 219 Ekaterine Kaanadze Tbilisi State University Faculty of Exact and Natural Sciences Chavchavadze St.1, 0128 Tbilisi, Georgia ekaterine kaani@yahoo.com