On Approximative Frames in Hilbert Spaces

Transcription

1 Palestie Joural of Mathematics Vol. 3() (014), Palestie Polytechic Uiversity-PPU 014 O pproximative Frames i Hilbert Spaces S.K. Sharma,. Zothasaga ad S.K. Kaushik Commuicated by kram ldroubi MSC 010 Classificatios: 4C15, 438. Keywords ad phrases: Frame, pproximative Frames. bstract geeralizatio of frames called approximative frames has bee itroduced ad studied. Results o the existece of approximative frames have bee proved. lso, a characterizatio of approximative frame i terms of approximative frame operator is give. Fially, a Paley-Wieer type perturbatio result for approximative frames has bee obtaied. 1 Itroductio Frames for Hilbert spaces were formally itroduced i 195 by Duffi ad Schaeffer [17] who used frames as a tool i the study of o-harmoic Fourier series. Daubechies, Grossma ad Meyer [16], i 1986, reitroduced frames ad observed that frames ca be used to fid series expasios of fuctios i L (R). Frames are geeralizatios of orthoormal bases i Hilbert spaces. The mai property of frames which makes them so useful is their redudacy. Represetatio of sigals usig frames, is advatageous over basis expasios i a variety of practical applicatios. May properties of frames make them very useful i characterizatio of fuctio spaces ad other fields of applicatios such as sigal ad image processig [8], filter bak theory [7], wireless commuicatios [19] ad sigma - delta quatizatio [6]. For more literature o frame theory, oe may refer to [5, 9, 10, 14]. Recetly, may geeralizatios of frames have bee itroduced ad studied. Casazza ad Kutyiok [1] itroduced the otio of frames of subspaces or fusio frames to have may more applicatios of frames i sesor etworks ad packet ecodig. Li ad Ogawa [1] itroduced the otio of pseudo-frames i Hilbert spaces usig two Bessel sequeces. Christese ad Eldar [15] gave aother geeralizatio of frames called oblique frames. Later o, Su [] itroduced a more geeral cocept, called geeralized frames or g frames for Hilbert spaces ad proved that oblique frames, pseudo frames ad fusio frames are special cases of g frames. Fusio frames ad g frames are also studied i [1, 3, 0, ]. Very recetly, Gǎvrua [18] itroduced aother geeralizatio of frames called frames for operators (K-frames). ldroubi [] itroduced two methods for geeratig frames for a Hilbert space H. The first method uses bouded operators o H ad the other oe uses bouded liear operators o l to geerate frames for H. He also characterized all the mappigs that trasform frames ito other frames. Recetly, i various areas of applied mathematics, computer sciece, ad electrical egieerig, sparsity has played a key role. Choosig a suitable basis (or a frame), may types of sigals ca be represeted by oly a few o-zero coefficiets usig sparse sigal processig methods. I this paper, we itroduce otio of approximative frame, as aother geeralizatios of frames, which is costructed keepig i mid that it has sparsity i its ature ad with the help of examples it has bee show that the class of approximative frames is larger tha the class of frames i the sese that a o-frame ca be made a approximative frame. lso, we geeralize some basic results o existece ad perturbatio of approximative frames. Preiaries Throughout the paper, H will deote a ifiite dimesioal Hilbert space, {m } a ifiite icreasig sequece i N ad K will deote the scalar field R or C. Defiitio.1. sequece {x } i a Hilbert space H is said to be a frame for H if there exist costats ad B with 0 < B < such that x x, x B x, x H. (.1) =1

2 pproximative Frames 149 The positive costats ad B, respectively, are called lower ad upper frame bouds for the frame {x }. The iequality (.1) is called the frame iequality for the frame {x }. frame {x } is said to be tight if it is possible to choose, B satisfyig iequality (.1) with = B as frame bouds. Parseval if = B = 1 as frame bouds. exact if removal of ay arbitrary x reders the collectio {x } i o loger a frame for H. sequece {x } H is called a Bessel sequece if it satisfies upper frame iequality i (.1). If {x } is a frame for H, the the bouded liear operator T : l (N) H, T ( {α } ) =1 = α x, {α } =1 l (N) is called the pre-frame operator or the sythesis operator. The adjoit operator is give by =1 T : H l (N), T (x) = {x, x } =1, x H T is called the aalysis operator. By composig T ad T, we obtai frame operator S : H H, S(x) = T T (x) = x, x x, x H. Defiitio.. ([14]) The pseudo-iverse of a operator U with closed rage is defied as the uique operator U satisfyig =1 N U = R U, R U = N U ad UU x = x, x R U. 3 pproximative Frames We begi this sectio with the followig defiitio of approximative frame Defiitio 3.1. Let H be a Hilbert space ad {x,i },,,m be a sequece i H, where {m } be a icreasig sequece of positive itegers. The, {x,i },,,m is called a approximative frame for H if there exist positive costats ad B with 0 < B < such that x x, x,i B x, x H. (3.1) The positive costats ad B are called lower ad upper approximative frame bouds. approximative frame {x,i },,,m is said to be tight if we ca choose = B as approximative frame bouds. Parseval if we ca choose = B = 1 as approximative frame bouds. sequece {x,i },,,m of iequality (3.1) is satisfied. is said to be a approximative Bessel sequece if righthad side Regardig the existece of approximative frames, we give the followig examples: Example 3.. Let H be a Hilbert space ad {e } be a orthoormal basis for H. Defie a sequece {x,i },,, i H by { e i, if i = 1,,, x,i = e r, if i > ad r i mod. The, {x,i },,, ad B =. is a approximative frame for H with approximate frame bouds = 1 Next, we give a example of a Parseval approximative frame.

3 150 S.K. Sharma,. Zothasaga ad S.K. Kaushik Example 3.3. Let H be a Hilbert space ad {e } be a orthoormal basis for H. Defie a sequece {x,i },,, i H by x,1 = e 1, x,i+1 = 1 1 i e i, i = 1,,, 1, N, x,i = e i+1, i = 1,,,, N. i + 1 The, {x,i },,, is a Parseval approximative frame for H. Remark 3.4. Oe may observe that if a Hilbert space H has a frame, the it also has a approximative frame. Ideed, if {x } is a frame for Hilbert space H with lower ad upper frame bouds ad B respectively. The for x,i = x i, i = 1,,, ; N, {x,i },,, is a approximative frame for H with the same lower ad upper approximative frame bouds ad B. Ifact, we have x, x,i = x, x i, x H. If we have a Bessel sequece for a give Hilbert space H which does ot satisfy lower frame iequality the i order hadle such sequeces couple of ways are to add more elemets to the sequece. to sparse the elemets of the sequece i such a way that it forces the sequece to satisfy the lower frame iequality ad at the same time keeps the upper frame iequality itact i some sese. s the otio of approximative frames has sparsity i its behavior, due to which it is possible to make a Bessel sequece which is ot a frame for H a approximative frame for H. I the followig examples, we shall demostrate this costructio. Example 3.5. Let H be a Hilbert space ad {e } be a orthoormal basis for H. Recall that {e + e +1 } is a Bessel sequece i H which is ot a frame for H. Defie a sequece {x,i } i=0,1,,, The, we have i H by Hece, {x,i } i=0,1,,, ad B = 5. x,0 = e 1, x,i 1 = e i, i = 1,,,, N, x,i = e i + e i+1, i = 1,,,, N. x x, x,i 5 x, x H. i=0 is a approximative frame for H with approximate frame bouds = 1 Now, we sparse the elemets of a Bessel sequece (costructed by Casazza ad Christese [11]) which is ot a frame for H ad costruct a approximative frame for H. Example 3.6. Let H be a Hilbert space ad {e } be a orthoormal basis for H. Defie a sequece {x } by x = e, N. The it is elemetary to observe that {x } is ot a frame for H. Defie a sequece {x,i } (+1),,, x,1 = e 1, x, = x,3 = e x,4 = x,5 = x,6 = e 3. The, {x,i } (+1),,, = B = 1., 3, i H by x ( 1), +1 = x, ( 1) + = = x, (+1) = e, N. is a tight approximative frame for H with approximative frame boud

4 pproximative Frames 151 Let {m } be a icreasig sequece of positive itegers. Defie the space The, l ( { l 1,,,m ( ) = {α,i },,,m } : α,i K, α,i <. 1,,,m ) is a Hilbert space with the orm iduced by the ier-product give by {α,i },,,m, {β,i },,,m = α,i β,i, {α,i },,,m, {β,i },,,m Next, we give a characterizatio for a approximative Bessel sequece. l 1,,,m ( ). Theorem 3.7. Let {m } be a icreasig sequece of positive itegers ad {x,i },,,m be a sequece i H. The, {x,i },,,m is a approximative Bessel sequece with boud B if ad oly if the operator T {m} : l ( 1,,,m ) H defied by T {m}({α,i },,,m ) = α,i x,i, {α,i },,,m l 1,,,m ( ) is well-defied bouded liear operator from l 1,,,m ( ) ito H with T {m} B. Proof. Let {α,i },,,m m p m q α p,i x p,i Thus l ( 1,,,m ). The, for p, q N, p > q, we have α q,i x q,i = sup g =1 ( m p m q α p,i x p,i α q,i x q,i ), g [ m p m q ] sup α p,i x p,i, g + α q,i x q,i, g g =1 [( m p sup g =1 ( m q [( m p B ) 1 ( m p ) 1 α p,i x p,i, g + ) 1 ( m q ) 1 ] α q,i x q,i, g m α,i x,i exists i H. Therefore T {m} : l ( α p,i ) 1 + ( m q 1,,,m ) H is a well-defied oper- ator. It is easy to verify that T {m} is liear ad for each {α,i },,,m have T {m }({α,i },,,m ) = sup g =1 = sup g =1 Hece T {m} is bouded operator with T {m} < B. ) 1 α q,i ]. T {m}({α,i },,,m m α,i x,i, g ( m ) 1 B α,i B {α,i },,,m Coversely, the adjoit of liear operator T {m} is defied by T{m (x) = {x, x },i},,,m, x H.. l ( ), g 1,,,m ), we

5 15 S.K. Sharma,. Zothasaga ad S.K. Kaushik Note that, x, T {m}({α,i },,,m ) H = x, α,i x,i = α,i x, x,i H = {x, x,i },,,m, {α,i },,,m H Hece x, x,i = T {m } (x) T {m } x, x H. be a approxi- Let {m } be a icreasig sequece of positive itegers ad {x,i },,,m mative frame for H. The the operator T {m} : l ( ) H give by 1,,,m T {m}({α,i },,,m ) = α,i x,i, {α,i },,,m l 1,,,m ( ) is bouded by Theorem 3.7. The operator T {m} is called ({m }-) pre-approximative frame operator or ({m }-) approximative sythesis operator, the adjoit operator T{m is called ({m } }- )approximative aalysis operator. By composig T {m} ad T{m }, we obtai the ({m }-)approximative frame operator S {m} : H H give by S {m}(x) = T {m}t {m } (x) = x, x,i x,i, x H. I the ext result we discuss some properties of ({m })-approximative frame operator Theorem 3.8. Let {m } be a icreasig sequece of positive itegers ad {x,i },,,m be a approximative frame for H with approximative frame bouds ad B ad approximative frame operator S {m}. The S {m} is a bouded, ivertible, self-adjoit ad positive operator. Further, {S 1 {m (x },i)} 1,, m is also a approximative frame for H with approximative frame bouds B 1, 1 ad approximative frame operator S 1 Proof. Clearly S {m} is bouded ad {m }. S {m} = T {m} T {m } T {m } B. Sice S{m = ( } T {m} T{m }) = T{m} T{m = S } {m }, the operator S {m} is self-adjoit. lso S{m}(x), x = x, x,i x,i, x So, by approximative frame iequality (3.1), we have = x, x,i, x H. I(x), x S {m}(x), x B I(x), x, x H, where, I is the idetity operator. This gives I S {m} B I. Thus, S {m} is a positive operator ad 0 I B 1 S {m} B B I

6 pproximative Frames 153 This gives I B 1 S {m} < 1. Hece S {m} is ivertible. Further, ote that x, S 1 {m (x },i) = B S 1 {m } (x) S 1 {m } (x,i), x,i B S 1 {m } x, x H. Sice {S 1 {m (x },i)} 1,, m is a Bessel sequece, the approximative frame operator for {S 1 {m (x },i)} 1,, m is well-defied. lso, we have B 1 I S 1 {m 1 } I. This gives B 1 x x, S 1 {m (x },i) 1 x, x H. Thus, {S 1 {m (x },i)} 1,, m B 1 ad 1. Fially, is a approximative frame for H with approximative frame bouds x, S 1 {m (x },i) S 1 {m (x },i) = S 1 {m } ( S 1 = S 1 {m } S {m } S 1 {m } (x) = S 1 {m } (x). {m } (x), x,i x,i ) Hece the approximative frame operator for approximative frame {S 1 {m (x },i)} 1,, m is S 1 {m. } We call the approximative frame {S 1 {m (x },i)} 1,, m {x,i },,,m. as the dual approximative frame for Corollary 3.9. Let {m } be a icreasig sequece of positive itegers ad {x,i },,,m a approximative frame for H with the frame operator S {m}. The x = x, S 1 {m (x },i) x,i, x H. (3.) be Proof. Follows i view of Theorem 3.8. Next we give a ecessary ad sufficiet coditio for a approximative frame Theorem Let {m } be a icreasig sequece of positive itegers ad {x,i },,,m be a sequece i H. The, {x,i },,,m is a approximative frame for H if ad oly if T {m} : l 1,,,m ( ) H defied by T {m}({α,i },,,m ) = α,i x,i, {α,i },,,m l 1,,,m ( ) is a well-defied bouded liear operator o l 1,,,m ( ) oto H. Proof. By Theorem 3.8, T {m} is a well-defied, bouded liear operator ad S {m}(= T {m} T{m ) } is surjective. Thus, T {m} is surjective. Coversely, suppose that T {m} is a well-defied mappig o l 1,,,m ( ) oto H. For each N, defie a sequece of bouded liear operators T m : l ( ) H by 1,,,m T m ({α,i },,,m ) = α,i x,i, {α,i },,,m l 1,,,m ( ). Clearly T m T {m} poit-wise as. So by Baach-Steihaus Theorem, T {m} is bouded. lso, the adjoit operator of T {m} give by T{m : H } l 1,,,m ( ) is T {m (x) = }

7 154 S.K. Sharma,. Zothasaga ad S.K. Kaushik {x, x,i },,,m, x H. Therefore Let T {m } : H l ( x, x,i = T {m } (x) T {m} x, x H. 1,,,m ) be the pseudo iverse of T {m}. The x = T {m} T {m } (x) = (T {m ) },i(x) x,i, x H. Thus x 4 = (T {m ) },i(x) x,i, x T {m x } x, x,i, x H. Hece, {x,i },,,m is a approximative frame for H. I the ext result, we prove that if a sequece is a approximative frame for a dese subset of a Hilbert space, the it is also a approximative frame for the Hilbert space with the same bouds. Theorem Let {m } be a icreasig sequece of positive itegers, {x,i },,,m sequece i H ad there exist positive costats ad B such that be a x x, x,i B x, for all x i a dese subset V of H. The, {x,i },,,m B. is a approximative frame for H with approximative frame bouds ad Proof. Suppose that there exist x H such that x, x,i > B x. The, there exists some N such that x, x,i > B x Sice V is dese i H, there exist y V such that y, x,i > B y. This a cotradictio. Further, we kow that x T {m } (x), x V. (3.3) Sice T{m } is bouded ad V is dese i H, it follows that (3.3) holds for all x H. Fially, we give a characterizatio of approximative frame i terms of a bouded liear operator.

8 pproximative Frames 155 Theorem 3.1. Let {m } be a icreasig sequece of positive itegers, {x,i },,,m be a approximative frame for H with approximative frame bouds ad B. The {T (x,i )},,,m is a approximative frame for H if ad oly if there exists a positive costat γ such that adjoit operator T satisfies Proof. For each x H, we have T x γ x, x H. lso, γ x T x T x, x,i = x, T (x,i ). T x, x,i B T x, x H. Thus, {T (x,i )} is a approximative frame for H. Coversely, assume that {T (x,i )} is approximative frame for H with approximative lower frame boud. The, for each x H x T x, x,i B T x. Note: I case {m } = {} ad x,i = x i, i = 1,,,, N, Theorem 3.1 reduces to a result due to ldroubi []. The followig result states that if a Hilbert space has a approximative frame, the it also has a Parseval approximative frame. Theorem Let {m } be a icreasig sequece of positive itegers, {x,i },,,m be a approximative frame for H with {m } approximative frame operator S {m} ad S 1/ {m } be the square root of S 1 {m }. The {S 1/ {m x },i} 1,,,m is a Parseval approximative frame. Proof. Straight forward. 4 Perturbatio of pproximative Frames Christese[13] gave a versio of Paley-Wieer Theorem for frames. We give a similar versio for approximative frames. Theorem 4.1. Let {m } be a icreasig sequece of positive itegers ad {x,i },,,m be a approximative frame for H with bouds ad B. ssume that {y,i },,,m be a sequece ( ) i H such that there exist costats λ, µ 0 such that λ + µ < 1 ad m m α,i (x,i y,i ) λ α,i x,i + µ {α,i },,,m for all fiite scalar sequeces {α,i },,,m. (4.1) The, {y,i },,,m is also a approximative frame for H with approximative frame bouds ( ( )) ( ) 1 λ + µ ad B 1 + λ + µ B.,

9 156 S.K. Sharma,. Zothasaga ad S.K. Kaushik Proof. By Theorem 3.7, T {m} B. lso, for all fiite scalar sequeces {α,i },,,m, we have α,i y,i = α,i (x,i y,i ) + α,i x,i Let p, q N with p > q. The (1 + λ) α,i x,i + µ {α,i },,,m. m p m q m p m q α,i y,i α,i y,i (1 + λ) α,i x,i + (1 + λ) α,i x,i Sice {α,i },,,m l ( 1,,,m ) ad + µ {α,i },,,m. m m α,i x,i exists, α,i y,i exists. Thus the ({m }-) pre-approximative frame operator U {m} : l 1,,,m ( m U {m}({α,i },,,m ) = This gives α,i y,i is well-defied. So, we have ) H defied by α,i y,i (1 + λ) α,i x,i + µ {α,i },,,m. U {m }({α,i },,,m ) (1 + λ) T {m }({α,i },,,m ) ( (1 + λ) ) B + µ + µ {α,i },,,m {α,i },,,m. Therefore by Theorem 3.7, {y,i },,,m is a approximative Bessel sequece for H with ( ) boud B 1 + λ + µ B. Defie T {m : H l 1,,,m } ( ) by T {m (x) = T } {m (T } {m }T{m } ) 1 (x) { } = x, T {m}t{m (x },i), x H =1,, m The T {m } (x) = x, ( T {m}t{m }) 1(x,i ) m m Sice α,i x,i ad 1 x, x H. iequality (4.1) holds for all {α,i },,,m l 1,,,m ( ), we have T {m }{α,i },,,m α,i y,i exist for all {α,i },,,m l ( U {m}{α,i },,,m λ T {m}{α,i },,,m l ( 1,,,m ), the 1,,,m ). lso, for each {α,i },,,m + µ {α,i },,,m. (4.)

10 pproximative Frames 157 Note that for each x H, T {m}t {m } (x) = x ad Usig (4.), we have U {m}t {m } (x) = (T {m x) },iy,i = x, (T {m}t{m } ) 1 (x,i ) y,i = T {m } (x). x U {m}t {m (x) λ x + µ T } {m (x) } ( λ + µ ) x, x H. ( ) This gives U {m}t {m 1 + λ + µ }. Sice λ + µ < 1, the operator U {m}t {m is } ivertible ad lso, every x H ca be writte as (U {m}t {m ) 1 } ( 1 λ + µ ). x = U {m}t {m (U } {m }T {m ) 1 } (x) = (U {m}t {m ) 1 } (x), (T {m}t{m } ) 1 (x,i ) y,i. Therefore m x 4 = This gives (U {m}t {m ) 1 } (x), (T {m}t{m } ) 1 (x,i ) 1 (U {m }T {m ) 1 } (x) 1 Hece, {y,i },,,m 1 [ ( )] x 1 λ + µ y,i, x y,i, x is a approximative frame for H. y,i, x, x H. [ ( 1 λ + µ )] x. y,i, x Fially, we give the followig sufficiet coditio for a perturbed sequece to be a approximative frame for H. Theorem 4.. Let {m } be a icreasig sequece of positive itegers ad {x,i },,,m be a approximative frame for H with approximative frame bouds ad B. Let x 0 0 be ay elemet i H ad {α,i },,,m a approximative frame for H if be ay sequece of scalars. The, {x,i + x 0 α,i },, m α,i < x 0. is

11 158 S.K. Sharma,. Zothasaga ad S.K. Kaushik Proof. Let y,i = x,i + α,i x 0, i = 1,,, m, N. The, for ay x H, we have x, x,i y,i = x, α,i x 0 α,i x x 0. Therefore, {x,i + x 0 α,i },, m. is a approximative frame for H if m α,i x 0 < ckowledgemet The authors thak the referee(s) for there valuable suggestios ad critical remarks for the improvemet of the paper. The authors also thak Dr. Vireder, Ramjas College, Uiversity of Delhi, Delhi, INDI for his useful commets ad suggestio o the first draft of this paper. Refereces [1]. bdollahi ad E. Rahimi, Some results o g- frames i Hilbert spaces, Turk. J. Math. 34, (010). []. ldroubi, Portraits of Frames, Proce. mer. Math. Soc., 13, (1995). [3]. Khosravi ad M.M. zadaryai, Fusio Frames ad G-Frames i Tesor Product ad Direct Sum of Hilbert Spaces, pplicable alysis ad Discrete Mathematics, 6, (01). [4] R. Bala, P.G. Casazza ad D. Edidi, O sigal recostructio without phase, ppl. Comp. Harm. al., 0, (006). [5] J. Beedetto ad M. Fickus, Fiite ormalized tight frames, dv. Comp. Math., 18, (003). [6] J. Beedetto,. Powell ad O. Yilmaz, Sigma-Delta quatizatio ad fiite frames, IEEE Tras. Iform. Theory, 5, (006). [7] H. Bolcskel, F. Hlawatsch ad H.G. Feichtiger, Frame - theoretic aalysis of over sampled filter baks, IEEE Tras.Sigal Process., 46, (1998). [8] E.J. Cades ad D.L. Dooho, New tight frames curvelets ad optimal represetatio of objects with piecewise C sigularities, Comm. Pure ad ppl. Math., 56, (004). [9] P.G. Casazza, The art of frame theory, Taiwaese J. of Math., 4(), (000). [10] P.G. Casazza, Custom buildig fiite frames, Wavelet frames ad operator theory, Cotemp. Math., 345, 61-86, mer. Math. Soc. Providece, RI (004). [11] P.G. Casazza ad O. Christese, Frames ad Schauder bases, I pproximatio Theory: I Memory of. K. Verma", , (eds. Govil, N. K., Mohapatra, R. N., Nashed, Z., Sharma,., Szabados, J.). Marcel Dekker, New York, [1] P.G. Casazza ad G. Kutyiok, Frames of subspaces, i Wavelets, Frames ad Operator Theory (College Park, MD, 003), Cotemp. Math., 345, mer. Math. Soc., Providece, RI, (004). [13] O. Christese, Paley-Wieer theorem for frames, Proc. mer. Math. Soc., 13, (1995). [14] O. Christese, itroductio to Frames ad Riesz Bases, Birkhäuser, 003. [15] O. Christese ad Y.C. Eldar, Oblique dual frames with shift-ivariat spaces, ppl. Compt. Harmo. al., 17(1), (004). [16] I. Daubechies,. Grossma ad Y. Meyer, Pailess o-orthogoal expasios, J. Math. Physics, 7, (1986). [17] R.J. Duffi ad.c. Schaeffer, class of o-harmoic Fourier series, Tras. mer. Math. Soc., 7, (195). [18] L. Gǎvrua, Frames for operators, pp. Comput. Har. al., 3(1), (01). [19] R.W. Heath ad.j. Paulraj, Liear dispersio codes for MIMO systems based o frame theory, IEEE Tras. Sigal Process. 50, (00). [0]. Khosravi ad K. Musazadeh, Fusio frames ad g-frames, J. Math. al. ppl., 34, (008). [1] S. Li ad H. Ogawa, Pseudo frames for subspaces with applicatios, J. Fourier al. ppl., 10(4), (004). [] W. Su, G-frames ad g-riesz bases, J. Math. al. ppl., 3(1), (006).

12 pproximative Frames 159 uthor iformatio S.K. Sharma, Departmet of Mathematics, Kirori Mal College, Uiversity of Delhi, Delhi , INDI. Zothasaga, Departmet of Mathematics, Uiversity of Delhi, Delhi , INDI. S.K. Kaushik, Departmet of Mathematics, Kirori Mal College, Uiversity of Delhi, Delhi , INDI. Received: October 0, 013. ccepted: February 10, 014.