Joural of Advaced Research i Pure Mathematics Olie ISSN: 1943-2380 Vol. 3, Issue. 1, 2010, pp. 104-110 doi: 10.5373/jarpm.473.061810 O Orlicz N-frames Reu Chugh 1,, Shashak Goel 2 1 Departmet of Mathematics, M.D. Uiversity, Rohtak, Idia. 2 Departmet of Mathematics, B.S.A. Istitute of Techology ad Maagemet, Faridabad (M.D. Uiversity, Rohtak), Idia. Abstract. Frames with respect to some sequece space, amely X d -frames were itroduced ad studied by Casazza et al. [1]. We study frames with respect to Orlicz sequece spaces ad called them Orlicz N-frames. A ecessary ad sufficiet coditio ad also a sufficiet coditio for the stability of Orlicz N-frames have bee give. Also, Orlicz N-Bessel sequeces have bee studied ad a sufficiet coditio, i terms of Orlicz N-Bessel sequeces, for a sequece to be a Orlicz N-frame has bee obtaied. Fially, bouded liear operators o l (N), have bee used to give a ecessary ad sufficiet coditio for a sequece to be a Orlicz N-frame. Keywords: Frames; N-frames; Bessel sequeces. Mathematics Subject Classificatio 2010: 42C15. 1 Itroductio Frames are mai tools for use i sigal processig, image processig, data compressio ad samplig theory etc. Today eve more uses are beig foud for the theory such as optics, filter baks, sigal detectio, as well as the study of Besov spaces, Baach space theory etc. Earlier, Fourier trasform has bee a major tool i aalysis for over a cetury. It has a serious lackig for sigal aalysis i which it hides i its phases iformatio cocerig the momet of emissio ad duratio of a sigal. What was eeded was a localized time-frequecy represetatio which has this iformatio ecoded i it. I 1946, Deis Gabor filled this gap ad formulated a fudametal approach to sigal decompositio i terms of elemetary sigals. O the basis of this developmet, i 1952, Duffi ad Schaeffer [5] defied frames for Hilbert spaces to study some problems i oharmoic Fourier series. The idea of Correspodece to: Reu Chugh, Departmet of Mathematics, M.D. Uiversity, Rohtak, Idia. Email: chughreu@yahoo.com Received: 18 Jue 2010, accepted: 21 September 2010. http://www.i-asr.com/jourals/jarpm/ 104 c 2011 Istitute of Advaced Scietific Research
Reu Chugh ad Shashak Goel 105 Duffi ad Schaeffer did ot geerate much geeral iterest outside of o harmoic Fourier series. But after the ladmark paper of Daubechies, Grossma ad Meyer [4], i 1986, the theory of frames bega to be more widely studied. Coifma ad Weiss [3] itroduced the otio of atomic decompositio for fuctio spaces. Feichtiger ad Gröcheig [6] exteded the otio of atomic decompositio to Baach spaces. Gröcheig [7] itroduced a more geeral cocept for Baach spaces called Baach frame. Baach frames were further studied i [2, 8]. Frames with respect to some sequece space X d, amely, X d -frames were itroduced ad studied by Casazza, Christese ad Stoeva [1]. I the preset paper, we study frames with repsect to Orlicz sequece spaces ad obtai a ecessary ad sufficiet coditio. Also, a sufficiet coditio for the stability of Orlicz N-frames has bee give. Further, Orlicz N-Bessel sequece has bee studied ad a sufficiet coditio, i terms of Orlicz N-Bessel sequece, for a sequece to be a Orlicz N-frame has bee give. Fially, bouded liear operators o l (N), have bee used to give a ecessary ad sufficiet coditio for a sequece to be a Orlicz N-frame. 2 Prelimiaries Throughout this paper E will deote a Baach space over the scalar field K(R or C), E, the first cojugate space of E, E d a associated Baach space of scalar-valued sequeces idexed by N. A Orlicz fuctio N is a cotiuous, o-decreasig ad covex fuctio defied for t 0 such that N(0) = 0 ad lim t N(t) =. If N(t) = 0 for some t > 0, the N is said to be a degeerate Orlicz fuctio. To ay Orlicz fuctio N, we ca associate the space give by l (N) = x = x i } : ( ) } xi N <, for some α > 0 α i The space l (N) with orm x i } = if α > 0 : ( ) } x N 1 is a Baach space, α called Orlicz sequece space. The space h (N) = x = x i } : ( ) } x N <, for all α > 0 α is a subspace of l (N). Further, the space l (N) with each of the followig orms give by (i) x } N = if α > 0 : ( ) } x N < α
106 O Orlicz N-frames (ii) x } N = sup x y : N( y ) 1 } is a Baach space. For more basic cocepts regardig Orlicz sequece spaces, oe may refer to [9]. 3 Mai Results We begi this sectio with the followig defiitio of Orlicz N-frames i which the coefficiets of the buildig blocks are take from the Orlicz sequece space l N. Defiitio 3.1. A sequece f } E is called Orlicz N-frame for E, if (i) f (x)} l (N), x E (ii) there exists positive costats A ad B with 0 < A B < such that A x E f (x)} l(n) B x E, x E (3.1) The positive costats A ad B are called bouds for the Orlicz N-frame f }. Let U : E l (N) ad T : l (N) E be defied by U(x) = f (x)}, x E T (c }) = c i f i, c } l (N) The operator U is called aalysis operator ad T is called sythesis operator. Regardig stability of Orlicz N-frames, we have the followig result i=1 Theorem 3.1. Let f } E be a Orlicz N-frame for E ad let g } E be such that g (x)} l (N) for all x E. The, g } is a Orlicz N-frame for E if ad oly if there exist a costat K > 0 such that (f g )(x)} l(n) K mi f (x)} l(n), g (x)} l(n) }, x E Proof. Let A f, B f ; A g, B g, respectively be frame bouds for the Orlicz N-frames f } ad g }. The, by usig frame iequalities for these frames, we get ( (f g )(x)} l(n) 1 + B ) g f (x)} l(n), x E A f Similarly, we obtai Choose K = (f g )(x)} l(n) ( 1 + B f A g ( 1 + B ) ( g or 1 + B ) f accordig as A f A g ) g (x)} l(n), x E
Reu Chugh ad Shashak Goel 107 mi f (x)} l(n), g (x)} l(n) } is f (x)} l(n) or g (x)} l(n). Coversely, let C f ad D f are bouds for the Orlicz N-frame f } i E. The, for all x E, we have Therefore, C f x E f (x)} l(n) (f g )(x)} l(n) + g (x)} l(n) (1 + K) g (x)} l(n) (1 + K)( (f g )(x)} l(n) + f (x)} l(n) ) (1 + K) 2 f (x)} l(n) (1 + K) 2 D f x E C f (1 + K) x E g (x)} l(n) D f (1 + K) x E, x E Hece, g } is a Orlicz N-frame for E with bouds C f (1 + K) ad D f (1 + K). I the followig theorem, we give a sufficiet coditio for a sequece to be a Orlicz N-frame for E. Theorem 3.2. Let f } E be a Orlicz N-frame for E with bouds A ad B. Let g } E be such that g (x)} l (N), x E. If there exist o-egative costats λ, µ, ν ad ξ such that maxλ, µ, ν, ξ} (i) (1 < A, where maxλ, µ, ν, ξ} < 1. maxλ, µ, ν, ξ}) (ii) (f g )(x)} 2 l (N) λ f (x)} 2 l (N) + 2µ f (x)} l(n) g (x)} l(n) + ν g (x)} 2 l (N) + ξ x 2 E, x E the, g } is a Orlicz N-frame for E with bouds A ( maxλ, µ, ν, ξ})(1 + A) 1 + ad B + ( maxλ, µ, ν, ξ})(1 + B) maxλ, µ, ν, ξ} 1 maxλ, µ, ν, ξ} Proof. Let η = maxλ, µ, ν, ξ}. The (ii) may be restated as: (f g )(x)} l(n) η f (x)} l(n) + g (x)} l(n), + x E }, x E (3.2) Now, g (x)} l(n) f (x)} l(n) + (f g )(x)} l(n) f (x)} l(n) + η f (x)} l(n) + g (x)} l(n) + x E } (usig (3.2))
108 O Orlicz N-frames This gives, (1 η) g (x)} l(n) [(1 + η)b + ] x E, x E Similarly, (1 + η) g (x)} l(n) [(1 η)a ] x E, x E Therefore, [(1 η)a η 1 + η ] x E g (x)} l(n) [ (1 + η)b + η (1 η) ] x E, Hece, g } is a Orlicz N-frame for E ad with desired bouds. x E Defiitio 3.2. A sequece f } E is said to be a Orlicz N-Bessel sequece if it satisfy upper Orlicz N-frame iequality (3.1). Regardig Orlicz N-Bessel sequeces, we have the followig observatios: (I) A Orlicz N-frame for E is a Orlicz N-Bessel sequeces. The coverse eed ot be true. Ideed, let N(t) = t ad let f } E is defied by f 1 (x) = 0, f (x) = ξ, > 1, x = ξ } E. The, f } is a Orlicz N-Bessel sequece. But f } is ot a Orlicz N-frame. (II) A sequece f } E is a Orlicz N-Bessel sequece if ad oly if the coefficiet mappig T : E l (N) give by T (x) = f (x)}, x E is bouded. (III) For each i 1, 2,..., k}, if f i, } is a Orlicz N-Bessel sequece with boud M i, the [ k i=1 λ if i, ] is a Orlicz N-Bessel sequece with boud k i=1 λ i M i, where λ i s are ay scalars. The ext result also gives a sufficiet coditio for a give sequece to be o Orlicz N-frame. Theorem 3.3. Let f } E be a Orlicz N-frame for E ad with bouds A, B. Let g } E be such that g (x)} l (N), x E ad let f + g } be a Orlicz N-Bessel sequece for E ad with boud K < A. The g } is a Orlicz N-frame for E with bouds A K ad B + K. Proof. I view of the frame iequality for the Orlicz N-frame f } ad the fact that K is a Orlicz N-Bessel boud of the Orlicz N-Bessel sequece f + g }, we have (A K) x E f (x)} l(n) (f + g )(x)} l(n) g (x)} l(n) f (x)} l(n) + (f + g )(x)} l(n) (B + K) x E, x E Hece, g } is a Orlicz N-frame for E ad with desired frame bouds.
Reu Chugh ad Shashak Goel 109 Remark 3.1. Towards the coverse of Theorem 3.3, we observe that if f } ad g } are Orlicz N-frames for E, the f +g } is a Orlicz N-Bessel sequece for E. Ifact, if f } ad g } are Orlicz N-frames for E ad W : l (N) l (N) is a liear homeomorphism such that W (f (x)}) = g (x)}, x E, the f + g } is a Orlicz N-Bessel sequece for E with bouds K = mi T I + W, V I + W 1 }, where T, V : E l (N) are the coefficiet mappig give by T (x) = f (x)}, V (x) = g (x)}, x E ad I deotig a idetity map o l (N). Let f } E be a Orlicz N-frame for E. Let g } E be such that g (x)} l (N), x E. Let W : l (N) l (N) be a bouded liear operator such that W (f (x)}) = g (x)}, x E. The g } i geeral, ot a Orlicz N-frame for E. Ideed, let E = l 1, ad let N-fuctio be give by N(t) = t. Defie f } E be give by f (x) = ξ, x = ξ } E. The, x E = f (x)} l(n), x E. Defie W : l (N) l (N) by W (f (x)}) = g (x)}, x E, where g = f +1, N. The, g } is ot a Orlicz N-frame for E. I view of the above discussio, we obtai a ecessary ad sufficiet coditio for a sequece g } i E to be a Orlicz N-frame for E. Theorem 3.4. Let f } E be a Orlicz N-frame. Let g } E be such that g (x)} l (N) ad let W : l (N) l (N) be a bouded liear operator such that W (f (x)}) = g (x)}, x E. The g } is a Orlicz N-frame for E if ad oly if W (f (x)}) l(n) K L(g (x)}) l(n), x E where K is a positive costat ad L : l (N) l (N) is a bouded liear operator such that L(g (x)}) = f (x)}, x E. Proof. Let A f ad B f be the frame bouds for the Orlicz N-frame f }. The, we have Therefore KA f x E K f (x)} l(n) = K L(g (x)}) l(n) W (f (x)}) l(n) (= g (x)} l(n) ) W B f x E, x E KA f x E g (x)} l(n) W B f x E, x E. Hece g } is a Orlicz N-frame for E. Refereces [1] P. Casazza, O. Christese, D.T. Stoeva. Frame expasios i separable Baach spaces. Joural of Mathematical Aalysis ad Applicatios, 2005, 307: 710-723. [2] O. Christese, C. Heil. Perturbatio of Baach frames ad atomic decompositios. Math. Nachr, 1997, 185: 33-47.
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