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Continuity of Gabor Frame Operators on Weighted Wiener Amalgam Spaces

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Continuity of Gabor Frame Operators on Weighted Wiener Amalgam Spaces International Journal of Mathematics Trends and Technology- Volume 20 Number 1- April 2015 Continuity of Gabor Frame Operators On Weighted Wiener Amalgam Spaces Mr. Vishad Tiwari#1, Dr. J.K. Maitra*2, , Dr. Ashish Kumar#3 #Research Scholar, Department of Mathematics, R.D.University Jabalpur,India. *Reader, Department of Mathematics, R.D.University Jabalpur,India. #Computer Programmer, Department of Mathematics, R.D.University Jabalpur,India. Abstract— We prove eight theorems for the continuity of Gabor In the present paper we prove eight theorems, which provide Frame operators defied on weighted Wiener amalgam spaces, generalizations to the corresponding results of Walnut (loc. which generalize the corresponding results of Walnut (1992) on cit.). The paper is divided into eleven sections. In the first four 푳풑-spaces. sections, we present a number of known definitions and Keywords— Gabor Frames, Gabor expansions, Wiener concepts for use in the sequel. The remaining seven sections Amalgam spaces and time frequency analysis. are devoted to the study of continuing properties of Gabor frame operators and their inverses on suitable Wiener amalgam I. INTRODUCTION spaces. This paper paves the way for the study continuing properties of Gabor frame operators and their inverses on more Denis Gabor, in his famous paper [6] entitled "Theory general modulation spaces. of Communication" initiated the study of a signal representation as a series formed of translations and II. PRELIMINARIES 2/2 modulations of the Gaussian function 푔(푥)휋−1/4푒−푥 . It is well known that the use of this function in time-frequency Let {푓푖 }푖∈퐼 be a collection of vectors in a Hilbert spaces 퐻. analysis is interesting for minimizing uncertainty principle The set {푓푖 }푖∈퐼 is called a frame for the space 퐻 provided inequality. This type of representations, using more general there exist positive constants 퐴 and 퐵 satisfying the window functions 푔, play a vital role in the study of signal condition. 2 2 2 analysis and quantum mechanics both. In the mean time, theory 퐴 ∥ 푓 ∥ ≤ ∑ | < 푓, 푓푖 > | ≤ 퐵 ∥ 푓 ∥ 2.1 of frames was introduced by Duffin and Schaeff [2]and, later 푖∈퐼 on, it was observed that there was a close relationship between for all 푓 ∈ 퐻. wavelet theory and the theory of frames. In particular, the connection between Gabor theory, signal analysis and the The numbers 퐴 and 퐵 known as lower and upper frame theory of frames in Hilbert spaces was studied in much detail b bounds respectively. In case 퐴 = 퐵, then {푓푖 }푖∈퐼 is called a a number of research workers including A.J.E.M. Janssen, I. tight frame. The frame is said to be exact provided it ceases to Daubechies, Christopher Heil, D.F. Walnut and J.J. Bendetto be a frame when even a single element is removed from the (for details and other references [9] and [FG 89]). above collection. A sequence {푓푖}푖∈퐼 satisfying the condition Feichtinger and Gr표̈chenig [FG 89], on the other hand, using a ∑ | < 푓, 푓 > |2 < ∞ more general theory of coorbit spaces, have investigated a very 푖 푖∈퐼 general concept of frames applicable to various types of for all 푓 ∈ 퐻 is called a Bessel sequence. On account of the Banach spaces and Hilbert spaces. Also, it is known that if the uniform boundedness principle, there exists an upper frame window function is well-localized in the time-frequency plane, bound for a Bessel such that then the Gabor representations hold good for general 2 2 modulation spaces too. ∑ | < 푓, 푓푖 > | ≤ 퐵 ∥ 푓 ∥ , ∀푓 ∈ 퐻. 푖∈퐼 Wiener amalgam spaces, as a generalization to function spaces, If {푓푖}푖∈퐼 is a Bessel sequence, then an operator 푇 defined by 2 were introduced by Feichtinger [3]. Recently, these spaces 푇: 푙 (퐼) → 퐻, have been found highly useful for the study of time-frequency analysis (cf. [7] and references there). 푇(푐푖)푖∈퐼 = ∑ 푐푖 푓푖, 푖∈퐼 Walnut [11] has proved a number of interesting results is a bounded and linear operator, which is known as a preframe concerning the contributing of the Gabor frames operators and operator, where 푝 inverse Gabor frames operators on 퐿 -spaces, 1 ≤ 푝 ≤ ∞, 푙2(퐼) = {(푓 ) : ∑ | 푓 |2 < ∞} and on weighted Wiener amalgam spaces. 푖 푖∈퐼 푖 푖∈퐼 ISSN: 2231-5373 http://www.ijmttjournal.org Page 1 International Journal of Mathematics Trends and Technology- Volume 20 Number 1- April 2015 2 It is well known that 푙 (퐼) is a Hilbert space with respect to the ∥ 푓 ∥∞,푤= esssup푥∈푅{|푓(푥)|푤(푥)} < ∞. 푝 inner product Since 푤 is moderate, 퐿푤 is translation invariant (for the proof see [10], pp. 22-23). < {푓푖 }, {푔푖} >= ∑ 푓푖 , 푔푖, 푖∈퐼 Let 푄 be a fixed non-empty compact set in 푅. We denote by 푝 푝 푔푖 being the conjugate of 푔. 푊(퐿 , 퐿푤)(푅), 1 ≤ 푝 ≤ ∞, the Wiener amalgam space under The adjoint operator 푇∗ of 푇 is given by the norm (for details see [3]); ∗ 2 ∗ 푇 : 퐻 → 푙 (퐼), 푇 푓 = {< 푓, 푓푖 >푖∈퐼. ∗ 푝 푝 If 푆 is the composition operator of 푇 and 푇 , then we have ∥ 푓 ∥ 푝 푝 =∥ 푓|푊(퐿 , 퐿 ) ∥=∥∥ 푓 ⋅ 휒 ∥ ∥ 푊(퐿 ,퐿푤) 푤 푄+푥 푝 푝,푤 푆: 퐻 → 퐻, 푆 푓 = 푇푇∗푓. = (∫( ∫| 푓(푦)푝휒 (푦 − 푥)푑푦)1/푝푤푝(푥)푑푥)1/푝, 푄 푅 푅 푖. 푒. 푆푓 = ∑ < 푓, 푓푖 > 푓푖 . 푖∈퐼 where 휒푄 is the characteristic function of 푄. 푆 is known as frame operator. Hence it is clear that ∗ 푝 푝 ∥ 푆 ∥≤∥ 푇푇 ∥ It is known that the definition of the space 푊(퐿 , 퐿푤)(푅) is ≤∥ 푇 ∥∥ 푇∗ ∥ independent of the choice of 푄 (cf. [3]) and , since 푤 is 2 푝 푝 ≤∥ 푇 ∥ = 퐵 moderate 푊(퐿 , 퐿푤)(푅) is translation invariant (for the detail Also, weseethat푆∗ = (푇푇∗)∗ see [Heil 90], p. 33). = 푇푇∗ = 푆 ⇒ 푆 is a self-adjoint operator. IV. GABOR FRAME OPERATORS ON 푳(푹) It is well known that 푆 is invertible and satisfies the condition 퐵−1퐼 ≤ 푆−1 ≤≤ 퐴−1퐼, Gabor expansion of a function 푓 ∈ 퐿2(푅) in terms of a single 퐼 being the identity operator. generating function 푔 ∈ 퐿2(푅) by translations (time-shifts) −1 Also, it is known that {푆 푓푖} is a frame for the Hilbert space and modulation (frequency-shifts) of 푔 is given by 퐻, which is known as the dual frame of {푓푖 }푖∈퐼. ∑ 푓(푡) = 푐푚,푛 푔푚,푛(푡), (4.1) By virtue of the above definitions, we see that the function 푓 푚,푛∈푍 can be recovered in the form where 푓 = 푆푆−1푓 2휋푖푚푏푡 푔푚,푛(푡) = 푔(푡 − 푛푎)푒 −1 = ∑ < 푓, 푆 푓푖 > 푓푖, ∀ 푓 ∈ 퐻, = 푇푛푎푀푚푏푔, 푖∈퐼 푍 being the set of all integers. oras The Gabor coefficients 푐푚,푛 =< 푓, 푔푚,푛 > for all 푚, 푛 ∈ 푍 −1 푓 = 푆푆 푓 are given the analysis mapping 푇푔 such that −1 푇 : 푓 → {< 푓, 푔 > . = ∑ < 푓, 푓푖 > 푆 푓푖, ∀ 푓 ∈ 퐻, 푔 푚,푛 푚,푛 ∗ 푖∈퐼 The adjoint 푇푔 of 푇푔 is given by the sums in the above expression being convergent in the space 푇∗: {푐 → ∑ 푐 푔 . 퐻. 푔 푚,푛 푚,푛 푚,푛 푚,푛 Daubechies, on the other hand, has shown that (cd., [D 90]) if ∗ It is well known that 푇푔 and 푇 both are bounded linear {푓푖}푖∈퐼 is a frame for 퐻 with the frame operators 푆 , then −1/2 operators. {푆 푓푖}푖∈퐼 is a tight frame found 1. ∗ Hence the Gabor frame operator 푆 ≡ 푆푔 = 푇 ∘ 푇 is given by III. WEIGHTED WIENER AMALGAMS ON 푹 푆푓 = ∑ < 푓, 푔푚,푛 > 푔푚,푛. 푚,푛∈푍 + Let 푤: 푅 → 푅 be a function such that (i) 푤(0) = 1. Daubechies [D 90] has shown that the Gabor frame (ii) 푤(푥 + 푦) ≤ 푤(푥) 푤(푦) for all 푥, 푦 ∈ 푅 2 {푔푚,푛}푚,푛∈푍 for 퐿 (푅) exists and its dual frame {푔̃푚,푛}푚,푛∈푍 Then 푤 is called a submultiplication weight function on 푅. also is a Gabor frame, where 푔̃ = 푆−1푔 . Walnut [Wal 92, A locally integrable function 푤: 푅 → 푅+ is known as 푚,푛 pp. 499-500] has proved that {푆−/1푔 } frame with moderate weight function if there exists a submultiplictive 푚,푛 푚,푛∈푍 bound 1. weight satisfying the condition: 푤(푥 + 푦) ≤ 푤(푥) 푤(푦); ∀푥, 푦 ∈ 푅. V. CONTINUITY OF GABOR FRAME OPERATORS. Throughout this paper we assume that 푤 is a moderate weight function on 푅. In a recent paper, Walnut [Wal 92, Theorem 3.1] has 푝 ∞ 1 We denote by 퐿푤(푅), 1 ≤ 푝 ≤ ∞, the Banach space of all proved that if 푔 ∈ 푊 = 푊(퐿 , 퐿 )(푅); 푎, 푝 > 0 and complex-valued functions on 푅 under the norm 1 ≤ 푝 ≤ ∞, then there exists a constant 퐶 depending on 푔, 푎 ∞ 푝 푝 푝 1/푝 and 푏 such that for all 푓 ∈ 퐶푐 (푅) ∥ 푓 ∥푝,푤=∥ 푓|퐿푤 ∥= (∫| 푓(푥) 푤 (푥)푑푥) < ∞. 푅 ∥ 푆푓 ∥푝≤ 퐶 ∥ 푓 ∥푝 and In case 푝 = ∞, we denote by 퐿∞ (푅) the Banach space of all 푤 ∥ 푆푓 ∥푊(퐿∞,퐿1)≤ 퐶 ∥ 푓 ∥푊(퐿∞,퐿1), measurable functions 푓 with respect to the norm ∞ where 퐶푐 is the space of functions with compact support 퐶. ISSN: 2231-5373 http://www.ijmttjournal.org Page 2 International Journal of Mathematics Trends and Technology- Volume 20 Number 1- April 2015 In part (b) of the above theorem, Walnut has proved amalgams Since 푅̂ is isometric to 푅, we shall write 푅 in space of 푅̂.
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