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 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. 푝 1/푝 푝 1/푝 = (∫( ∫| 푓(푦) 휒푄(푦 − 푥)푑푦) 푤 (푥)푑푥) , 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 {푓 } . 푖 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 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. results for the weighted norms ∥ 푓 ∥푝,푤푝 and ∥ 푓 ∥푊,푤푠 , In this section we prove the following theorem, which provide 푠 a generalization of the corresponding results of Walnut (cf. where 푤푠 denotes the weight function (1 + |푥|) or 푒푠|푥|, 푠 > 0. [11], Theorem 3.2 (a) and (b)) as in the case of Theorem 5.1: In this section our aim is to generalize the above results of ∞ 1 Walnut for the Wiener amalgams norms ∥ 푓 ∥ 푝 푝 , 푤 Theorem 6.1: If 푔̂ ∈ 푊(퐿 , 퐿 )(푅); 푎, 푏, > 0 and 1 ≤ 푝 ≤ 푊(퐿 ,퐿푤) ∞, then there exists a constant 퐶 depending only on 푔, 푎 and being a moderate weight function on 푅. Precisely, we prove 푏 such that the following ∧ ∥ (푆푓) ∥ 푝 푝 ≤ 퐶 ∥ 푓̂ ∥ 푝 푝 푊(퐿 퐿푤) 푊(퐿 퐿푤) ∞ 1 ̂ ∞ Theorem 5.1 If 푔 ∈ 푊퐿 , 퐿 (푅), 푎, 푏 > 0 and 1 ≤ 푝 ≤ ∞, for all 푓 ∈ 퐶푐 (푅). then there exists a constant 퐶 depending only on 푔 푎 and 푏 We shall use the following lemma in the proof of our theorem : such that Lemma 6.2. If 푔̂ ∈ 푊(퐿∞, 퐿1)(푅) and 푎, 푏 > 0, then ∥ 푆푓 ∥ 푝 푝 ≤ 퐶(푏, 푤) ∥ 푓 ∥ 푝 푝 (5.1) ∧ −1 ̂ 푊(퐿 ,퐿푤) 푊(퐿 ,퐿푤) (푆푓) = 푎 ∑ 푇푗/푎 푓 ⋅ 퐺̃푗, ∞ for all 푓 ∈ 퐶푐 (푅). 푗∈푍 It may be mentioned here that in case 푤 = 1 our result in (5.1) 푤ℎ푒푟푒 reduce to (3.1.1) of walnut, while for the local norm ∥ 푓 ∥∞ 퐺̃푗 = ∑ 푇푚푏 (푔̂푇푗/푎푔̂). and global norm ∥ 푓 ∥1 the result (3.1.2) of Walnut becomes a 푚∈푍 particular case of the result in (5.1). Also, it is clear that for Also, we have 푤 = 푤 the part (b) of the Theorem 3.1 (b) of Walnut is a 푠 ̃ particulars case of our results in Theorem 5.1. ∑ 푤 (푗/푎) ∥ 퐺푗 ∥∞< ∞. We shall use the following lemma in the proof of our theorem: 푗∈푍 ∞ 1 for 푔̂푤 ∈ 푊(퐿 , 퐿 )(푅). Lemma 5.2. If 푔 ∈ 푊(퐿∞, 퐿1)(푅) and 푎, 푏 > 0, then for all The proof follows on the lines of Walnut [Wal 92, Lemma 2.3]. 푓 ∈ 퐿2(푅), we have −1 Proof of Theorem 6.1. On account of the relation (6.2) and 푆푓 = 푏 ∑ 푇푗/푏 푓 ⋅ 퐺푗, ∞ 2 denseness of the space 퐶푐 (푅) in 퐿 (푅), we obtain 푗∈푍 ∧ −1 ̂ ∥ (푆푓) ∥ 푝 푝 ≤ 푎 ∑ ∥ 푇푗/푎푓 ⋅ 퐺̃ ∥ 푝 푝 , where 퐺푗 is the discrete auto-correlation function given by 푊(퐿 퐿푤) 푊(퐿 퐿푤) 푗∈푍 ∑ 퐺푗 = 푇푛푎 (푔푇푗/푏푔̄), −1 ≤ 푎 ∥ 푓̂ ∥ 푝 푝 ∑ 푤 (푗/푎) ∥ 퐺̃ ∥ 푛∈푍 푊(퐿 퐿푤) 푗 ∞ 푔̄ being the conjugate function of 푔. 푗∈푍 ∑ ∞ 1 = 퐶 ∥ 푓̂ ∥ 푝 푝 Also, we have 푗∈푍 푤 (푗/푏) ∥ 퐺푗 ∥∞< ∞ for 푔푤 ∈ 푊(퐿 , 퐿 ). 푊(퐿 퐿푤) For the proof see [Wal 90, Lemma 2.1 and 2.2]. 푓표푟 −1 ̃ ∞ 2 퐶 = 푎 ∑ 푤 (푗/푎) ∥ 퐺푗 ∥∞. Proof of Theorem 5.1. Since 퐶푐 (푅) ⊂ 퐿 (푅), the result in ∞ 푗∈푍 (5.2) holds true for all 푓 ∈ 퐶푐 (푅). Thus we see that This complete the proof of the theorem. −1 ∥ 푆푓 ∥ 푝 푝 ≤ 푏 ∑ ∥ 푇 푓 ⋅ 퐺 ∥ 푝 푝 푊(퐿 ,퐿푤) 푗/푏 푗 푊(퐿 ,퐿푤) 푗∈푍 VII. APPROXIMATION OF FUNCTIONS BY THEIR −1 GABOR EXPANSIONS. ≤ 푏 ∑ ∥ 퐺 ∥ ∥ 푇 푓 ∥ 푝 푝 푗 ∞ 푗/푏 푊(퐿 ,퐿푤) 푗∈푍 Heil and Walnut [9] have shown that a necessary −1 ≤∥ 푓 ∥ 푝 푝 푏 ∑ ∥ 퐺 ∥ ⋅ 푤(푗/푏). condition for the generation of a frame by {푔푚,푛}푚,푛 ∈ 푍 is 푊(퐿 ,퐿푤) 푗 ∞ 푗∈푍 퐴 ≤ ∑ | 푇 푔(푥)|2 ≤ 퐵 foralmostall푥 ∈ 푅, Walnut (loc. cit., Lemma 2.2) has shown that the series on the 푛푎 푛∈푍 right-hand side is convergent. Hence, writing where 퐴, 퐵 > 0 are some constants. −1 ∞ 1 퐶(푏, 푤) = 푏 ∑ ∥ 퐺푗 ∥∞⋅ 푤(푗/푏). Using an analogous for 푔 ∈ 푊(퐿 , 퐿 )(푅), Walnut has proved 푗∈푍 that The proof of the theorem follows: 2푏 퐵 − 퐴 + 2퐶(푏) ‖푓 − 푆푓‖ ≤ ‖푓‖푝, 1 ≤ 푝 ≤ ∞, In case 푝 = ∞, the proof follows in a similar way. 퐵 + 퐴 푝 퐵 + 퐴 where 푏 > 0 and 퐶(푏) is a constant depending on 푔 and 푎. VI. CONTINUITY IF THE FOURIER TRANSFORM OF 푺풇. He has obtained three other similar approximations in the Let 푓 be the Fourier transform operator on the space 푝 spaces 푊(퐿∞, 퐿1)(푅), 퐿 (푅) and weighted 푊(퐿∞, 퐿1)(푅) of tempered distribution 푆′(푅) and 퐹푓 = 푓̂. We denote by 푤푠 with the weight function 푤 . 푝 the space of all tempered distributions such that 푠 퐹퐿 (푅) 푓 푝 ̂ ̂ 푎̉ 푡푓 ∈ 퐿 (푅), 푅 being the dual group of 푅. We define the In this section our aim is to generalize the above mentioned norm on 퐹퐿푝(푅) by 푝 result of Walnut in the Wiener amalgam space 푊(퐿푝, 퐿 )(푅). ̂ 푤 ∥ 푓 ∥퐹퐿푝=∥ 푓 ∥푝. Precisely, we prove the following:

ISSN: 2231-5373 http://www.ijmttjournal.org Page 3

International Journal of Mathematics Trends and Technology- Volume 20 Number 1- April 2015

there exists a 푏0 > 0 depending only on 푔 and 휂 such that

∑ 푤 (푗/푏) ∥ 퐺 ∥ < 푎−1휂 Theorem 7.1 If 푔 ∈ 푊(퐿∞, 퐿1)(푅) satisfies the condition 푗 ∞ 푗≠0 퐴 ≤ ∑ | 푇 푔(푥)|2 ≤ 퐵 a. e. 푥, 푎 > 0, 푗∈푍 푛푎 (7.1) for every 0 ≤ 푏 ≤ 푏 and every 0 < 푎 ≤ 1. 푛∈푍 0 then for each 푏 > 0, there exists a constant 퐶(푏) depending Now, choosing 휂 → 0 as 푏 → 0,, the result in (7.2) follows. on 푔 and 푎 such that 2푏 VIII. ‖푓 − 푆푓‖ 퐵 + 퐴 푝 푝 푊(퐿 ,퐿푤) In this section we prove the following theorem, which 퐵 − 퐴 + 2퐶(푏, 푤) (7.2) generalize all the four results of Walunt (cf. [11], p.499): ≤ ‖푓‖ 푝 푝 , 1 ≤ 푝 퐵 + 퐴 푊(퐿 ,퐿푤) ≤ ∞, Theorem 8.1 If 푔̂ = 푊(퐿∞, 퐿1)(푅), 푏 < 0 and there exist where some constants 퐴′, 퐵′ > 0 satisfying the condition lim 퐶 (푏, 푤) = 0 (7.3) ′ 2 ′ 푏→0 퐴 ≤ ∑ | 푇푚푏푔̂(휉)| ≤ 퐵 a. e. 휉, (8.1) 푚∈푍 It may be mentioned here that all the four results in Theorem then, for each 푎 > 0, there exists a constant 퐶̃(푎) depending 3.3 of Walnut [11] are particular cases of our theorem for on 푔 and 푏 such that special values of 푝 and 푤 = 푤푠. The condition (7.1) ensure 2푎 ∧ that 푆푓 is well defined (cd [loc. cit., p-495). ‖(푓 − ′ ′ 푆푓) ‖ 퐵 + 퐴 푝 푝 푊(퐿 ,퐿푤) (8.2) 퐵′ − 퐴′ + 2퐶̃(푎) Proof of the Theorem Using (5.2), we have ̂ ≤ ′ ′ ‖푓‖ 푝 푝 , 2푏 퐵 + 퐴 푊(퐿 ,퐿푤) ‖푓 − 푆푓‖ 퐵 + 퐴 푝 푝 and 푊(퐿 ,퐿푤) lim 퐶̃ (푎, 푤) = 0. (8.3) 푎→0 2 Proof of the Theorem: Using Lemma 6.2, we obtain = ‖푓 − 퐺 ⋅ 푓 퐵 + 퐴 0 2푎 ∧ ‖(푓 − ′ ′ 푆푓) ‖ 퐵 + 퐴 푝 푝 푊(퐿 ,퐿푤) 2 − ∑ 푇푗/푏 푓퐺푗‖ 2 퐵 + 퐴 ‖ ̂ ̃ 푗≠0 = ‖푓 − ′ ′ 퐺0 ⋅ 푓 푗∈푍 푝 푝 퐵 + 퐴 푊(퐿 ,퐿푤) 2 ≤ ‖(1 − 퐺 ) 푓‖ 0 푝 퐵 + 퐴 푊(퐿푝,퐿 ) 2 푤 − ∑ 푇 푓̂퐺̃ ‖ 퐵′ + 퐴′ 푗/푏 푗 푗≠0 2 푝 + ‖ ∑ 푇 푓퐺 ‖ 푗∈푍 푊(퐿푝,퐿 ) 퐵 + 퐴 푗/푏 푗 푤 푗≠0 2 푗∈푍 푝 ≤ ‖(1 − 퐺̃ ) 푓̂‖ 푊(퐿푝,퐿 ) ′ ′ 0 푝 푤 퐵 + 퐴 푊(퐿푝,퐿 ) 2 2 푤 ≤ ‖1 − 퐺0‖ ∥ 푓 ∥ 푝 푝 + 퐵 + 퐴 푊(퐿 ,퐿푤) 퐵 + 퐴 2 ∞ ̂ ̃ 푝 + ‖ ∑ 푇푗/푏 푓퐺푗‖ ∥ 푓 ∥푊(퐿푝,퐿 ) 퐵′ + 퐴′ 푤 푗≠0 푗∈푍 푝 푝 ⋅ ∑ ∥ 퐺푗 ∥∞ 푤(푗/푏) 푊(퐿 ,퐿푤) 푗≠0 2 푗∈푍 ≤ ‖1 − ′ ′ 퐺0‖ 2 2퐵 퐵 + 퐴 ∞ ≤ max { 1 − , − 1} ∥ 푓 ∥ 푝 푝 2 푊(퐿 ,퐿 ) ̂ 푝 ̂ 푝 퐵 + 퐴 퐵 + 퐴 푤 ∥ 푓 ∥푊(퐿푝,퐿 )+ ′ ′ ∥ 푓 ∥푊(퐿푝,퐿 ) 2퐵 푤 퐵 + 퐴 푤 + ∥ 푓 ∥ 푝 푝 ∑ 푤 (푗/푏) ∥ 퐺푗 ∥∞. ̃ 퐵 + 퐴 푊(퐿 ,퐿푤) ⋅ ∑ 푤 (푗/푏) ∥ 퐺푗 ∥∞ 푗≠0 푗≠0 푗∈푍 푗∈푍 퐵 − 퐴 2퐶(푏) ′ ′ 푝 푝 2퐴 2퐵 = ∥ 푓 ∥푊(퐿푝,퐿 )+ ∥ 푓 ∥푊(퐿푝,퐿 ) 퐵 + 퐴 푤 퐵 + 퐴 푤 ≤ max { 1 − ′ ′ , ′ ′ 퐵 − 퐴 + 2퐶(푏, 푤) 퐵 + 퐴 퐵 + 퐴 ̂ 푝 = ∥ 푓 ∥ 푝 푝 , − 1} ∥ 푓 ∥푊(퐿푝,퐿 ) 퐵 + 퐴 푊(퐿 ,퐿푤) 푤 2 푤ℎ푒푟푒 + 퐵′ + 퐴′ 퐶(푏, 푤) = ∑ ∥ 퐺푗 ∥∞ 푤(푗/푏). ∥ 푓̂ ∥ 푝 푝 ∑ 푤 (푗/푏) ∥ 퐺̃ ∥ . 푗≠0 푊(퐿 ,퐿푤) 푗 ∞ 푗∈푍 푗≠0 Walnut [loc. cit., p. 486] has shown that for any given 휂 > 0, 푗∈푍

ISSN: 2231-5373 http://www.ijmttjournal.org Page 4

International Journal of Mathematics Trends and Technology- Volume 20 Number 1- April 2015

퐵′ − 퐴′ + 2퐶̃(푎, 푤) of the theorem is complete. = 퐵′ + 퐴′ ′ ∥ 푓 ∥ 푝 푝 , Proof of theorem 9.2 Let 퐶̃(푎, 푤) < 퐴 in Theorem 8.1. Then 푊(퐿 ,퐿푤) 푤ℎ푒푟푒 we have 퐵′ − 퐴′ + 2퐶̃(푏, 푤) 퐶̃(푎, 푤) = ∑ 푤 (푗/푏) ∥ 퐺̃ ∥ . < 1. 푗 ∞ 퐵′ + 퐴′ 푗≠0 ̂ ∞ ̂ 푗∈푍 Hence, for 푓 ∈ 퐶푐 (푅), we obtain Also, as in Lemma 2.3 of Walnut (loc. cit.), for a given 휂 > 0, 2푏 ∧ ‖[(퐼 − 푆) 푓] ‖ ≤∥ 푓̂ ∥ 푝 푝 . ′ ′ 푊(퐿 ,퐿푤) there exists 푎0 > 0 depending only on 푔 and 휂 such that for 퐵 + 퐴 푝 푝 푊(퐿 ,퐿푤 every 푎 ≤ 푎 ≤ 푎0 and every 푎 < 푏 ≤ 1, we have 2푏 ⇒ ‖퐼 − 푆‖ < 1. ⇒ 푆 is bounded and invertible on ∑ 푤 (푗/푏) ∥ 퐺̃ ∥ < 푏−1휂. 퐵′+퐴′ 푗 ∞ 푝 푝 ̂ 푗≠0 푊(퐿 , 퐿푤)(푅). 푗∈푍 Hence, as in the proof of the Theorem 9.1, we have ∞ Now, choosing 휂 → 0 as 푎 → 0, we get 2푎 2푎 ̃ −1 ̂ 푝 ̂ lim 퐶 (푎, 푤) = 0. ∥ 푆 푓 ∥푊(퐿푝,퐿 )= ‖ ∑ (퐼 − 푆) 푓‖ 푎→0 푤 퐵′ + 퐴′ 퐵′ + 퐴′ 푘=0 푝 푝 Thus the theorem holds true. 푊(퐿 ,퐿푤 ≤ 퐶 ∞ IX. CONTINUITY OF INVERSE GABOR FRAME 2푎 2푎 ∥ 푓̂ ∥ 푝 푝 ‖ ∑ (퐼 − )‖ OPERATOR 푊(퐿 ,퐿푤 퐵′ + 퐴′ 퐵′ + 퐴′ 푘=0 ∞ In this section our objective is to prove two theorems 퐶 being a positive constant not necessarily the same at each for the continuity of the inverse Gabor frame operators occurrence. corresponding to the Theorem 5.1 and 6.1. We prove the By virtue of convergence of the series on the right hand side the following theorems: proof of the theorem is complete.

Theorem 9.1 If 푔 ∈ 푊(퐿∞, 퐿1) and satisfying the condition X. RECONSTRUCTION OF 풇 BY ITERATION (7.1), then for all 푏 > 0 sufficiently small, 푆−1 exists and is 푝 푝 On account if the relation (2.1) the series associated continues on 푊(퐿 , 퐿푤)(푅), 1 ≤ 푝 ≤ ∞. with 푆푓 in (2.2) is unconditionally convergent for every Theorem 9.2 If 푔̂ ∈ 푊(퐿∞, 퐿1) and satisfies the condition 푓 ∈ 퐻. This entails that 푆 is a positive and bounded linear (8.1) then , for all 푎 > 0 sufficiently small, 푆−1 exists and is operators. If 퐼 is theidentity operators, It is known that (cf. [2]) continuous on 퐹푊(퐿∞,퐿1),1 ≤ 푝 ≤ ∞. Proof of Theorem 9.1 퐴퐼 ≤ 푆 ≤ 퐵퐼. Assuming 퐶(푏, 푤) < 퐴 in Theorem 7.1, we see that Hence, as pointed out by Walnut [Wal 92, p. 480], we have 퐵 − 퐴 2 퐵 − 퐴 퐵 − 퐴 + 2퐶(푏, 푤) 퐼 ≤ 퐼 − 푆 ≤ 퐼 < 1. 퐵 + 퐴 퐵 + 퐴 퐵 + 퐴 퐵 + 퐴 2푏 퐵−퐴 ∞ ⇒∥ 퐼 − 푆 ∥≤ < 1 Thus, for 푓 ∈ 퐶푐 , we have 퐵+퐴 퐵+퐴 2푏 ⇒ 푆 is a bounded and invertible operators. ‖(퐼 − 푆) 푓‖ ≤∥ 푓 ∥ 푝 푝 푝 푊(퐿 ,퐿 ) −1 퐵 + 퐴 푊(퐿푝,퐿 ) 푤 ⇒ 푆 can be represented in the form of Neumann series: 푤 ∞ which implies that 2 푏 푘 2푏 푆−1 = ∑ (퐼 − 푆) 푝 푝 퐵 + 퐴 퐵 + 퐴 ‖(퐼 − 푆) |푊(퐿 , 퐿푤)‖ < 1. 푘=0 퐵 + 퐴 푘 2 ∞ 2 Hence we may choose 푑 number 푑 > 0 such that ⇒ 푓 = ∑푘=0 (퐼 − 푆) 푆푓. 2푏 퐵+퐴 퐵+퐴 ‖퐼 − 푆‖ < 푑 < 1. The function 푓 can also be recovered using the iteration 퐵 + 퐴 푝 푝 process: → 푆 is bounded and invertible on 푊(퐿 , 퐿푤)(푅). 2 −1 → 푆 can be represented by the Neumann Series: 푓0 = 푆푓, (10.1) ∞ 퐵 + 퐴 2푏 2푏 푘 푆−1 = ∑ (퐼 − 푆) 퐵 + 퐴 퐵 + 퐴 2 2 푘=0 푓푛+1 = 푓푛 + 푆푓 − 푆푓푁 and its sum convergence to 푆−1 in the operator norm. 퐵 + 퐴 퐵 + 퐴 It is known that 푓 → 0 in 퐻 as 푛 → ∞ (cf. [11], pp. Thus we see that 푛 ∞ 496-497). −1 2푏 2푏 In this section we prove the following theorem, which includes ∥ 푆 푓 ∥ 푝 푝 = ‖ ∑ (퐼 − 푆) 푓‖ 푊(퐿 ,퐿푤) 퐵 + 퐴 퐵 + 퐴 both the results of Walnut in Theorem 4.3 as particular cases: 푘=0 푝 푝 푊(퐿 ,퐿푤 ∞ 2푏 2푏 Theorem 10.1 Let 푔푤 ∈ 푊(퐿∞, 퐿1)(푅) nd for some 푎 > 0 푝 ≤ 퐶 ∥ 푓 ∥푊(퐿푝,퐿 ‖ ∑ (퐼 − )‖ 푤 퐵 + 퐴 퐵 + 퐴 the relation (7.1) holds with 퐴 > 0. If {푓푛} is a sequence 푘=0 ∞ defined recursively (10.1), 푓 ∈ 푊(퐿푝, 퐿푝 )(푅) and 푏 > 0 is Since the series on the right hand side is convergent, the proof 푤

ISSN: 2231-5373 http://www.ijmttjournal.org Page 5

International Journal of Mathematics Trends and Technology- Volume 20 Number 1- April 2015 sufficiently small, then lim ∥ 푓 − 푓 ∥ 2 = 0, 1 ≤ 푝 ≤ ∞. 푛 푆푓 (10.2) 푛→∞ 퐵+퐴

Proof We assume that 퐶(푏, 푤) < 퐴. Hence, by Theorem 7.1, we have 2푏 ∥ 퐼 − 푆 ∥< 1. (10.3) 퐵 + 퐴

푝 푝 We now defined an operator 푇 on 푊(퐿 , 퐿푤)(푅) such that 2푏 2푏 푇휙 = 휙 + 푆푓 − 푆휙. 퐵 + 퐴 퐵 + 퐴

REFERENCES

[1] I. Daubechies. The Wavelet transforms time-frequency localizatioin and singnal analysis. IEEE Trans. Tnform. Theory 39(1990), 961-1005. [2] R. J. Duffin and A. C. Scharffer. A Class of non-harmonic Fourier Series. Trans. Amer. Math. Soc. 72 (1952), 341-366. [3] H. G. Feichtinger. Banach Convolution Algebras of Wieners Type Proc. Conf. "Functios, Series, Operators", Budpest, 1980. [4] H. G. Feichtinger and K. Gröchenig. Banach Spaces Related to Interable Group Representations and their Atomic Decompositions I.J.Manatsh. Math. 86(1989), 307-340. [5] H. G. Feichtinger and T Ströhmer. Gabor Analysis And Alogorthms: Theory And Aapplications.Birkhaüser,Boston,1998. [6] D. Gabor. Theory of Communication. J. IEE (London) 93 (III) (1946), 429-457. [7] K Gr표̈chnig, C. Heil and K. Okoudjou, Gabor analysis in weighted amalgam spaces. Sampling Theiry in Signal and Image Processing Vol 1, No. 3 (2002), 225-259. [8] C. Heil. Wiener amalgam spaces in gerneralized harminic analysis and Wavelet theiry. Ph.D. Theis, Univ. Maryland (U.S.A.), 1989. [9] C. Heil and D.F. Walnut. Continuous and discrete wavelet transforms. SIAM Rev. 31(1989), 628-666. [10] D. Walnut. Weyl-Heisenberg wavelet expansions: Existence and stability in weighted spaces. Ph.D. Theis, Univ. Maryland, (U.S.A.), 1989. [11] [Wal 92] D.F. Walnut. Continuity properties of the Gabor operator. J. Math. Anal. & Applns. Vol. 165, No. 2(1992), 479-504.

ISSN: 2231-5373 http://www.ijmttjournal.org Page 6