Functional Analysis Techniques as a Mathematical tools in Wavelets Analysis

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1 Saer El-Sater, et. al., JOURNAL OF PHYSICS VOL. NO. Feb. (0) PP. 0-7 Functional Analysis Tecniques as a Matematical tools in Wavelets Analysis Saer El-sater, Ramaan Sale Tantawi an A. Abel-afiez Abstract In te frame of te functional analysis tecniques, wavelets is briefly escribe, wat wavelets are, ow to use tem, wen we o nee tem, wy tey are preferre, an were tey ave been applie. It is sown wat specific features of functions can be reveale by tis analysis. Tey provie unique information about scales at ifferent locations. In particular, tey are use for analysis of patterns in te pase space of very ig multiplicity events Keywors: Functional Analysis; Daubecies Wavelets; Haar Wavelets. W I. INTRODUCTION avelets ave become a great matematical tool in many investigations. Tey are use in tose cases were te result of te analysis of a particular signal soul contain not only te list of its typical frequencies (scales) but also nowlege of te efinite local coorinates were tese properties are important. In particular, te seconary Manuscript receive Jan, 0 an accepte Feb. 4, 0. Pysics Dept., Faculty of Science, El-Baat University, Syria Matematics Dept., Faculty of Science, Zagazig University, Egypt. Experimental Nuclear Pysics Dept., NRC, AEA, Egypt., Temporary Aress: Matematics Dept., Faculty of Eucation / Al-Maara- Haramout University, Yemen. abel_afiez@yaoo.com particle istributions witin te available pase space for te very ig multiplicity events can be stuie an te corresponing patterns efine by te particle correlations foun. Te wavelet basis is forme by using ilations an translations of a particular function efine on a finite interval. Commonly use wavelets generate a complete ortonormal system of functions wit a finite support. Tat is wy, by canging te scale (ilations), tey can istinguis te local caracteristics of a signal at various scales, an by translations tey cover te wole region in wic it is stuie. Due to te completeness of te system, tey also allow for te inverse transformation to be properly one. If one can as suc a question; ow te igmultiplicity event woul loo if only correlations of a efinite scale are left in tis event. Tis is emonstrate below. Te locality property of wavelets gives a substantial avantage over te Fourier transform, wic provies us only wit nowlege of te global frequencies (scales) of te object uner investigation because te system of basic functions use (sine, cosine, or imaginary exponential functions) is efine over an infinite interval. It as been proven tat any function can be written as a superposition of wavelets, an tere exists a numerically stable algoritm to compute te coefficients for suc an expansion. Moreover, tese coefficients completely caracterize te function, an it is possible to reconstruct it in a numerically stable way by nowing tese coefficients. Moreover, in practical calculations, teir irect form is not even require, an only te numerical values of te coefficients of te functional equation are use. Tis is a very important proceure, calle multiresolution analysis, wic gives rise to te multiscale local analysis of te signal an fast numerical algoritms. Eac scale contains an inepenent nonoverlapping set of information about te signal in te form of wavelet coefficients, wic are etermine from an iterative proceure calle te fast wavelet transform. In combination, tey provie its complete analysis an simplify te iagnosis of te unerlying processes. Te extene review on wavelets can be see in []. In tis article functional analysis tecniques will be escribe in terms of te Haar an Daubecies wavelet. Copyrigt Researcpub.org. All rigts Reserve. 0

2 Saer El-Sater, et. al., JOURNAL OF PHYSICS VOL. NO. Feb. (0) PP. 0-7 II. THEORY Every signal can be caracterize by its average (over some intervals) values (tren) an by its variations aroun tis tren. Let us call tese variations fluctuations inepenently of teir nature, be tey of ynamic, stocastic, psycological, pysiological, or any oter origin. Wen processing a signal, one is intereste in its fluctuations at various scales because from tese one can learn about teir origin. Te goal of wavelet analysis is to provie tools for suc processing. In fact, pysicists ealing wit experimental istograms analyze teir ata at ifferent scales wen averaging over ifferent size intervals. Tis is a particular example of a simplifie wavelet analysis treate in tis section. To be more efinite, let us consier te situation wen an experimentalist measures some function f (x) witin te interval 0 x, an te best resolution obtaine wit te measuring evice is limite to /6 of te wole interval. Tus, te result consists of 6 numbers representing te mean values of f(x) in eac of tese bins an can be plotte as a 6-bin istogram sown in Fig.. It can be represente by te following formula [-4] 5 4, 0 f ( x) s ( x) () 4, Were s 4, = f(/6)/4, an ϕ 4, is efine as a step lie bloc of te unit norm (i.e., of eigt 4 an wit /6, wit 4, normalization fixe by x ) let us write te normalize sums an ifferences for an arbitrary level j as [4-6]. sj-, j, j, j-, j, j, [ s s ], [ s s ] () or for te bacwar transform s [ s ], j, j, j, j, [ s ] () j, j, Since, for te yaic partition consiere, tis ifference as opposite signs in te neigboring bins of te previous fine level, we introuce te function ψ wic is an, respectively, in tese bins an te normalize functions ψ j, = j/ ψ( j x ). Tis allows us to represent te same function f(x) as 7 7 f ( x) s,, ( x),, ( x) (4) 0 0 One procees furter in te same manner to te sparser levels,, an 0 wit averaging one over te interval lengts /4, /, an, respectively. Tis is sown in te subsequent rawings in Fig.. Te sparsest level wit te mean value of f over te wole interval enote as s 0,0 provies [7,8] f ( x) s 0,0 ( x) 0,0 0, 0,0, 0,0 ( x) (x) 7 0 0,,, ( x), ( x) Te functions ϕ j, (x) an ψ j, (x) are normalize an all te above representations of te function f(x) [Eqs. () (5)] are matematically equivalent an te functions ϕ 0,0 (x) an ψ 0,0 (x) are sown in Fig. Equation (5) is representing te wavelet analyze function irectly reveals te fluctuation structure of te signal at ifferent scales j an various locations present in a set of coefficients j,, wereas te original form () ies te fluctuation patterns in te bacgroun of a general tren. In practical applications, te latter wavelet representation is preferre because, for rater smoot functions, strongly varying only at some iscrete values of teir arguments, many of te ig-resolution coefficients in relations similar to Eq. (5) are close to zero (compare to te informative coefficients) an can be iscare. Bans of zeros (or close to zero values) inicate tose regions were te function is fairly smoot. III. FUNCTIONAL ANALYSIS TECHNIQUES FOR HAAR WAVELETS To analyze any signal, one soul, first of all, coose te corresponing basis, i.e., te set of functions to be consiere as functional coorinates. In most cases, we will eal wit signals represente by square integrable functions efine on te real axis. For nonstationary signals, for example, te location of tat moment wen te frequency caracteristics ave abruptly been cange is crucial. Terefore, te basis soul ave a compact support; i.e., it soul be efine on a finite region. Wavelets ave tis property. Neverteless, wit tem it is possible to span te wole space by translation of te ilate versions of a efinite function. Tat is wy every signal can be ecompose in a wavelet series (or integral). Eac frequency component is stuie wit a resolution matce to its scale. (5) Copyrigt Researcpub.org. All rigts Reserve.

3 Saer El-Sater, et. al., JOURNAL OF PHYSICS VOL. NO. Feb. (0) PP. 0-7 Let us try to construct functions satisfying te above criteria. An eucate guess woul be to relate te function ϕ(x) to its ilate an translate version. Te simplest linear relation wit M coefficients is [9] M 0 ( x) (x ) (6) wit te yaic ilation an integer translation. At first sigt, te cosen normalization of te coefficients wit te extracte factor loos somewat arbitrary. Actually, it is efine a posteriori by te traitional form of fast algoritms for teir calculation an normalization of functions ϕ j, (x), ψ j, (x). Te integer M efines te number of coefficients an te lengt of te wavelet support, is given as x (x) (x ) an te normalization conition cosen as: Te function (x) obtaine from te solution to tis equation is calle a scaling function [0-]. If te scaling function is nown, one can form a moter wavelet (or a basic wavelet) ψ(x) accoring to [4] M 0 (7) ( x) g (x ) (8) Were, Te simplest example woul be for M = wit two nonzero coefficients equal to / i.e., te equation leaing to te Haar scaling function H (x): (x) H(x) (x -) H (9) H One easily gets te solution to tis functional equation H (x) = (x). (-x). Were θ(x) is te Heavisie step function equal to at positive arguments an 0 at negative ones. Te aitional bounary conition is H (0) =, H () = 0. Tis conition is important for te simplicityof te wole proceure of computing te wavelet coefficients, wen two neigboring intervals are consiere. Te moter wavelet is (x) (x) (- x) -(x -) (- x) (0) H wit bounary values efine as ψ H (0) =, ψ H (/) =, ψ H () = 0. Tis is te Haar wavelet nown since 90 an use in functional analysis[5,6]. Bot te scaling function H (x) an te moter wavelet ψ H (x) are sown in Fig.. In figure a istogram an its wavelet ecomposition is sown. Te initial istogram is sown in figure (e) wic correspons to te level j = 4 wit 6 bins (Eq. ()). Te intervals are labele on te abscissa axis on teir leftan sies. For te level j = te mean values over two neigboring intervals of te previous level are sown on te left-an sie. Tey correspon to eigt terms in te first sum in Eq. (4). On te rigt-an sie, te wavelet coefficients, are sown. For te levels j =,, 0 are obtaine in a similar way. In figure Te Haar scaling function ϕ H (x) ϕ 0,0 (x) (to te left) an moter wavelet ψ H (x) ψ 0,0 (x) (to te rigt). IV FUNCTIONAL ANALYSIS TECHNIQUES FOR DAUBECHIES WAVELETS Here, we sow ow te tecniques of te functional analysis wors in practice wen applie to te problem of fining te coefficients of any filter an g. Tey can be irectly obtaine from te efinition an properties of te iscrete wavelets. Tese coefficients are efine by relations (6) an (8) [7]. x) (x ), ( x) g (x ) () ( Were an te ortogonality of te scaling function efine as: x (x) (x - m) 0 () leas to te following equation for te coefficients m () Te ortogonality of wavelets to te scaling functions, x (x) (x - m) 0 (4) 0m Copyrigt Researcpub.org. All rigts Reserve.

4 Saer El-Sater, et. al., JOURNAL OF PHYSICS VOL. NO. Feb. (0) PP. 0-7 (a) (b) (c) Copyrigt Researcpub.org. All rigts Reserve. 0

5 Saer El-Sater, et. al., JOURNAL OF PHYSICS VOL. NO. Feb. (0) PP. 0-7 IV. () Fig. A istogram an its wavelet ecomposition. (e) Fig. Te Haar scaling function (to te left) an moter wavelet (to te rigt). Copyrigt Researcpub.org. All rigts Reserve.

6 Saer El-Sater, et. al., JOURNAL OF PHYSICS VOL. NO. Feb. (0) PP. 0-7 gives te equation g aving a solution of te form m 0 (5) 0 -, -,, ( 4 ) () g (-) M-- (6) Tus te coefficients g for wavelets are irectly efine by te scaling function coefficients. Anoter conition of te ortogonality of wavelets to all polynomials up to te power (M ), efining teir regularity an oscillatory beavior [8-0]. x x n (x) 0, n 0,,..., M - (7) Provies te relation n g Giving rise to 0 n (-) 0 (8) (9) Taing formula (6) in consieration, te normalization conition x (x) (0) can be rewritten as anoter equation for : () For M= in equations (), (9) an () we get te following system of equations: () In case of te minus sign for, correspons to te wellnown filter 0 ( 4 ( 4 ), ( 4 ), (- 4 ), ) (4) Tese coefficients efine te simplest D4 (or ψ) wavelet from te famous family of ortonormal Daubecies wavelets wit finite support. It is sown in Fig. by te otte curve wit te corresponing scaling function sown by te soli curve. Some oter iger ran wavelets are also sown tere. It is clear from tis figure (especially for D4) tat wavelets are smooter at some points tan at oters. V. CONCLUSIONS Te functional analysis as a matematical construction of te wavelet transformation an its utility in practical applications attract researcers from bot pure an applie science. We especially empasize ere tat wavelet analysis of multi particle events in ig energy particle an nucleus collisions proposes a completely new approac to te effective event-by event stuy of patterns forme by seconary-particle locations witin te available pase space. Te newly foun patterns ave alreay sown some specific ynamical features not iscovere before. One can expect oter surprises wen very ig multiplicity events obtaine in etectors wit goo acceptance become available for analysis. By solving tis system of equations we got Copyrigt Researcpub.org. All rigts Reserve.

7 Saer El-Sater, et. al., JOURNAL OF PHYSICS VOL. NO. Feb. (0) PP. 0-7 Fig.. Daubecies scaling functions (left curves) an wavelets (rigt curves) for M =, 4. REFERENCES []. I. M. Dremin, Pysics of Atomic Nuclei, Vol. 67, No., 004, pp From Yaernaya Fizia, Vol. 67, No., 004, pp []. I. M. Dremin, O. V. Ivanov, an V. A. Necitailo, Usp. Fiz. Nau 7, 465 (00) [Pys. Usp. 44, 447 (00)]. []. Y. Meyer an R. Coifman, Wavelets, Caleron Zygmun an Multilinear Operators (Cambrige Univ. Press, Cambrige, 997). [4]. Y. Meyer, Wavelets: Algoritms an Applications (SIAM, Pilaelpia, 99). [5]. Progress in Wavelet Analysis an Applications, E. byy. Meyer an S. Roques (Eitions Frontie` res, Gif-sur-Yvette, 99). [6]. K. Hernanez an G. Weiss, A First Course on Wavelets (CRC, Boca Raton, 997). [7]. G. Kaiser, A Frienly Guie to Wavelets (Birauser, Boston, 994). [8]. Wavelets: An Elementary Treatment of Teory an Applications, E. byt. Koornwiner (Worl Sci., Singapore, 99). [9]. Wavelets in Pysics, E. by J. C. Van en Berg (Cambrige Univ. Press, Cambrige, 998). PHYSICS OF ATOMIC NUCLEI Vol. 68 No DREMIN [0]. S. Mallat, A Wavelet Tour of Signal Processing (Acaemic Press, New Yor, 998). []. G. Erlebacer, M. Y. Hussaini, an L. M. Jameson, Wavelets Teory an Applications (Oxfor Univ. Press, Oxfor, 996). []. Wavelets in Meicine an Biology, E. bya. Alroubi an M. Unser (CRC, Boca Raton, 994). []. R. Carmona,W.-L. Hwang, an B. Torresani, Practical Time-Frequency Analysis (Acaemic Press, San Diego, 998). [4]. E. A. DeWolf, I. M. Dremin, an W. Kittel, Pys. Rep. 70, (996). Copyrigt Researcpub.org. All rigts Reserve.

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