ABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS
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1 Novi Sad J. Math. Vol. 45, No. 1, 2015, ABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS Mirjana Vuković 1 and Ivana Zubac 2 Ddicatd to Acadmician Bogoljub Stanković for his 90th birthday Abstract. W analyz quasiasymptotic bounddnss of distributions and thir wavlt transforms, in gnral, as wll as for a class of α xponntially boundd distributions and thir wavlt transforms in particular. Th main ida of this papr is to us, instad of th quasiasymptotic bhaviour, th notion of quasiasymptotic bounddnss. In this way w obtain nw Ablian typ thorms for th wavlt transform of distributions with diffrnt growth. AMS Mathmatics Subjct Classification (2010): 46F12, 42C40 Ky words and phrass: Wavlt transform; distributions: tmprd; α-xponntially; quasiasymptotic bounddnss; Ablian typ thorms 1. Introduction Th wavlt transform is a powrful tool for studying local proprtis of functions and gnralizd functions. Usually, wavlt analysis prsnts two main important faturs [1, 2, 9], th wavlt transform as a tim-frquncy analysis tool, and wavlt analysis as part of approximation thory [5]. Hr w considr th first fatur of th thory. For a mor gnral approach, w rfr to [3, 4]. This papr is a part of th Mastr s thsis of th scond author [12]. Th quasiasymptotics is usd to analyz th pointwis proprtis of tmprd distributions (s [3, 4, 10] and rfrncs thr). By th gnral thory of tmprd distributions, w ar limitd to comparison by th functions of th form k α L(k), rsp., ε α L(ε), whr L is Karamata s rgulary varying function [7]. Morovr, th limit distribution has to b th homognous of ordr α, (s [10]). W rfr to [7] for th proprtis of rgularly varying functions L. Our main ida of this papr in obtaining th Abl typ thorms is to us, instad of th quasiasymptotic bhaviour, th notion of quasiasymptotic bounddnss. This nabls us to hav mor frdom in th sns that w should not considr only th spac of tmprd distributions and th comparison function ρ(k) k α L(k), rsp., ε α L(ε). In this way w obtain nw Abl typ thorms for th wavrlt transform of distributions with diffrnt growth. 1 Acadmy of Scincs and Arts of Bosnia and Hrzgovina, Sarajvo, -mail: mvukovic@anubih.ba 2 Faculty of Mchanical Enginring and Computing, Univrsity of Mostar, -mail: ivana.fsr@gmail.com
2 202 Mirjana Vuković, Ivana Zubac 2. Dfinitions W us th standard notation (cf. [11, 10, 1, 8]): N is th st of positiv intgrs including zro, R + is a st of positiv ral numbrs, its complmnt is R (similarly on has R + ) and H R R +. Th Schwartz spacs of smooth compactly supportd and rapidly dcrasing tst functions ar dnotd by D (R) and S (R), rspctivly; thir dual spacs, th spacs of distributions, and tmprd distributions ar D (R) and S (R), rspctivly. W us th Fourir transform of a function s L 1 (R): s F s, (F s) (ω) iωt s A function g L 1 (R) L (R) with + + iωt s (t) dt, ω C. g (t) dt 0 is calld a wavlt. Rcall, [1], th wavlt transform of a function s L p (R), 1 p, is dfind as W g s (b, a) g b,a s + ( 1 t b a g a ) s (t) dt, (b, a) H. Th function g is usually calld mothr wavlt or analyzing wavlt, and functions g b,a, (b, a) H ar wavlts. For g, s L 1 (R) L 2 (R) w can calculat th wavlt cofficints in th Fourir spacs as W g s (b, a) g b,a s 1 2π ĝ b,a ŝ, (b, a) H. Lt f (t) S (R) and ρ (k) b a positiv continous function for k > 0. W say that f (t) has quasiasymptotic bhavior at infinity (rsp., at zro) rlatd to a positiv function ρ (k) in S(R), in th sns of convrgnc in S (R), if 1 f (kt) g (t), k, ρ (k) ( ) 1 f (εt) g (t), ε 0+. ρ (ε) Morovr, on can considr ths notions in D (R), (cf. [10]), s 3.1 blow. 3. Ablian typ thorms Th quasiasymptotic bounddnss of tmprd distributions and th wavlt transform of tmprd distributions supportd by [0, ) was considrd in [6, 9]. In th first subsction w considr th quasiasymptotic bounddns for tmprd distributions on th whol ral lin. On can transfr all th rsults to th n dimnsional cas H n+1 R n R +. This will not b considrd in this papr.
3 Abl typ thorms for th wavlt transform Ablian thorm for lmnts of D (R) Lt H b a spac of tst functions on R with th convrgnc structur so that D(R) is dns in it and th inclusion mapping D(R) H is continuous. This implis that H is a subspac of th spac of distributions D (R). Lt f H and ρ(k), k > a > 0, (ρ(ε), 0 < ε < ε 0 < 1), b a positiv and continous function. W say that f is a quasiasymptotically boundd function with rspct to ρ at infinity, (at zro) ovr H if (3.1) f(k )/ρ(k), (f(ε )/ρ(ε)) is boundd in H in th wak sns. Our first thorm (s [12]) is a simpl gnralization of th vry wll known on for tmprd distributions. Thorm 3.1. Lt a function f D (R) b a quasiasymptotically boundd at zro with rspct to a continuous positiv ral valud function c (ε) (rsp. at infinity with c (k)), that is f (εx), φ (x) C φ c (ε), ε 0 (rsp. f (kx), φ (x) C φ c (k), k ). whr φ D(R) and C φ > 0 dpnds of φ. Lt g D (R) b a mothr wavlt. Thn th wavlt transform for f is a boundd function at 0 (rspctivly ), i.., thr is a C C (g) such that W g f (x, x) c (x) C, x 0, (rsp. W g f (x, x) c (x) C, x ). ( ) t εx Proof. Not if g D, thn t ḡ εy, t R is in D. According to th dfinition of th wavlt transform, w hav ( ) W g f (εx, εy) 1 t εx c (ε) f (t), c (ε) εy ḡ, x, y H. εy If w put t εs, w obtain W g f (εx, εy) c(ε) f (εs) c (ε), 1 ( ) s x y ḡ C. y For x 1, y 1 (and latr ε x), w hav W g f (x, x) c (x) C. In th cas k, w hav th sam proof.
4 204 Mirjana Vuković, Ivana Zubac 3.2. Ablian thorm for α-xponntially distributions Lt α (0, 2). Our main objctiv is th quasiasymptotic bounddns ovr th dual spac of th spac of tst functions with th α xponntial dcras dfind as follows. { } S α (R) ϕ C (R) γ k (ϕ) sup k x α ϕ (r) (x) < for vry k > 0 r k,x R Clarly S α S β if 0 < α < β. Th convrgnc structur is clar if on knows wll th Schwartz spac S(R). Namly, γ k, k N, is a squnc of norms which dfins th topology. Clarly, D(R) is dns in it. This spac is an xampl of th Glfand - Shilov typ spacs K Mp (R) in th spcial cas whn M p is a smooth function M p (x) m > 0, x R, and M p (x) p x α for x > 1. Th following structural thorm can b provd in a standard way (cf. [8], Thorm 8.2.8). Thorm 3.2. For vry f S α thr xist p > 0, m N and boundd functions F j, j 0, 1,..., m so that f j0 ( p x α F j (x)) (j). Now w considr th bounddns of th lmnts of this spac. Thorm 3.3. Evry f S α is quasiasymptotically boundd with rspct to ρ (k) ovr th spac S α+2.. Proof. Lt φ S α+2. Thn f (kx), φ (x) m ( p α F j ( ) ) (j) (kx), φ (x) j0 j0 ( p α F j ( ) ) (j) (x), 1 ( x ) j0 p k α+2 k φ k ( 1) j p x α F j (x) 1 ( x ), j0 p k α+2 k j+1 φ(j) k ( 1) j p kx α F j (kx), φ (j) (x). k j Thr xist constants C j > 0, j 1,..., m, such that f (kx) p k α x α F j (kx), φ (x) C j φ (j) (x) dx. k j j0 R
5 Abl typ thorms for th wavlt transform Dividing th last intgral into two parts, w obtain, with anothr C j > 0, f (kx), φ (x) C j j0 + x >1 x <1 pkα x α k j pkα+2 φ (j) (x) dx (p+1) x α+2 φ (j) (x) dx. p x α+2 x α+2 Sinc r t r 2 + t 2 w obtain k α x α k 2α + x 2α. According to this inquality and assumption α (0, 2) it follows that (3.2) pk α x α pk α+2 + p x α+2. Now, with sutabl constants D j, w finally obtain f (kx), φ (x) which finishs th proof. D j + D j j0 j0 x >1 D j + D j x >1 p(kα+2 + x α+2 ) p(kα+2 + x α+2 ) dx <, x α+2 dx x α+2 Now w show that th wavlt transformation of any distribution from S α is α + 2-xponntially boundd. Thorm 3.4. Lt f S α, α < 2 and g C such that g S α+2 is a mothr wavlt. Thn W g f (x, x) <, x > x 0 > 0. p x α+2 Proof. It is known that f is quasiasyptotically boundd with p x α+2 p > 0, so from Thorm 3.3 w hav ( ) W g f (x, x) f (t) 1 s x, p x α+2 p x α+2 x n ḡ < C, x for som i.. ( ) W g f (x, x) f (t) 1 s x, p x α+2 p x α+2 x n ḡ <. x
6 206 Mirjana Vuković, Ivana Zubac Rfrncs [1] Holschnidr, M., Wavlts, an Analysis Tool. Th Clarndon Prss, Oxford Univrsity Prss, Nw York [2] Daubchis, I., Tn Lcturs on Wavlts. SIAM, Phyladlphia [3] Drozhzhinov, Yu.N., Zavyalov, B.I., Taubrian thorms for gnralizd functions with valus in Banach spacs. Izv. Math. 66 (2002), [4] Drozhzhinov, Yu.N., Zavyalov, B.I., Multidimnsional Taubrian thorms for Banach-spac valud gnralizd functions. Sb. Math. 194 (2003), [5] Gröchnig, K., Foundation of Tim-Frquncy Analysis. Birkhausr, Boston [6] Sanva, K., Buckovska, A., Asymptotic bhaviour of th distributional wavlt transform at 0. Math. Balkanica (N.S.) 18 (2004), [7] Bingham, N.H., Goldi, C.M., Tugls, J.L., Rgular variation. Encyclopdia of Mathmatics and its Applications 27, Cambridg Univrsity Prss, Cambridg [8] Pilipovic, S., Stankovic, B., Prostori distribucija. Srpska akadmija nauka i umjtnosti, Novi Sad [9] Pilipovic, S., Vindas, J. Multidimnsional Taubrian Thorms for Vctor- Valud Distributions. Publ. Inst. Math. Bograd 95 (2014), [10] Pilipovic, S., Stankovic, B., Vindas, J., Asymptotic bhavior of gnralizd functions. Sris on Analysis, Applications and Computation, 5. World Scintific Publishing Co. Pt. Ltd., Hacknsack, NJ [11] Vladimirov, V.S., Gnralizd functions in mathmatical physics. Mir Publishrs, Moscow [12] Zubac (Milinković Rosić), I., Prilozi toriji malotalasnih transformacija. MA tsis, Faculty of Philosophy, Univrsity of East Sarajvo Rcivd by th ditors January 18, 2015
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