Beurling's theorem for the Bessel–Struve transformThéorème de Beurling pour la transformée de Bessel–Struve

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Harmonic analysis

Beurling’s theorem for the Bessel–Struve transform

Théorème de Beurling pour la transformée de Bessel–Struve

Azzedine Achak,Radouan Daher, Hind Lahlali

DepartmentofMathematics,FacultyofSciencesAïnChock,UniversityofHassanII,Casablanca,Morocco

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received19April2014

Acceptedafterrevision11September2015 Availableonlinexxxx

PresentedbytheEditorialBoard

TheBessel–Struvetransformsatisfiessomeuncertaintyprinciplesinasimilarwaytothe EuclideanFouriertransform.Beurling’stheoremisobtainedfortheBessel–Struvetransform F^αB,S.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

r é s um é

La transformé de Bessel–Struve satisfait quelques principes d’incertitude de manière similaire au casde latransformée de Fourier euclidienne. Lethéorème de Beurlingest obtenupourlatransforméedeBessel–Struve.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

1. Introduction

There are manyknown theorems that state that a function and its classical Fouriertransform on R ^{cannot both be} sharply localized. That it is impossible for a non-zero function and its Fourier transform to be simultaneously small.

Hardy[5],Miyachi[9],CowlingandPrice [3], andBeurling[1]for exampleinterpreted thesmallnessassharp pointwise estimatesforintegrabledecayoffunctions. Beurling’stheorem,whichwasfoundby Beurlingand hisproofwas published muchlaterbyHörmander,saysthatforanynon-trivialfunction f∈^L2(R),theproduct f(x)f(y)isneverintegrableonR² withrespecttothemeasuree^|x||y|dxdy,wheref standsfortheFouriertransformof f.Afar-reachinggeneralizationofthis resulthasbeenrecentlyprovedbyBonami,DemangeandJaming[2].Theyprovedthat

Theorem1.1.Iff∈^L2(R)^{satisfiesforanintegerN}

R

R

|^f(x)||_f(y)|

(1+ |x| + |y|)^Ne|x||y|dxdy<∞,

thenf isoftheform f(x)=^P(x)e^−bx2,whereP isapolynomialofdegreestrictlylowerthan ^N−₂¹andb isapositiveconstant.

E-mailaddress:[email protected](H. Lahlali).

http://dx.doi.org/10.1016/j.crma.2015.09.013

1631-073X/©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

ManyauthorshaveestablishedtheanalogousofBeurling’stheoreminothervarioussettingofharmonicanalysis(seefor instance[7,8]).ThepurposeofthispaperistoestablishananalogousofBeurling’stheoremfortheBessel–Struvetransform.

Theoutlineofthecontentofthispaperisasfollows.

Section2isdedicatedtosomepropertiesandresultsconcerningtheBessel–Struvetransform.

Section3isdevotedtotheBeurling’stheoremfortheBessel–Struvetransform.

2. Bessel–Struvetransform

WeconsidertheBessel–Struveoperatorlα, α>−¹₂^,definedonC^∞(R)by

l_αu(x)=^d2u

dx²(x)+²α+¹ x

du

dx(x)−^du dx(0)

.

Forλ∈C^,thedifferentialequation:

l_αu(x)=λ²u(x)

u(0)=¹,u(0)=√^λ(α+1) π(α+³₂)

(1)

possessesauniquesolutiondenotedα(λ).Thiseigenfunction,calledtheBessel–Struvekernel,isgivenby:

_α(λx)=^jα(iλx)−^ihα(iλx), x∈R.

jα andhα arerespectivelythenormalizedBesselandStruvefunctionsofindex α.Thesekernelsaregivenasfollows:

j_α(z)= (α+¹)

+∞

k=0

(−1)^k_z

2 2k

k! (^k+α+¹)

and

h_α(z)= (α+1) +∞

k=0

(−¹)^k_z

2 2k+¹

k+³₂

k+α+³₂.

Let p∈^[1,+∞^{],wedenotebyLp}α(R)thespaceofreal-valuedfunctions f measurableonR^suchthat

fp,α=

⎛

⎝

R

|f(x)|^pdμ_α(x)

⎞

⎠

1 p

<+∞ if p<+∞

where

dμα(x)=^A(x)dx andA(x)= |^x|^2α+1, ^f_∞,α=^esssup

x∈R|^f(x)|<+∞ ^{if p}= ∞.

Thekernelα possessesthefollowingintegralrepresentation:

α(λx)= ²(α+¹)

√π(α₊¹₂) 1 0

(1−^t2)^α−12e^λxtdt, ∀^x∈R, ∀λ∈C.

TheBessel–Struvekernelα isrelatedtotheexponentialfunctionby

∀^x∈R, ∀λ∈C, α(λx)=Xα(e^λ.)(x),

whereXα istheBessel–Struveintertwiningoperator(see[4]).

Definition1.TheBessel–Struvetransformisdefinedon L¹α(R)by

∀λ∈R, F^α_B,S(f)(λ)=

R

f(x)_α(−ⁱλx)dμ_α(x).

Definition2.For f∈^L1α(R)withboundedsupport,theintegraltransformWα,givenby

W_α(f(x))= ²(α₊1)

√π(α+¹₂)|x|^2α+1

+∞

|x|

(y²−x²)^α−12y f(sgn(x)y)dy, x∈R\{0}

iscalledWeylintegraltransformassociatedwiththeBessel–Struveoperator.

Proposition1.(See[4].) WαisaboundedoperatorfromL¹α(R)toL¹(R),whereL¹(R)isthespaceofLebesgue-integrablefunctions.

Proposition2.(See[4].) Wehave∀^f∈^L1_α(R),F^α_B,S=F◦^Wα(f)whereF^{istheclassicalFouriertransformdefinedonL1}(R)^by F(g)(λ)=

R

g(x)e^−iλxdx.

Proposition3.(See[4].) Leta>0,theWeylintegraltransformverifies W_α(e₋a)=C e₋a

whereC= ^(α+1)

2√ πa^α+12

.

Proposition4.(See[4].)Leta>0andlet f beacontinuousfunctiononR^suchthat

∀^x∈R, |^f(x)| ≤^{Ce−ax2 (}∗⁾

ThenWα(f)isofclassC¹onR\{⁰}^andverifies

∀^x∈R\{⁰}, [^W1

2f](x)= −^xf(x)

and

∀α>¹

2, ∀^x∈R\{⁰}, [^Wαf](x)= −²αxW_α−1f(x).

Proposition5.(See[4].)For α=^k+¹₂^,k∈N^{,let f beacontinuousfunctionon}R^verifying(∗^).ThenWαisofclassC^k+1onR\{⁰} andwehaveVα◦^Wα(f)=^{f where}

V_αf(x)=(−¹)^{k+1 2}

2k+¹k! (2k+¹)!( ^d

dx²)^k+1(f(x)), x∈R^{∗ and d} dx²=¹

2 d dx.

Lemma1.(See[6].) Letg∈E(R\{⁰}),m andk aretwointegersnonnegative,wehave

∀^x∈R^∗, ( ^d

dx²)^k(x^mg(x))= k

i=0

b^k_ix^m−2k+ig⁽ⁱ⁾(x)

whereE(R)designatesthespaceofinfinitelydifferentiablefunctionsonR^andbk_i ^{areconstantsdependingoni,k andm.}

Lemma2.(See[6].)Letf beinD(R).WehaveVα◦(Wα(f))=^{f .}

Theorem2.1.(See[4].) Letf beameasurablefunctiononR^suchthat

^eafp,α<+∞ ^{and e}bF_B^α_,S(f)q,α<+∞ ⁽²⁾

forsomeconstantsa>0,b>0,1≤^p,q≤ +∞^{andatleastoneofp andq isfinite.Wehave}

1. Ifab≥¹₄^{,then f}=^0a.e.

2. Ifab< ¹₄,thenforallδ∈]^a,_4b¹[^{,thefunctionshavingtheform f}(x)=^P(x)e^−δx2 whereP isanevenpolynomialonR^satisfy relation(2).

3. Beurling’stheoremfortheBessel–Struvetransform Inthissection,wewillproveourresultmain.

Theorem3.1.LetN,k∈N^, α=^k+¹₂^{and f}∈^L2α(R).Then

R

R

|^f(x)||F_B^α_,S(f)(y)|

(1+ |x| + |y|)^{N e|x||y|}|x|^2α+1dxdy<∞ (3)

implies f(x)=^P(x)e^−rx2,wherer>0andP isanevenpolynomialofdegreestrictlylowerthan ^N−₂¹. Proof. Westartwiththefollowinglemma.

Lemma3.Wesupposethatf∈^L2_α(R)satisfies(3),thenf∈^L1_α(R).

Proof. Wemaysupposethat f=^{0 in L2}_α(R)^{.(3)andtheFubinitheoremimplythatforalmostevery y}∈R^,

|F^α_B,S(f)(y)|

(1+ |^y|)^N

R

|^f(x)|

(1+ |^x|)^Ne|x||y||^x|^2α+1dx<∞.

SinceF_B^α_,S(f)=^{0,thereexists y}0∈R^{, y}0=^{0 suchthat}F^α_B,S(f)(y0)=^0.

Therefore

R

|f(x)|

(1+ |^x|)^Ne|x||y0||x|^2α+1dx<∞.

Since e^|x||y0|

(1+ |^x|)^{N 1 forlarge}|^x|^{,itfollowsthat}

R

|^f(x)||^x|^2α+1dx<∞^. 2

ThislemmaandProposition 1implythat Wα(f)iswelldefinedalmosteverywhereonR^.ByProposition 1wecanfind aconstantC>0 suchthat

R

|^Wα(f)(x)|^dx≤^C

R

|^f(x)||^x|^2α+1dx,

thus

R

R

|^Wα(f(x))||F_B^α_,S(f)(y)| (1+ |^x| + |^y|)^{N e}

|^x||^y|dxdy≤^C

R

R

|^f(x)||F_B^α_,S(f)(y)| (1+ |^x| + |^y|)^{N e}

|^x||^y||^x|^2α+1dxdy

<∞.

ItfollowsfromProposition 2that

R

R

|^Wα(f(x))||F◦^Wα(f)(y)|

(1+ |x| + |y|)^{N e|x||y|dxdy}<∞.

According toTheorem 1.1,wecandeducethatforallx∈R, Wα(f)(x)=^P(x)e^−rx2,wherer>0 and P isapolynomialof degreestrictlylowerthan ^N−₂¹.

HenceapplyingLemma 1,wecanfindconstantsbssuchthat f(x)=^Vα◦^Wα(f)(x)=

|^s|<^N−₂¹

b_sx^se^−rx2.

Then it follows from Lemma 3 that

R

|

|^s|<^N−1₂

bsx^s|^e−rx2|^x|^2α+1dx<∞^{. Then for some constants a}∈]⁰,r[^{, we have}

^eaf1,α<+∞^{.Ontheotherhand,}

F_B^α_,S(f)(y)=F◦^Wα(f)(y)=F(P(x)e^−rx2)=^R(y)e^−y

2 4r

where RisapolynomialofdegreedegP.Thenforsomeconstantsb∈]⁰,¹₄r[^,wehaveebF_B^α_,S(f)2,α<+∞^.Accordingto Theorem 2.1,wecandeducethat f(x)=^P(x)e^−rx2,where P isanevenpolynomialofdegreestrictlylowerthan ^N−₂¹. 2

References

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19(2002)22–35.

[3]M.G.Cowling,J.F.Price,GeneralizationsofHeisenbergInequality,LectureNotesinMathematics,vol. 992,Springer,Berlin,1983,pp. 443–449.

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j.ajmsc.2012.09.003.

[5]G.H.Hardy,AtheoremconcerningFouriertransform,J.Lond.Math.Soc.8(1993)227–231.

[6]L.Kamoun,S.Negzaoui,OntheharmonicanalysisassociatedtotheBessel–Struveoperator,arXiv:1011.5394v1.

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[9]A.Miyachi,AgeneralizationofHardy,in:HarmonicAnalysisSeminarHeldatIzunagoakaShizuoka-ken,Japan,1997,pp. 44–51.