A new Weber type integral equation related to the Weber-Titchmarsh problem

0.06 MB·

by  Semyon Yakubovich

A NEW WEBER TYPE INTEGRAL EQUATION
RELATED TO THE WEBER-TITCHMARSH PROBLEM
S.YAKUBOVICH
7
1
0
2

n ABSTRACT. Wederivesolvability conditions andclosed-formsolutionfortheWebertypeintegral equation,
a related tothefamiliar Weber-Orrintegral transforms andtheoldWeber-Titchmarshproblem (posedinProc.
J Lond.Math.Soc.22(2)(1924),pp.15,16),recentlysolvedbytheauthor.Ourmethodinvolvespropertiesofthe

5 inverseMellintransformofintegrablefunctions.TheMellin-Parsevalequalityandsomeintegrals,involvingthe
1 Gausshypergeometricfunctionareused.

]
A
C Recently,theauthorgavesolvabilityconditionsandclosed-formsolutionfortheclassicalWeberequation
. [7]
h
t ¥
ma Cn (xx ,ax )g(x )dx = f(x), (1)
Z0
[ where f(x)isagivenfunctionon[a,¥ ), a>0,g(x),x R+shouldbedeterminedandthekernel
∈
1 Cn (a ,b )=Jn (a )Yn (b ) Yn (a )Jn (b ) (2)
−
v
8 involvesBesselfunctionsofthefirstandsecondkindJn (z),Yn (z), n C[1],Vol. II.Itwassolvedformally
∈
0 byTitchmarshin1924andposedasanopenproblem(see[3],p. 15.)todescribeaclassofcomplex-valued
0 functionsg(x), x R ,whichcanbeexpandedintermsofthefollowingrepeatedintegral
+
4 ∈
0 x ¥ ¥
1. g(x)= Jn2(ax)+Yn2(ax)Za Cn (xt,xa)tZ0 Cn (tx ,ax )g(x )dx dt, x>0. (3)
0 Expansion(3)is related to the familiar Weber-Orr integralexpansionsof an arbitraryfunction f(x) as re-
7
peatedintegrals
1
Xiv: f(x)=Z0¥ Jn2t(aCtn)(+xtY,na2t()at)Za¥ Cn (x t,at)x f(x )dx dt, (4)
ar f(x)=Za¥ Cn (xt,xa)tZ0¥ Jn2(Caxn ()t+x ,Yan2x()ax )x f(x )dx dt, (5)
which are differentfrom (3). Our method is based on the use of the Mellin transform [4]. Precisely, the
MellintransformisdefinedinLm ,p(R+), 1<p 2bytheintegral
≤
¥
f∗(s)= f(x)xs−1dx, (6)
Z0
Date:January17,2017.
2000MathematicsSubjectClassification. Primary44A15,44A35,33C10;Secondary33C05,45E99.
Keywordsandphrases. Weber-Orrintegraltransforms,Mellintransform,Besselfunctions,Gauss’shypergeometricfunction.
1
2 S.Yakubovich
being convergentin mean with respect to the norm in L (m i¥ ,m +i¥ ), q= p/(p 1). Moreover, the
q
Parsevalequalityholdsfor f Lm ,p(R+), g L1 m ,q(R+) − −
∈ ∈ −
¥ 1 m +i¥
Z0 f(x)g(x)dx= 2p iZm i¥ f∗(s)g∗(1−s)ds. (7)
−
TheinverseMellintransformisgivenaccordingly
1 m +i¥
f(x)= f (s)x sds, (8)
2p iZm i¥ ∗ −
−
wheretheintegralconvergesinmeanwithrespecttothenorminLm ,p(R+)
¥ 1/p
f m ,p= f(x) pxm p−1dx . (9)
|| || (cid:18)Z0 | | (cid:19)
In particular, letting m =1/p we get the usual space L (R ). We will modify the definition of a special
1 +
classoffunctionsrelatedtotheMellintransform(6)anditsinversion(8),whichwasintroducedin[5],[6].
Indeed,wehave
Definition1. DenotebyM 1(L )thespaceoffunctions f(x), x R , beingrepresentablebyinverse
− c +
Mellintransform(8)ofintegrablefunctionsF(s) L (c)onthevertic∈allinec= s C:m =Res=c .
1 0
ThespaceM 1(L )withtheusualoperations∈ofadditionandmultiplicationb{y∈scalarisalinearvec}tor
− c
space. IfthenorminM 1(L )isintroducedbytheformula
− c
1 +¥
f = F(c +it) dt, (10)
M−1(Lc) 2p Z ¥ | 0 |
(cid:12)(cid:12) (cid:12)(cid:12) −
thenitbecomesaBanachspace. (cid:12)(cid:12) (cid:12)(cid:12)
Definition2. Let m =0, c ,c R besuchthat2signc +signc 0. By M 1 (L )we denotethe
6 1 2∈ 1 2≥ c−1,c2 c
spaceoffunctions f(x),x R+,representableintheform(8),wheresc2ep c1|s|F(s) L1(c).
∈ ∈
ItisaBanachspacewiththenorm
1
(cid:12)(cid:12)f(cid:12)(cid:12)Mc−11,c2(Lc)= 2p Zcep c1|s||sc2F(s)ds|.
Inparticular,lettingc =c =0w(cid:12)(cid:12)eg(cid:12)(cid:12)etthespaceM 1(L ). Moreover,itiseasilyseentheinclusion
1 2 − c
Md−1,1d2(Lc)⊆Mc−1,1c2(Lc)
when2sign(d c )+sign(d c ) 0.
1 1 2 2
− − ≥
Usingthistechniqueweprovedthefollowing
Theorem1[7]. Leta>0, n C, 0<Ren <1/2,g(x) M 1(L )withc= s C: 1<Res<0 .
∈ ∈ 0−,1 c { ∈ − }
Then for almost all x>0 expansion (3) holds, where the inner and outer integralsare understoodin the
impropersense.
Theseresultswillbeappliedtosolvetheso-calledWebertypeintegralequation
¥
j (l )[Jn (xl )Yn +1(al ) Yn (xl )Jn +1(al )]dl = f(x), x>a>0 (11)
Z0 −
intheclassM 1(L ). ThekeyingredientwillbealsopropertiesforderivativeofBesselfunctions,namely,
0−,1 c
(see[1],Vol. II)
d
n n
x∓ x± Jn (x)= Jn 1(x), (12)
(cid:20) dx (cid:21) ± ∓
Webertypeintegralequation 3
d
n n
x∓ x± Yn (x)= Yn 1(x), (13)
(cid:20) dx (cid:21) ± ∓
andthefollowingintegral,whichisadirectconsequenceofrelation(2.13.15.4)in[2],Vol. 2,namely,
¥ 2 san +1 cos(pn )
Fn (x,s)=Z0 l −s[Jn (xl )Yn +1(al )−Yn (xl )Jn +1(al )]dl = p −x2−s+n G (s/2) G (−n −1)G (1+n −s/2)
s s a2 2 sxn +sG (n +1)G ( s/2)
×2F1(cid:18)1−2, 1+n −2; 2+n ; x2(cid:19)− p−a1+n G (1+n +−s/2)
s s a2 2 san +1 p s G (n +1 s/2)G (1 s/2)
×2F1(cid:18)−n −2, −2; −n ; x2(cid:19)−p −x1+n −scos(cid:16) 2 (cid:17) −G (2+n ) −
s s a2
F 1+n , 1 ; 2+n ; , 1<Res<0, x>a, (14)
×2 1(cid:18) −2 −2 x2(cid:19) −
where G (z) is Euler’s gamma-function and F (a,b;c;x) is Gauss’s hypergeometric function [1], Vol. 1,
2 1
havinganintegralrepresentationastheEulerintegral
G (c) 1
F (a,b;c;x)= ub 1(1 u)c b 1(1 xu) adu, (15)
2 1 G (b)G (c b)Z0 − − − − − −
−
for instance, underconditionsRea>0, Rec>Reb>0 x [0,1). Moreover,representation(15)givesus
∈
thefollowinguniformestimatefortheGaussfunction,whichwillbeusedbelow
G (c) 1
F (a,b;c;x) uReb 1(1 u)Re(c b) 1(1 xu) Readu
|2 1 |≤(cid:12)(cid:12)G (b)G (c−b)(cid:12)(cid:12)Z0 − − − − − −
(cid:12) (cid:12)
(cid:12) (cid:12) G (c)
(1 x) ReaB(Reb, Re(c b)) , (16)
≤ − − − (cid:12)G (b)G (c b)(cid:12)
(cid:12) − (cid:12)
whereB(a,b)isEuler’sbeta-function[1],Vol. 1.Let 1<Re(cid:12)n < 1/2.The(cid:12)n,denotingbythesameletter
(cid:12) (cid:12)
− −
Cvariouspositiveconstants,whichcanoccur,weobtain
s s a2 G (2+n )
F 1 , 1+n ; 2+n ; Cx2 Res(x2 a2)Res/2 1 ,
(cid:12)2 1(cid:18) −2 −2 x2(cid:19)(cid:12)≤ − − − (cid:12)G ((2(1+n ) s)/2)G ((2+s)/2)(cid:12)
(cid:12) (cid:12) (cid:12) − (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)2F1(cid:18)−n −2s, −2s; −n ; ax22(cid:19)(cid:12)(cid:12)≤Cx−2n −Res(x2−a2)n +R(cid:12)es/2(cid:12)G ( s/2G)G(−(n n)+s/2)(cid:12), (cid:12)
(cid:12) (cid:12) (cid:12) − − (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) s s(cid:12) a2 (cid:12) (cid:12)
F 1+n , 1 ; 2+n ; Cx2(1+n ) Res(x2 a2)Res/2 1 n
(cid:12)2 1(cid:18) −2 −2 x2(cid:19)(cid:12)≤ − − − −
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) G (2+(cid:12) n )
.
×(cid:12)G ((2 s)/2)G ((2(1+n )+s)/2)(cid:12)
(cid:12) − (cid:12)
Theseestimatesallowtoprovetheco(cid:12)nvergenceoftheintegral(11)asa(cid:12)nimproperone.Infact,writingitas
(cid:12) (cid:12)
N
lim j (l )[Jn (xl )Yn +1(al ) Yn (xl )Jn +1(al )]dl ,
N ¥ Z0 −
→
4 S.Yakubovich
wetakej fromthesubspaceM 1 (L ) M 1(L )withc= s C: 1<Res<0 . So, accordingto
1−/2,1 c ⊂ 0−,1 c { ∈ − }
Definition2j isgivenbyintegral(8)ofsomefunctionF (s)fromtheweightedL -space. Thenchanging
1
theorderofintegrationbyFubini’stheoremforeachfixedN andusing(14),itbecomes
N 1 m +i¥
Z0 j (l )[Jn (xl )Yn +1(al )−Yn (xl )Jn +1(al )]dl = 2p iZm i¥ F (s)Fn (x,s)ds
−
1 m +i¥ ¥
−2p iZm i¥ F (s)ZN l −s[Jn (xl )Yn +1(al )−Yn (xl )Jn +1(al )]dl ds, x>a.
−
Wewillprovethat
m +i¥ ¥
lim F (s) l −s[Jn (xl )Yn +1(al ) Yn (xl )Jn +1(al )]dl ds=0, x>a. (17)
N ¥ Zm i¥ ZN −
→ −
Todothis,weappealtotheasymptoticbehaviorofBesselfunctionsatinfinity[1],Vol. IItofindforfixed
x>a
2 1
Jn (xl )Yn +1(al )−Yn (xl )Jn +1(al )=−pl √xa(cid:20)cos(l (x−a))+O(cid:18)l (cid:19)(cid:21), l →¥ .
Substitutingthisexpressioninto(17)andintegratingbypartsintheinnerintegralwithrespecttol itgives
¥
l −s[Jn (xl )Yn +1(al ) Yn (xl )Jn +1(al )]dl =O (s +1)N−m −1 ,
ZN − | |
(cid:0) (cid:1)
and
m +i¥ ¥
F (s) l −s[Jn (xl )Yn +1(al ) Yn (xl )Jn +1(al )]dl ds
(cid:12)(cid:12)Zm −i¥ ZN − (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) m +i¥ (cid:12)
CN−m −1 F (s) (s +1)ds 0, N ¥
≤ Zm i¥ | | | | | |→ →
−
underassumptionm +1>0.Henceweprovedtheequality
¥ 1 m +i¥
Z0 j (l )[Jn (xl )Yn +1(al )−Yn (xl )Jn +1(al )]dl = 2p iZm i¥ F (s)Fn (x,s)ds, (18)
−
wheretheintegralintheright-handsideof(18)convergesabsolutely. Indeed,from(14),(16)andStirling’s
asymptoticformulaforthegamma-functionatinfinity[1],Vol. I,wehave
F(x,s) Cxm Ren ep s/2 s m , x>a.
− || −
| |≤ | |
Therefore,
m +i¥ m +i¥
Zm i¥ |F (s)Fn (x,s)ds|≤Cxm −Ren Zm i¥ |F (s)|ep |s|/2|sds|=C||j ||M1−/12,1(Lc)xm −Ren .
− −
Moreover,ittendstozerowhenx ¥ whenm Ren <0.
Let f M 1(L )withc= s→C:Res=g−>1/2 . Returningtointegralequation(11)andobserving
∈ 0−,1 c { ∈ }
thatduetotheabsoluteanduniformconvergenceoftheintegral(8)anditsderivativewithrespecttox x >
0
0functions f,j arecontinuouslydifferentiableon[a,¥ )andR , respectively,weactwiththediffe≥rential
+
operator xn d x n onitsbothsides. Hence,employing(12),(13),weobtain
dx −
(cid:2) (cid:3)
Webertypeintegralequation 5
¥ d
lj (l )[Yn +1(xl )Jn +1(al ) Jn +1(xl )Yn +1(al )]dl = xn x−n f(x). (19)
Z0 − (cid:20) dx (cid:21)
Itisallowedowingtotheuniformconvergencebyx a >aoftheintegralwithrespecttol in(19)ifwe
0
keepfunctionlj (l )inthesamespaceM 1(L ),i.e≥.
0−,1 c
1 m +i¥
lj (l )= 2p iZm i¥ Y (s)l −sds, (20)
−
where sY (s) L (c), c= s C: 1< m <0 . In fact, recalling the asymptotic behavior of Bessel
1
| | ∈ { ∈ − }
functionsatinfinityandintegratingbyparts,wefind
¥
(cid:12)(cid:12)ZN lj (l )[Yn +1(xl )Jn +1(al )−Jn +1(xl )Yn +1(al )]dl (cid:12)(cid:12)
1 (cid:12)(cid:12) ¥ sin(l (x a)) m +i¥ Y (s)l s 1dsdl +CN m 1 m (cid:12)(cid:12)+i¥ Y (s)ds
≤ p 2√xa(cid:12)ZN − Zm i¥ −− (cid:12) − − Zm i¥ | |
(cid:12) − (cid:12) −
≤O(N−m −1(cid:12)(cid:12))+p 2(a0−1a)√a0a(cid:12)(cid:12)ZN¥ cos(l (x−a))Zm(cid:12)(cid:12)m−+i¥i¥ Y (s)(s+1)l −s−2dsdl (cid:12)(cid:12)
O(N m 1)+ N−m −1(cid:12)(cid:12) m +i¥ Y (s)(s +1)ds 0, N ¥ , (cid:12)(cid:12)
≤ − − p 2(a0 a)√a0aZm i¥ | | | | | |→ →
− −
wherethedifferentiationundertheintegralsignisallowedviatheabsoluteanduniformconvergence.
Meanwhile,bythesamereasonsforx a,wehave(see(8))
≥
(cid:20)xn ddxx−n (cid:21)f(x)=−21p iZgg+i¥i¥ (s+n )F(s)x−s−1ds, x≥a,
andittendstozerowhenx ¥ viatheestimate −
→
d 1
xn x n f(x) Cx g 1 f , g >
(cid:12)(cid:20) dx − (cid:21) (cid:12)≤ − − M0−,11(Lc) 2
(cid:12) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
aswellas (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)ddx x−n f(x) (cid:12)≤Cx−g−n −1 f M0−,11(Lc)→0, x→¥ .
(cid:12) (cid:2) (cid:3)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
This meansthat equations(cid:12)(11), (19)are(cid:12)equivalent. H(cid:12)e(cid:12)nc(cid:12)(cid:12)e employingTheorem1, the uniquesolution of
(cid:12) (cid:12)
equation(19)hastheform
1 ¥ d
j (l )=−Jn2+1(al )+Yn2+1(al )Za Cn +1(l t,l a)tn +1dt(cid:2)t−n f(t)(cid:3)dt, l >0. (20)
Itcan be written in a differentformwith the integrationby parts, eliminatingouterintegratedtermssince
t1/2f(t)=o(1), t ¥ andtakingintoaccount(12),(13).Hence,
→
1 ¥ d
j (l )= Jn2+1(al )+Yn2+1(al )Za tf(t)(cid:20)t−1−n dtt1+n (cid:21)Cn +1(l t,l a)dt
l ¥
= Jn2+1(al )+Yn2+1(al )Za tf(t)[Jn (l t)Yn +1(al )−Yn (l t)Jn +1(al )]dt, l >0.
6 S.Yakubovich
Wesummarizeourresultsbythefollowing
Theorem2. Let 1<Ren < 1/2, f M 1(L )withc= s C:Res=g >1/2 , j M 1 (L )
− − ∈ 0−,1 c { ∈ } ∈ 1−/2,1 c
andlj (l ) M 1(L )with c= s C: 1<Res<Ren . Thenj is theuniquesolutionoftheWeber
∈ 0−,1 c { ∈ − }
typeintegralequation(11),givenbytheformula
l ¥
j (l )= Jn2+1(al )+Yn2+1(al )Za tf(t)[Jn (l t)Yn +1(al )−Yn (l t)Jn +1(al )]dt, l >0.
Acknowledgments
TheworkwaspartiallysupportedbyCMUP[UID/MAT/00144/2013],whichisfundedbyFCT(Portugal)
with national (MEC) and European structural funds through the programs FEDER, under the partnership
agreementPT2020.TheauthorthanksMarkCraddockforpointingouttheWebertypeequationforpossible
applicationstohisattention.
REFERENCES
1. A.Erde´lyi,W.Magnus,F.OberhettingerandF.G.Tricomi,HigherTranscendentalFunctions,Vols.IandII,McGraw-Hill,New
York,LondonandToronto(1953).
2. A.P.Prudnikov,Yu.A.BrychkovandO.I.Marichev,IntegralsandSeries:Vol.1:ElementaryFunctions,GordonandBreach,New
York(1986);IntegralsandSeries:Vol.2:SpecialFunctions,GordonandBreach,NewYork(1986);Vol.3:MoreSpecialFunctions,
GordonandBreach,NewYork(1990).
3. E.C.Titchmarsh,Weber’sintegraltheorem,Proc.Lond.Math.Soc.,22(2),(1924),15-28.
4. E.C.Titchmarsh,AnIntroductiontotheTheoryofFourierIntegrals,ClarendonPress,Oxford(1937).
5. VuKimTuan, O.I.Marichev andS.Yakubovich, Compositionstructureofintegral transformations, J.Soviet Math., 33(1986),
166-169.
6. S. Yakubovich and Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions. Mathematics and its
Applications,287.KluwerAcademicPublishersGroup,Dordrecht(1994).
7. S.Yakubovich,OntheWeberintegralequationandsolutiontotheWeber-Titchmarshproblem.Arxiv.1612.05455.
DEPARTMENTOFMATHEMATICS,FACULTYOFSCIENCES,UNIVERSITYOFPORTO,CAMPOALEGRESTR.,687;4169-007
PORTO,PORTUGAL
E-mailaddress: [email protected]