Convolution product formula for associated homogeneous distributions on R

Transcript

1 Cnvlutin prduct frmula fr assciatd hmgnus distributins n R Ghislain R. FRANSSENS Blgian Institut fr Spac Arnmy Ringlaan 3 B-80 Brussls Blgium ghislain.franssns@arnmy.b Octbr 009 Abstract Th st f assciatd hmgnus distributins AHs basd n R H R cnsists f th distributinal gnralizatins f pwr-lg functins with dmain in R. This st cntains th majrity f th n-dimnsinal distributins n typically ncuntrs in physics applicatins. Rcnt wrk dn by th authr has shwn that th st f AHs n R admits a clsd cnvlutin structur H R prvidd that critical cnvlutin prducts ar dfind by a functinal xtnsin prcss. In this papr th gnral cnvlutin prduct frmula fr H R is drivd. Cnvlutin f AHs n R is fund t b assciativ xcpt fr crtain critical tripl prducts. Ths critical prducts ar shwn t b nn-assciativ in a minimal and intrsting way. Kywrds. Gnralizd functin Assciatd hmgnus distributin Cnvlutin Ral lin. AMS 000 MSC 46F0 46F 46F30. Intrductin Lt H R dnt th st f assciatd hmgnus distributins AHs basd n i.. with supprt in th ral lin R [4] [7]. Th lmnts f H R ar th distributinal gnralizatins f pwr-lg functins with dmain in R [0] []. A cmprhnsiv intrductin t AHs basd n R is givn in [4]. Fr structur thrms f AHs basd n R s [5]. Th st H R is imprtant fr th fllwing rasns. i H R cntains th majrity f th n-dimnsinal distributins n typically ncuntrs in physics applicatins such as th dlta distributin δ th ta distributin η a nrmalizd Cauchy s principal valu th stp distributins ± svral s calld psud-functins gnratd by taking Hadamard s finit part f crtain divrgnt intgrals assciatd Risz krnls gnralizd Hisnbrg distributins all thir gnralizd drivativs and primitivs and many familiar thrs []. ii H R cntains th krnls f th fractinal intgratin/drivatin pratrs n R. iii H R is a subst f th tmprd distributins S R and just as S R is als H R clsd undr Furir transfrmatin [0]. Ths rasns strngly mtivat th cnstructin f a cnvlutin structur fr H R. In tw prvius paprs by th authr [6] and [7] it was shwn hw th cnvlutin prduct can b dfind n H R and th rsulting cnvlutin structur was dntd by H R. Mr spcifically it was first fund in [6] that if f a and g b ar AHs with dgrs f hmgnity a and b whs supprt is nt bundd n th sam sid n can still cnstruct thir cnvlutin prvidd th rsulting dgr f hmgnity ab is nt a natural numbr. Th

2 cass fr which a b N ar calld critical cnvlutin prducts. In [7] it was thn shwn that critical cnvlutin prducts gnrat partial distributins i.. squntially cntinuus linar functinals nly dfind n a subspac S r f Schwartz spac S f smth functins f rapid dscnt [4] [7]. Sinc S is a Frécht spac [3 p. 84] [9 Appndix] S is lcally cnvx [3 p. 9] and th cntinuus xtnsin vrsin f th Hahn-Banach thrm applis [3 p. 56]. This thrm nsurs that in this cas partial distributins can b xtndd t distributins dfind n th whl f S hnc t a tmprd distributin [3 p. 90]. Basd n this justificatin critical cnvlutin prducts wr dfind in [7] as gnrally nn-uniqu xtnsins in H R. Cmbining th rsults frm [6] and [7] thn shws that H R is clsd. Onc such a structur bcms availabl crtain cnvlutin quatins dfind vr th whl lin.g. diffrntial quatins with cnstant cfficints intgral quatins with pwrlg krnls tc. rduc t algbraic quatins with rspct t th structur H R. Rcall that th classical fractinal intgratin/drivatin calculus is nly dfind vr half-lins [6]. Mrvr a substructur f H R can b idntifid that srvs t justify th distributinal fractinal intgratin/drivatin calculus vr th whl ral lin a rsult t b prsntd lswhr. By cmbining th structur H R and prprty iii abv tgthr with th gnralizd cnvlutin thrm [7 p. 9] [9 p. 0] nw als dfins a clsd multiplicatin prduct fr AHs n R. This givs us a multiplicatin structur H R. which is ismrphic t H R undr Furir transfrmatin. Th structur H R. is furthr studid in th frth cming papr [8]. Th imprtanc f H R. stms frm th fact that it nw bcms pssibl fr th first tim t giv a rigrus maning t distributinal prducts such as δ δ.δ and many intrsting thrs as a distributin fr instanc it is fund in [8] that δ cδ c C arbitrary. Attmpts t dfin a multiplicatin prduct fr th whl st f distributins such as in [] using rgularizatin squncs and passag t th limit r [] using th Furir transfrm fail t prduc maningful prducts f crtain AHs as a distributin.g. fr δ s [3 Chaptr ]. Othr apprachs allw t dfin a multiplicatin prduct fr a diffrnt class f gnralizd functins G such as in Clmbau s wrk which blngs t nn-standard analysis [3] and whr δ G but his δ has n assciatin in. Th hr adptd dfinitin fr multiplicatin f AHs maks that H R. is intrnal t Schwartz distributin thry in th sns that all ur prducts ar in H R S R R and this uniqu fatur maks ur multiplicatin prduct stand ut frm th abv apprachs. Furthr ur multiplicatin prduct fr AHs is nt in cntradictin with Schwartz impssibility thrm [5] fr rasns xplaind in [7] and [8]. Th structurs H R and H R. hav imprtant cnsquncs fr varius aras in th applid scincs. Fr instanc H R. maks it pssibl t us distributins frm H R in nn-linar mdls. istributinal prducts invlving.g. th dlta distributin th stp distributins tc. ccurring in fr instanc nn-linar shck wav prpagatin mdls nw acquir a rigrus maning within classical distributin thry. Furthr th st H R f n-dimnsinal AHs can b xtndd undr suitabl pullbacks alng scalar functins t highr dimnsinal AHs invariant undr crtain cntinuus grups. Th s cnstructd highr dimnsinal AHs inhrit thir algbraic structur frm th n-dimnsinal structurs H R and H R.. Ths invariant highr dimnsinal AHs play an imprtant rl in th thry f partial diffrntial quatins. Crtain particular AHs invariant undr O n rduc t th Laplac pratr fr a particular valu f thir dgr f hmgnity and ar thus imprtant in ptntial thry. Th inhritd cnvlutin structur thn prvids th justificatin fr algbraically slving Pissn s quatin and vn cmplx dgr gnralizatins f it. Ths AHs that ar invariant undr O 3 i.. th Lrntz grup play a rl in wav thry and mr gnrally ths invariant undr O p q ar th tls that svrally simplify th intgratin

3 f th gnralizd ultra-hyprblic quatin i.. th wav quatin in a flat univrs with p tim dimnsins and q spac dimnsins. Th lattr O p q-invariant substs f AHs ar f paramunt imprtanc in advancd analysis.g. fr dvlping Cliffrd Analysis vr psud- Euclidan spacs. Th lattr has dirct applicatins in physics and fr p and q 3 can b rgardd as a functin thry that is tailr-mad t slv prblms in lctrmagntism and quantum thry. In this papr w will mak th structur H R xplicit by driving th gnral cnvlutin prduct frmula fr AHs basd n R. In th fllwing sctins w first calculat a numbr f particular cnvlutin prducts f AHs which will b ndd t driv th final prduct frmula in a structurd way. Ths particular cnvlutin prducts ar intrsting and usful in thir wn right. Many f thm ar nw.g. Thrms and spcially 4. In sctin 5 w wrk ut th particular natur f th nn-assciativity f critical cnvlutin prducts. W us th ntatin and dfinitins intrducd in [4]. Cnvlutin prducts f δ k η k k! and sgn k! W hr cllct sm imprtant simpl prducts which will b ndd in subsqunt sctins. A dirct calculatin using th gnral rsults btaind in [6] and [7] prducs th symmtric cnvlutin prduct Tabl hlding k l N and with c C an arbitrary cnstant. Rcall that δ k is th gnralizd multiplicatin drivativ pratr f dgr k [4 q. 39]. Th pratr x l sgn l! is a gnralizd multiplicatin primitiv pratr f dgr l n R as is mad vidnt by ntry C..4 in Tabl if l < k. With ths bsrvatins mst rsults in Tabl ar asily undrstd. As an illustratin f hw ths rsults wr btaind w giv in Appndix th xplicit calculatin fr rsult C.3.3. Th wll-knwn rlatin C..3 shws that th gnralizd multiplicatin drivativs f a distributin mnmial fllw th sam rul as th rdinary drivativs f a functin mnmial. Th lss knwn rlatin C..4 with k l givs th gnralizd multiplicatin drivativs f a distributin mnmial f ppsit parity in a similar way. Rsult C.. rprducs a gnralizatin f th wll-knwn anti-invlutin prprty f th distributinal Hilbrt transfrmatin s als [6 q. 6]. Th rsults C..3 C..4 C.3.3 C.3.4 and C.4.4 dpnd n a linar cmbinatin f tw xtnsins which is rflctd by th ccurrnc f th arbitrary cnstant c. In particular fr k 0 C..3 and C..4 yild th Hilbrt transfrm pairs x l l! η x l sgn η l! x l c l! x l ln x c x l l! l!. Equatin shws that x l l N ar igndistributins f th Hilbrt pratr c C. This rsult als shws that th cmmnly statd prprtis fr th classical Hilbrt transfrmatin: i that th Hilbrt transfrmatin f a cnstant functin is zr and ii that th Hilbrt transfrmatin is a parity rvrsal pratin ar n lngr tru in gnral fr th distributinal Hilbrt transfrmatin. Th right-hand sid f is an l -th dgr primitiv f η. 3 Th cnvlutin prducts k! n z z ± W nd th fllwing spcial rsults t structur sm calculatins in th nxt sctin. 3

4 Prpsitin With z ± th hmgnus nrmalizd half-lin krnls givn by [4 q. 46] hlds that k n N and with c ± C arbitrary k! z n z ± k! n b b ± 0 b C\Z 3 zl 0 kl ± l n0 n>0 c ± l l N 4 k l! k! l n l l ± c ± k l! l Z 5 Prf. i Eq. 3. Using [4 qs. 50 and 5] and [6 q. 4] givs k N and z C\Z k! z ± k k k z ± 0. By [6 Thrm 6] thn hlds n N and z C\Z that k! n z z ± z n k! z ± 0. ii Eq. 4. ii. Lt k < l. W hav by [4 q. 47] l Z k Z [0l] k! l ± xk k! ±l δ l 0 by C..3 in Tabl. By 3 and [6 Thrm 6] thn hlds n N that k! z n z ± zl 0. ii. Lt l k. ii.. Fr n 0. W hav by a dirct calculatin using [4 q. 47] Nw with [4 q. 50] k l l kl 6 l k l kl. 7 k! l k k k l kl kl kl ; l k l! and k! l k k k l l kl kl kl l l k l!. 4

5 and ii.. Fr n Z. Frm [7 q. 7] fllws that k 0k;nl ;0 C 0n n0 cs n/ l sgn k l! S0n n0 α cs n/ l k l! q C n 0n nq z z zkl kl cs n q / z q z zkl q q S 0n nq z z zkl kl z q z zkl l nl;0k ;0 C n0 n0 cs n/ l sgn k l! Sn0 n0 α cs n/ l k l! q n C n0 nq z z zkl kl cs n q / z q z zkl q q S n0 nq z z zkl kl z q z. Hrin ar α ± C\ {i i} arbitrary and zkl C 0n nq S 0n nq nq 0 q n n q nq k0 nq k Enqk q Z [0n] n nq q q n n n nq nq 0 q n q k0 k Bnqk q Z [n] and C n0 nq S n0 nq nq 0 q n n q nq l0 l nq l Enql q Z [0n] n nq q n nq q n n nq nq q Z [n]. 0 q n q l0 l l B nql Only ths trms fr which n q is vn cntribut t th xprssins fr 0k;nl ;0 and. Fr n q vn hlds du t [4 q. 8] that th xprssins fr th cnstants nl;0k ;0 C 0n nq and Cn0 nq rduc t q Z [0n] C 0n nq qn C n0 nq qn Similarly du t [4 q. 9] hlds q Z [n] S 0n nq qn S n0 nq qn. 5

6 Thrfr and by using [4 qs. 50 5] w gt and k 0k;nl ;0 l sgn n0 k l! xkl k l! 0<n n z z zkl n0 kl 0<n n z z n0 S0n n0 α cs n/ zkl n0 S0n n0 α cs n/ l k l! l k l! z n z zkl n0 S0n n0 α x cs n/ kl k l! 8 l nl;0k n0 ;0 l sgn k l! xkl k l! 0<n kl z n z zkl kl n0 kl 0<n n z z n0 Sn0 n0 α cs n/ n0 Sn0 n0 α cs n/ zkl l k l! l k l! kl z n z zkl n0 Sn0 n0 α x cs n/ kl k l!. 9 Furthr k N l Z [0k] and with c c C arbitrary w hav by [7 q. 70] and 8 9 that k k z n z zl k 0k;nl ;0 k l c k l! z n z zkl n0 S0n n0 α cs n/ k l c k l! 0 and n z z zl k nl;0k ;0 kl l c k l! k z n z zkl l n0 Sn0 n0 α cs n/ k l c k l!. Nw using [4 q. 50] hlds that k! z n z zl k k k n z z z n z zl zkl k k n z z 6 zl

7 and k! z n z k zl k k z n z zl z n z zl k k z n z zkl. Substituting hrin th rsults 0 givs k! z n z zl n0 S0n n0 α cs n/ k l c k l! k! z n z zl l n0 Sn0 n0 α cs n/ k l c k l!. This can b rcast in th fllwing frm k! z n z n0 zl S 0n k! z n z zl l n>0 S 00 0 α k c n0 α cs n/ k c n0 n>0 S 00 0 α k c S n0 n0 α cs n/ k c l k l! l k l!. Fr n 0 th cnstants α and c bcm rlatd bcaus k l acquirs a uniqu valu du t th cmpact supprt f l. Similarly th cnstants α and c bcm rlatd bcaus l k acquirs a uniqu valu du t th cmpact supprt f l. Ths rlatins ar asily btaind frm quating 0 with 6 7. W btain S 00 0 α ± k c ±. Thn k! z n z ± zl ±l n0 n>0 c l ± k l! whrin w dfind c ± ± S 0n n0 α ± cs n/ S 00 0 α ±. iii Eq. 5. Using [4 q. 50] w hav l Z and k! n l l k! n l l k k k l n l zkl k k n l l n z z k k k l n l l n l k k z n z Similarly as fr 0 w nw btain l Z k k z n z n l l zkl n0 S0n n0 α cs n/ k c 7 zkl. 3 l k l! 4

8 and l n l k k z n z zkl l n0 Sn0 n0 α cs n/ k l c k l!. 5 Hrin ar α ± C\ {i i} and c ± C fur arbitrary cnstants. Substitutin f 4 5 in 3 givs k! n l l n0 S0n n0 α cs n/ k l c k l! k! n l l l n0 Sn0 n0 α cs n/ k l c k l!. Cntrary t cas ii.. abv fr n 0 th cnstants α ± and c ± ar n lngr rlatd sinc l ± ds nt has cmpact supprt s th prducts k l and l k ar als fr n 0 nn-uniqu xtnsins. Thrfr with arbitrary c ± C. k! n l l ± c ± 4 Th cnvlutin prducts l k l! k! n z z Th fllwing rsults will b ndd in th prduct frmula which is drivd in sctin 7. Prpsitin With n z z th nrmalizd parity krnls f th first kind givn by [4 qs ] hlds that k n N and with c c c C arbitrary and k! n b b 0 b C\Z 6 n0 l l/ l c l l N 7 n>0 c k l! k! n w w wl l k k! w n w wq c q q N 8 k q! k! n w w wq c q q N 9 k q! k! n w w wl l k k! n b b 0 b C\Z 0 n0 l l/ l c k! n w w wq c k! w n w wq c n>0 c l l N k l! q q N k q! q q N. 3 k q! 8

9 Prf. Lt k N. A. Fr z. i Eq. 6 b C\Z. With [4 qs. 50 and 85] k! b k k k b b cs /b kb k kb cs /b k k b b k Using [6 qs. 40 and 4] this rducs t k! b kb k kb cs /b k k kb k kb 0. Thn by [6 Thrm 6] and n N. ii Eq. 7 b l N. Frm 4 fllws that k! n b b 0. k! q w w w q w wl k<l0 l k q0 l q>0 c l k l! 4 with c C arbitrary. Put Using [4 q. 307] and 4 w gt E n x B n x n a q x q 5 q0 n b q x q. 6 q0 k! n w w wl k! i/n l l/ E n w i/ w w i l l/ B n w l l/ n q0 a qi/ nq i l l/ n q0 b qi/ nq l l/ n q0 a qi/ nq i l l/ n q0 b qi/ nq r sinc a 0 E n and E 0 i/ w w wl k! q w w w q w wl k<l0 l k q0 l q>0c k! n w w wl k<l 0 l k l l/ i/ n E n n>0 l c l c l k n0 l l/ l c n>0 c 9 l k l! l k l! l k l!

10 with arbitrary c c c C. iii Eq. 8 b q Z. Using [4 q. 9] 5 and C.3.3 in Tabl with arbitrary c ± c c C givs k! q q k! w w w w wq c x q q c c q k q! c c q r k! q cxkq with c C arbitrary. iv Eq. 9 b q Z. Using [4 q. 94] and C.3.4 in Tabl w gt k! q k! x q sgn q q! c q k q! with c C arbitrary. Applying [7 qs. 7 and 60] [4 qs. 303 and 305] and linarity t th rsults btaind in iii and iv yilds 8 9 n N and l Z. B. Fr z. i Eq. 0 b C\Z. With [4 qs. 50 and 86] k! b k k k b b sin /b kb k kb sin /b k k b b k Using [6 qs. 40 and 4] this rducs t k! b kb k kb sin /b k k kb k kb 0. ii Eq. b l N. Frm 4 fllws that k! q w w w q w wl k<l0 l k q0 l q>0 c l k l! 7 with c C arbitrary. Using [4 q. 307] 5 6 and 7 w gt k! n w w wl k! i/n l l/ E n w i/ i l l/ B n w l l/ n q0 a qi/ nq i l l/ n q0 b qi/ nq l l/ n q0 a qi/ nq i l l/ n q0 b qi/ nq i/ w w w w wl k! q w w w q w wl k<l0 l k q0 l q>0c 0. l k l!

11 r k! n w w wl k<l 0 l k l l/ i/ n E n n>0 l c l c n0 l l/ l c l l k n>0 c k l! with arbitrary c c c C. iii Eq. b q Z. Using [4 q. 99] and C.3.4 w gt k! q k! q cq. x q sgn q! l k l! with c C arbitrary. iv Eq. 3 b q Z. Using [4 q. 97] 5 and C.3.3 with arbitrary c ± c c C givs k! q q k! w w w w wq c x q q c c q k q! c c q r k! q cxkq with c C arbitrary. Applying [7 qs. 7 and 60] [4 qs. 304 and 306] and linarity t th rsults btaind in iii and iv yilds 3 n N and l Z. In particular fr l 0 Thrm givs with c C arbitrary. k! n w w w0 n0 n>0 c k! n w w w0 c k! 8 k!. 9 5 Assciativity f critical cnvlutin prducts W nw rturn t th qustin f assciativity in cas at last n f th cnvlutin prducts is critical. Thrm 3 Lt critical cnvlutin prducts b dfind by xtnsins as in [7 q. 69]. Thn with z ± th hmgnus nrmalizd half-lin krnls givn by [4 q. 46] hlds that i j k N and a b c C a i a j b b c k c a i a j b b c k c 30 c k c a i a j b b c k c a i a j b b. 3

12 Prf. I. Eq. 30. A. W first prf a b c a b c b a c. 3 i Fr b c n bc Z. i. a n bc / Z. W hav by dfinitin [7 q. 69] a b c a b;c ; with b;c ; cs b n bc n bc n bc z sin b z n bc zn z z bc zn bc rx n bc and r C arbitrary. By 3 w gt a b;c ; cs b a n bc n bc an bc sin b a z z n bc zn z z bc zan bc. W hav frm [6 q. 40] and using [6 Thrm 6] a z z lim zn bc b n bc b sina Substituting bth ths rsults givs a n bc n bc an bc sinb sina b ab sina sina b ab an bc n bc csa an bc n bc z z a b;c sin b ; sina an bc n sin a b bc an bc sina sin c sin a b sin a b c sin a b c which du t [6 q. 3] quals a b;c ; a b c. i. a n bc n Z. W hav by dfinitin [7 q. 69] a b c a b;c ; zan bc. with b;c ; cs b n bc n bc n bc z sin b z n bc zn z z bc zn bc r x n bc and r C arbitrary. Nw with a Z and using 4 5 w gt a b;c cs b a n bc ; n bc n sin b a z z n bc zn z z bc zn r x n 33

13 with r C arbitrary. W hav by dfinitin [7 q. 69] a n bc a;n bc ; 34 a z z 0a;n bc zn bc ;. 35 Frm th gnral xprssins [7 qs. 7 and 8 9] w gt th fllwing tw particular xprssins. a a;n bc ; cs a n n n z sin a z n bc zn z z rx zn n with r C arbitrary. Sinc a Z this rducs t with r 3 C arbitrary. b with a;c ; cs a n n n r3 x n 36 0a;n bc ; sin a C 0 x n sgn 0 n! S0 0 α x n n! C 0 z z n z z zn cs a z z n z z sin a S 0 C 0 S 0 z z n z z z z n z z zn zn zn C 0 q S 0 q q q l 0 q q c 0 l E ql l0 q c 0 q q q q 0 q l0 l q c 0 l B ql. Sinc a Z this rducs t 0a;n bc ; cs a C 0 z z n z z z z n z z S 0 zn zn. Mr xplicitly using c 0 q q and B B /6 w find that C 0 S 0 c 0 c 0 0 B. Thn c0 0 E 0 and 0a;n bc ; cs a z z. 37 zn 3

14 Substituting xprssins and 37 in 33 givs a b;c ; cs b cs a n 3 n n sin b cs a z z n zn z z cs b r r x n n! cs b cs a n n n sin b cs a z z n zn z z cs b r r cs b cs a x n n! r sinc a Z zn zn a b;c ; cs b cs a sin b sin a n n n sin b cs a cs b sin a z z n zn z z rx n zn with r C arbitrary. Idntificatin with ab;c ; cs a b n n n z sin a b z n bc zn z z zn rx n givs a b;c ; ab;c ;. ii Fr b c / Z but a b c n Z. W hav by [6 q. 3] and dfinitin [7 q. 69] sinc a b c a sinb c bc sinb sinb c bc sinc sinb c a;bc ; sinb sinb c n. Substituting hrin th xprssin a;bc ; cs a n n n z sin a z n bc zn z z with r C arbitrary givs a b c zn sinc cs a n sinb c n n n sinb sinb c sinc sin a z z sinb c n bc zn z z sinc sinb c rxn. 4 zn rx n n n

15 Fr a b c Z hlds that sin a c sinc cs a csc sin a n sinb. W can thrfr writ a b c csc sin a n sinb c n n csc sin a n sinb c n n sinc cs a n sinb c n n sin a c sinb c n n sinc sin a z z sinb c n bc zn z z zn csc sin a sinb c sinc sinb c r x n n!. Th scnd lin f this xprssin bcms Thn csc sin a n sinb c n n sinc cs a n sinb c n n sin a c sinb c n n sin a c n sinb c n n. a b c csc sin a n sinb c n n sinc sin a z z sinb c n bc zn z z csc sin a sin a c sinb c sinb c sinc sinb c r zn x n n!. Fr a b c Z hlds that csc sin a sinb c sinc sin a sinb c cs a b sin a b s w btain a b c cs a b n n n sin a b z z n bc zn z z r x n zn 5

16 with r C arbitrary. Idntificatin yilds again a b c ab;c ;. Cllcting rsults w hav nw prvd th first quality in 3. By symmtry in th paramtrs a and b th scnd quality in 3 immdiatly fllws. B. u t [7 q. 7] th gnral quatin 30 fllws. II. Eq. 3. Similarly. Thrm 3 cmbind with linarity shws that if critical cnvlutin prducts ar dfind as an xtnsin in th sns f [7] thn fm a fm b fm c 3 f b m fm a fm c 3 and f a m fm b fm c 3 fm a fm b fm c 3 H diffr by a distributin f th frm rx with r C arbitrary. Hwvr if an intrmdiat prduct is critical whil th final prduct is nt say a b N whil a b c / N thn th intrmdiat indtrminacy ds nt prpagat int th final prduct du t 3. Summarizing th cnvlutin prduct f critical tripls f AHs basd n R is nt assciativ. Hwvr its dviatin frm assciativity is asily quantifiabl and has th fllwing bautiful intrprtatin. If a b c N w d nt nd t pay attntin t th rdr in which th prduct fm a fm b fm c 3 is valuatd sinc w can always cmpnsat fr any ffct that a chang f rdr inducs by a chang f xtnsin f th final rsult. In thr wrds if a b c N calculating f a m f b m f c m 3 in diffrnt rdrs mrly rsult in diffrnt xtnsins f th critical tripl prduct. This rsult can als b statd as fllws. By cnsidring xtnsins f partial distributins as an quivalnc st thn Thrm 3 affirms that th cnvlutin prduct f critical tripls f quivalnc sts f AHs basd n R is assciativ. f b m Obviusly whn rgardd as a partial distributin dfind n S {k} th prduct fm a fm c 3 with a b c N acquirs a uniqu maning indpndnt f th rdr in which th cnvlutins ar takn bcaus thn assciativity hlds xactly fr th partial distributins. Lt us illustrat this by an xampl. Cnsidr th thr distributins δ and cmmnly usd t illustrat th lack f assciativity f th cnvlutin prduct fr distributins [7 p. 8]. On classically writs δ δ δ 0 0 δ δ undfind undfind. Rsults btaind in sm f ur prcding paprs allw us t cmplt th third xprssin. Frm [4 q. 5] fllws x and frm [7 q. ] with c C arbitrary s that ; ; x c x x x x c x x sgn c x cx and c C arbitrary. Using δ x w can nw rplac th abv classical argumnt with δ 0 c with c δ 0 c with c 0 δ 0 c with c C arbitrary. 6

17 Thus w can attribut t th prduct f this tripl in arbitrary rdr th fllwing maning δ 0 c fr sm cmplx cnstant c. In this xampl n f th factrs δ is f cmpact supprt and its prduct with any f th thr factrs givs ris t a distributin fr which th cnvlutin with th rmaining factr is uniquly dfind. This accidntly maks th prducts δ and δ uniqu and th cnstant c acquirs a uniqu valu in ths tw cass. Th third prduct hwvr δ is th mr intrsting as it rvals th gnral structur f th distributin δ. Fr th tripl δ and w still hav an intrmdiat critical prduct but th final prduct is nt critical sinc its dgr is / N. Nw w gt δ δ δ δ δ cx 0. Mr gnrally if nn f th thr factrs is f cmpact supprt and th rsulting dgr a b c is a natural numbr thn thir prduct in arbitrary rdr is a distributin that has maning as a nn-uniqu xtnsin and which w can rgard as an quivalnc st. 6 Cnvlutin prducts f m a a and m b b Th rsults btaind in this sctin ar fundamntal t prf th gnral cnvlutin prduct frmula in th nxt sctin. Thrm 4 With z n z th nrmalizd parity krnls f th first kind givn by [4 qs ] hlds that m n N and a b C a m a b n b z mn z zab 38 a m a b n b z mn z zab 39 a m a b n b z mn z zab 40 whrin m z z at z Z is rplacd by any xtnsin m z z and m z z at z Z is rplacd by any xtnsin m z z in th sns f [7 q. 60]. Prf. Th cass which d nt rquir xtnsins hav alrady bn prvd in [6 Thrm ]. W hr nly prf th rmaining cass whr a b r a b is a psitiv intgr f th in [6 Thrm ] xcludd parity typ. I. Cmbining with [4 qs and 98 99] givs p η p 4 p η p. 4 II. Using 4 4 calculat th fllwing p q N and with c c c C arbitrary cnstants. A. Eq. 38. i a C\Z and b q. W gt using assciativity hlding in this cas by [6 Thrm 0] a q a η p a η p a p ap. 7

18 ii a k N. By Thrm 3 and with [4 qs. 95 and ] w gt th fllwing. ii. Fr b q Z k q k k/ δ k k k/ η k η q k k/ δ k η k k/ η k η q k q c q k k/ η δ k k k/ δ δ k q k q c q k k/ η k k/ δ δ k q k q c q k k/ η k k/ δ k k/ kq k k/ kq k q c q k k/ k k/ η kq k q c q k kq k kq k q c q. k k/ k k/ kq u t 7 w can absrb any indtrminat trm ccurring in th right-hand sid in th lft-hand sid factr q by a chang f xtnsin. Thrfr k q k kq k kq. ii. Fr b q Z k q k k/ δ k k k/ η k q k k/ δ k k/ η δ k q k k/ δ k k/ η k k q c q k k/ k k/ kq k k q c q k kq k r by a chang f xtnsin f kq kq k k/ kq k k q c q k k/ kq k k/ k k/ η kq k k q c q k q k kq k kq Th cas k Z ] was alrady cvrd by [6 Thrm ]. iii a Z. W gt iii. fr a p Z p q η p η q η η p q c x pq δ c x pq pq pq c x pq 8.

19 r du t 8 and ntry C.3.3 f Tabl which lt us absrb any indtrminat trm ccurring in th right-hand sid in bth lft-hand sid factrs by a chang f xtnsin p q pq. iii. Fr a p Z p q p η q η p q c x pq η pq c x pq pq c x pq pq by a chang f xtnsin. iv Fr a b / Z : a b p Z. Using [4 q. 85] and [7 q. 70] w hav a b cs a/ cs b/ cs a/ cs b/ ab ab a b b a ab ab a;b ;0 b;a ;0 c c x p p!. Frm [7 q. 7] fllws a;b ;0 C 00 0 cs a x p sgn p! sin a z z z z and bcaus cs b cs a and sin b sin a b;a ;0 C 00 0 cs a x p sgn p! sin a z z z z S 00 0 α cs a zp S 00 0 α cs a zp. Thus with [4 q. 50] and bcaus cs b/ p sin a/ x p p! x p p! a b p sin a x p p! S 00 p z z z z p 0 c x p p! with c C arbitrary. Hnc a b B. Eq. 39. B. 0 α S 00 0 α cs a c c sin a z z z z zp c p. x p p! zp x p p! 9

20 i a C\Z. W gt using assciativity hlding in this cas by [6 Thrm 0] a q a η q η a q η aq aq. ii a k N. By Thrm 3 and with [4 qs. 95 and ] w gt th fllwing. ii. Fr b q Z k q k k/ δ k k k/ η k q k k/ δ k k/ η δ k q k k/ δ k k/ η k k q c q k k/ k k/ kq k k q c q k kq k r by a chang f xtnsin kq k k/ kq k k q c q k k/ kq k k/ k k/ η kq k k q c q k q k kq k kq Th cas k Z ] was alrady cvrd by [6 Thrm ]. ii. Fr b q Z k q k k/ δ k k k/ η k η q k k/ δ k η k k/ η k η q k q c q k k/ η δ k k k/ δ δ k q k q c q k k/ η k k/ δ δ k q k q c q k k/ η k k/ δ k k/ kq k k/ kq k q c q k k/ k k/ η kq. k k/ k k/ kq k q c q k kq k kq k q c q k kq k kq k k q c q. u t w can absrb any indtrminat trm ccurring in th right-hand sid in th lft-hand sid factr by a chang f xtnsin q k q k kq 0 k kq.

21 iii a Z. W gt iii. Fr a p Z p q η q η η p q c x pq δ pq c x pq pq c x pq ; η p r du t and ntry C.3.3 f Tabl which lt us absrb any indtrminat trm ccurring in th right-hand sid in bth lft-hand sid factrs by a chang f xtnsin p q pq. iii. Fr a p Z p q p η q η p q c x pq η pq c x pq pq c x pq pq by a chang f xtnsin. iv Fr a b / Z : a b p Z. Using [4 q. 85] and [7 q. 70] w hav a b sin a/ cs b/ sin a/ cs b/ Frm [7 q. 7] fllws ab ab a b b a a;b ;0 C 00 0 cs a x p sgn p! S00 sin a z z z z and bcaus cs b cs a and sin b sin a b;a ;0 C 00 0 cs a x p sgn p! S00 sin a z z z z ab ab a;b ;0 b;a ;0 c c 0 α cs a zp 0 α cs a zp. x p p! x p p! x p p! Thus with [4 qs. 50 and 97] and bcaus cs b/ p cs a/. a b p sin a x p p! S 00 p z z z z p 0 c x p p! 0 α S 00 0 α cs a c c sin a z z z z zp c x p p! zp x p p!

22 with c C arbitrary. Hnc a b p B. i b C\Z. W gt using assciativity which hlds in this cas by [6 Thrm 0] p b η p b η p b η. pb. pb ii b l N. By Thrm 3 and with [4 qs. 300 and ] w gt th fllwing. ii. Fr a p Z p l η p l l/ η l l l/ δ l p l l/ η η l l l/ η δ l l p c x pl p l l/ δ l δ l l/ δ l η l p c x pl p δ l l l/ δ l l/ η l p c x pl l l/ pl l l/ pl l l/ δ l l/ η l p c x pl l l/ l l/ pl l p c x pl l pl l r by a chang f xtnsin f ii. Fr a p Z pl p l l/ l l/ η pl l pc x pl p l l pl l pl p l p l l/ η l l l/ δ l p δ l l l/ η l l/ δ l l p c x pl l l/ pl l l/ pl l l/ η l l/ δ l l p c x pl l l/ l l/ η pl l l p c x pl l pl l pl l l p c x pl r by a chang f xtnsin f p pl l l. l l/ l l/ pl pl l pl.

23 Th cas l Z was alrady cvrd by [6 Thrm ]. iii b Z. W gt iii. Fr b q Z p q by a chang f xtnsin. iii. Fr b q Z p q q η p q c x pq η pq c x pq pq c x pq η p pq η p η q η η p q c x pq δ pq c x pq pq c x pq r by a chang f xtnsin f bth factrs in th lft-hand sid p q pq. iv Fr a b / Z : a b p Z. By cmmutativity this cas is quivalnt t th cas B. iv. C. Eq. 40. i a C\Z. W gt using assciativity which hlds in this cas by [6 Thrm 0] a q a η q η a q η aq aq. ii a k N. By Thrm 3 and with [4 qs. 300 and ] w gt th fllwing. ii. Fr b q Z k q k k/ η k k k/ δ k q k k/ η k k/ δ δ k q k k/ η k k/ δ k k q c q k k/ k k/ η kq k k/ kq k k q c q k k/ kq k k/ k k/ kq k k q c q k kq k kq k k q c q r by a chang f xtnsin f kq k q k kq 3 k kq.

24 Th cas k Z was alrady cvrd by [6 Thrm ]. ii. Fr b q Z k q k k/ η k k k/ δ k η q k k/ η k η k k/ δ k η q k q c q k k/ δ δ k k k/ η δ k q k q c q k k/ δ k k/ η δ k q k q c q k k/ δ k k/ η k k/ kq k k/ kq k q c q k k/ k k/ kq k q c q k kq k r by a chang f q iii a Z. iii. Fr a p Z kq k k/ k k/ η kq k q c q k q k kq k p q r by a chang f xtnsin f kq. p η q η p q c x pq η pq c x pq q pq c x pq ; p q pq. iii. Fr a p Z p q η p η q η η p q c x pq δ pq c x pq pq c x pq. r by a chang f bth factrs in th lft-hand sid p q pq. iv Fr a b / Z : a b p Z. Using [4 q. 85] and [7 q. 70] w hav a b sin a/ sin b/ sin a/ sin b/ ab ab a b b a ab ab a;b ;0 b;a ;0 c c x p p!. 4

25 Frm [7 q. 7] fllws a;b ;0 C 00 0 cs a x p sgn p! sin a z z z z and sinc cs b cs a and sin b sin a b;a ;0 C 00 0 cs a x p sgn p! sin a z z z z S 00 0 α cs a zp S 00 0 α cs a zp. Thus with [4 q. 50] and sinc cs b/ p sin a/ x p p! x p p! a b p sin a x p p! S 00 0 α S 00 0 α cs a c c sin a z z z z p z z z z zp c x p p! p 0 c x p p! with c C arbitrary. Hnc a b p. zp III. Applying th pratr a m b n and using jintly cntinuity at a b Z m n N as shwn in [5 Thrm 5] and [7 Thrm 7] rsults in th quatins Th distributins z and z ar knwn in fractinal calculus thry as th krnls f th cnvlutin pratrs that gnrat th Risz ptntial z ϕ and cnjugat Risz ptntial z ϕ f a functin ϕ S [6 p. 4]. Mr gnrally n culd call z m z ϕ th assciatd Risz ptntial and z m z ϕ th assciatd cnjugat Risz ptntial bth f rdr m f a functin ϕ S. Th cnvlutin prducts givn in Thrm 4 rval that ths assciatd krnls satisfy a bautiful algbraic prprty which is an ntirly nw rsult. In particular by using 0 η Thrm 4 yilds th fllwing Hilbrt transfrm pairs p N η z n z zp z n z 43 zp η z n z zp z n z zp ; 44 and x p p! η z n z zp z n z zp 45 η z n z zp z n z 46 zp with c C arbitrary. This cmplts th rsults givn in [6 qs ]. Equatins rval th nn-assciativity f th Hilbrt transfrmatin btwn n z z and n z z at z p z p as a cnsqunc f. 5

26 Crllary 5 With z n z x±i0 th nrmalizd cmplrnls givn by [4 qs ] hlds that m n N and a b C a m a xi0 b n b xi0 z mn z xi0 zab 47 a m a xi0 b n b xi a m a xi0 b n b xi0 z mn z xi0 zab 49 whrin z m z x±i0 at z Z is rplacd by any xtnsin z m z x±i0 in th sns f [7 q. 60]. Prf. Th crllary fllws frm using linarity [4 q. 350] and Thrm 4. Cmbining with [4 q. 350] givs k Z η k m k x±i0 i k m k x±i0. 50 Equatin 50 cmplts [6 q. 60]. 7 Cnvlutin prduct frmula Th nrmalizd parity rprsntatin [5 Thrm 4] can du t [6 qs ] and b writtn as z C m fm z α n z z n z 5 with cfficint distributins n0 α n z q n z δ q n z η H 0 and whrin th functins q n and q n n Z [0m] satisfy th prprtis [5 qs. 30 3]. In 5 it is undrstd that z n z at z k Z stands fr any xtnsin k n k and n z z at z k Z stands fr any xtnsin k n k. This als allws us t absrb any trm f th frm c c C pssibly prsnt in fm k in ithr k r k. W will us this in th fllwing. Thrm 6 Lt f a m g b n m α p a a p a 5 p0 n β q b q b b. 53 q0 Th cnvlutin prduct f fm a by with f a m and gn b a Ω a C b Ω b C and m n N is givn mn gn b γ t t0 m γ t a b z t z zab 54 p0 q0 n pqt α p β q. 55 In 54 z m z at a b k Z stands fr any xtnsin k m k and m z z at a b k Z stands fr any xtnsin k m k. 6

27 Prf. Th prpsitin fllws as a cnsqunc f linarity [4 q. 85 and 86] Thrm 3 and Thrm 4. Thrm 6 clarly shws that th cnvlutin prduct f tw assciatd hmgnus distributins is mst asily calculatd using th nrmalizd parity rprsntatin givn in [5 Thrm 4]. Undr th cnvlutin prduct th distributins a n a and if a k Z k n k bhav lik rdinary mnmials f dgr n undr rdinary multiplicatin. Th algbra f th cfficint distributins is an assciativ cnvlutin algbra vr R ismrphic t th bicmplx numbrs which cntains zr divisrs s [4 Sctin 5.6]. A Appndix A. Th cnvlutin prduct Lt k l C. By [4 q. 50] w hav k! xl l! k k k x l k! l! l kl kl kl l l k k l l l Using [7 q. 70] th mixd-supprt prducts hrin ar givn by k l l k;l ;0 c kl k l! k l;k ;0 c ll k l! with arbitrary c c C and th cnvlutin prduct thrfr bcms k. k! xl l! kl kl kl k k;l ;0 l l;k ;0 c c l. 56 k l! Althugh w d nt nd thm fr this calculatin it is intrsting t calculat th rsidus f k;l ; and l;k ;. Thy ar btaind frm [7 q. 0] as k;l ; sin k l ω k l! 0 l;k ; sin l l ω k l! 0. Thus a;b ; at a b Z ds xist as a distributin dfind n S which w dnt by k;l ;. Hr w thus hav an xampl f a critical cnvlutin prduct that w d nt nd l;k ; t xtnd. In this cas w culd dfin k l l k;l ; k;l ;0 57 k l;k ; l;k ;0 58 instad f using th gnral rul [7 q. 0]. In this way n nn-uniqunss is intrducd at this stag. 7

28 Th analytic finit parts ar btaind frm [7 q. ] as ;0 kl sgn k l! α /α α /α k l! k;l ;0 ll sgn k l! α /α α /α l;k l l k l! Ths analytic finit parts ar nt uniqu sinc thy dpnd n th arbitrary cmplx cnstant α α which paramtrizs th dirctin alng which th limit a b k l is takn s [7] and th indtrminacy is prprtinal t l. S ntwithstanding that w avidd th nn-uniqunss stmming frm th prcss f xtnding a partial distributin t a distributin dfind n S w still gt an indtrminacy frm th nn-uniqunss f th analytic finit part. W nw cntinu with ur calculatin. Substituting givs fr th mixd-supprt trms in 56 k k whrin w dfind c c c α/α l l l k xkl sgn x k l! α/α α /α kl kl kl Substituting bth ths rsults in 56 w btain r k! xl l! l c k l! α /α. By [4 q. 5] w hav xkl sgn x. k l! kl kl kl k k l l l xkl sgn x k l! xkl sgn x k l! l c k l! with c C arbitrary. Equatin 6 is rsult C.3.3 in Tabl. Rfrncs k k! x l l! c l k l! 6 [] W. Ambrs Prducts f distributins with valus in distributins J. Rin Angw. Math [] P. Antsik J. Mikusinski and R. Sikrski Thry f distributins. Th squntial apprach Elsvir Amstrdam 973. [3] J.F. Clmbau Nw Gnralizd Functins and Multiplicatin f istributins Math. Studis 84 NrthHlland Amstrdam 984. [4] G.R. Franssns Algbras f AHs basd n th lin I. Basics prprint: ALG.pdf 007. [5] G.R. Franssns Structur thrms fr assciatd hmgnus distributins basd n th lin Math. Mthds Appl. Sci [6] G.R. Franssns Th cnvlutin f assciatd hmgnus distributins n R Part I Appl. Anal

29 [7] G.R. Franssns Th cnvlutin f assciatd hmgnus distributins n R Part II Appl. Anal [8] G.R. Franssns Multiplicatin prduct frmula fr assciatd hmgnus distributins n R submittd 009. [9] G. Fridlandr and M. Jshi Intrductin t Th Thry f istributins nd d. Cambridg Univ. Prss Cambridg 998. [0] I.M. Gl fand and G.E. Shilv Gnralizd Functins Vl. I Acadmic Prss Nw Yrk 964. [] R.P. Kanwal Gnralizd Functins Thry and Tchniqu nd d. Birkhausr Bstn 998. [] W. Rudin Principls f Mathmatical Analysis McGraw-Hill Nw Yrk 976. [3] W. Rudin Functinal Analysis McGraw-Hill Nw Yrk 99. [4] L. Schwartz Théri ds istributins Vls. I&II Hrmann Paris 957. [5] Schwartz L. Sur l impssibilité d la multiplicatin ds distributins C. R. Math. Acad. Sci. Paris [6] S.G. Samk A.A. Kilbas and O.I. Marichv Fractinal Intgrals and rivativs Taylr/Francis Lndn 00. [7] A.H. Zmanian istributin Thry and Transfrm Analysis vr Nw Yrk

30 δ l η l x l l! δ k δ kl C.. η kl C.. k l η k δ kl C.. k l c k! c sgn k! x lk lk! x lk lk! l kl! x l sgn l! C..3 l<k δ kl k l C..3 l<k η kl k l C.3.3 c l C.3.4 kl! l sgn kl! c x lk sgn C..4 lk! x lk lk! l kl! Tabl : Sm particular cnvlutin prducts f hmgnus distributins. ln x c C..4 C