ULLETIN u. Maaysia Math. Soc. Secod Seies 22 999 - of the MALAYSIAN MATHEMATICAL SOCIETY The Neuti Poduct of the Distibutios ad RIAN FISHER AND 2 FATMA AL-SIREHY Depatet of Matheatics ad Copute Sciece Leiceste Uivesity Leiceste LE 7RH Egad e-ai: fb@e.ac.uk 2 P.O. o 32477 Jeddah Saudi Aabia e-ai: gcp328@kaau.edu.sa Abstact. The euti poduct of the distibutios ± ± 2 L ad 2 L. ad is evauated fo I the foowig we et N be the euti see va de Coput [] havig doai N { 2 L L} ad age the ea ubes with egigibe fuctios fiite iea sus of the fuctios : > 2 L ad a fuctios which covege to zeo i the oa sese as teds to ifiity. We ow et ρ be ay ifiitey diffeetiabe fuctio havig the foowig popeties: i ρ fo ii ρ iii ρ ρ iv ρ d. Puttig δ ρ fo 2 L it foows that { δ } is a egua sequece of ifiitey diffeetiabe fuctios covegig to the Diac deta-fuctio δ.
2. Fishe ad F. A-Siehy Now et D be the space of ifiitey diffeetiabe fuctios with copact suppot ad et D be the space of distibutios defied o D. The if f is a abitay distibutio i D we defie. f * δ f t f δ t fo 2 L. It foows that { f } is a egua sequece of ifiitey diffeetiabe fuctios covegig to the distibutio f. A fist etesio of the poduct of a distibutio ad a ifiitey diffeetiabe fuctio is the foowig see fo eape [2]. Defiitio. Let f ad g be distibutios i D fo which o the iteva a b f is the k-th deivative of a ocay suabe fuctio F i L p a b ad g k is a ocay suabe fuctio i L q a b with / p / q. The the poduct fg gf of f ad g is defied o the iteva a b by fg k i i k [ i ] i Fg k i. The foowig defiitio fo the o-coutative euti poduct of two distibutios was give i [3] ad geeaizes Defiitio. Defiitio 2. Let f ad g be distibutios i D ad et g g* δ. We say that the euti poduct f ο g of f ad g eists ad is equa to the distibutio h o the iteva a b if N i fg φ h φ fo a fuctios φ i D with suppot cotaied i the iteva ab. Note that if i fg φ h φ we sipy say that the poduct f g eists ad equas h. This defiitio of the euti poduct is i geea o-coutative. It is obvious that if the poduct f g eists the the euti poduct f ο g eists ad f g f ο g. Futhe it was poved i [3] that if the poduct fg eists by Defiitio the the poduct f ο g eists by Defiitio 2 ad fg f ο g. The et two theoes wee poved i [3].
3 The Neuti Poduct of the Distibutios ad Theoe. Let f ad g be distibutios ad suppose that the euti poducts f ο g ad f ο g eist o the iteva a b. The the euti poduct f ο g eists ad o the iteva ab. f ο g f ο g f ο g Theoe 2. The euti poduct o eists ad fo ± ± 2 L ad 2 L. We ow pove the foowig theoe: πcosec π o δ 2! Theoe 3. The euti poduct o eists ad cosec [ 2 ] π π o c ρ ψ δ 2 2! fo ± ± 2 L ad 2 L whee deotes the Gaa fuctio ad ψ Poof. We wi fist of a suppose that < <. The ad ae ocay stube fuctios ad Thus / * δ t δ t dt. fo 2 L ad so
4. Fishe ad F. A-Siehy d / / t δ / t δ t t d dt / t δ t υ tυ υ dυ t tδ / / t dt t dt t δ t dt 3 dt d whee the substitutio tv has bee ade deotes the eta fuctio ad i geea p p q p q q μ μ Makig the substitutio t y. We have μ / t t dt y ρ y dy δ 4 fo / t t δ t dt y ρ 2 L. I paticua whe it is easiy poved by iductio that y y dy yρ y dy 5 y ρ y dy! 2 6! y yρ y dy φ c ρ 2 7 fo 2 L whee i φ i 2 L
ad 5 The Neuti Poduct of the Distibutios ad c ρ t ρ t dt. Futhe puttig { ρ } > K sup we have u u du ρ K K ad so whe we have u du d / d K 8 Now et ϕ be a abitay fuctio i D. The whee < ξ < ad so Sice ϕ ϕ ϕ! ϕ ϕ ξ!! ϕ d.! ξ 9 d { } ϕ K d sup ϕ ξ
. Fishe ad F. A-Siehy 6 it foows fo equatios 3 to 9 that i N ϕ [ ] ½ c ϕ ρ φ. ½ ϕ Diffeetiatig the idetity μ μ μ patiay with espect to it foows that [ ] 2!! ad takig ogs ad diffeetiatig the idetity L gives. i i ψ ψ 2 I paticua we have.! φ 3 It ow foows fo equatios ad 3 that!! [ ].! cosec φ ψ π π 4
7 The Neuti Poduct of the Distibutios ad Futhe πcosec π! 5 ad equatio 2 ow foows fo equatios 4 ad 5 fo the case < <. Now et us suppose that equatio 2 hods whe k < < k ad 2 L whee k is a positive itege. This is tue whe k. Thus if k < < k it foows fo ou assuptio that π cosec π δ 2! ρ ψ ] o fo 2 L. It foows fo Theoe that [ ] o o 2 Thus π cosec π 2! [ ] 2 c ρ ψ δ cosec o π π δ 2! π cosec π 2! ρ ψ 2 ] δ. o π cosec π 2! ρ ψ 2 ] δ [ ψ ψ 2 ] δ π cosec π 2! π cosec π 2! ρ ψ ] δ sice fo equatio 2 we have ψ 2 ψ
8. Fishe ad F. A-Siehy ad so ψ ψ 2. Equatio 2 ow foows by iductio fo < 2 L ad 2 3 L. To cove the case we ote the poduct. eists by Defiitio ad. 6 fo a. Let us suppose that equatio 2 hods whe k < < k ad whee k is a positive itege. This is tue whe k. Thus if k < < k it foows fo ou assuptio that ρ ψ 2 ]. o 2 ½ π cosec π δ It foows fo equatio 6 ad Theoe that [ ] o o 2 o ½ π cosec π δ ½ [ 2 c ρ ψ 2 ] δ π cosec π o ½ [ 2 c ρ ψ ] δ π cosec π Equatio 2 ow foows by iductio fo < 2 L ad. Now et us suppose that equatio 2 hods whe k < < k ad 2 L whee k is a positive itege. This tue whe k. The fo a abitay fuctio ϕ i D we have φ ψ
9 The Neuti Poduct of the Distibutios ad whee ψ ϕ is aso i D. It foows fo ou assuptio with k < < k that N i ψ π cosec π 2! ρ ψ ] ψ ad so π cosec π 2! N i ϕ ρ ψ ] ϕ π cosec π 2! ρ ψ ] ϕ Equatio 2 ow foows by iductio fo copetig the poof of the theoe. > 2 L ad 2 L Cooay 3.. The euti poduct o eists ad cosec o π π ] ρ ψ δ 7 2! fo ± ± 2 ad 2 L. Poof. Equatio 7 foows o epacig by i equatio 2. Theoe 4. The euti poduct o eists ad cosec o π π ] ρ Ψ δ 8 2! fo ± ± 2 L ad 2 L. Poof. Diffeetiatig equatio patiay with espect to we get 2 cot cosec o o π π π δ 2!
. Fishe ad F. A-Siehy ad o usig equatio 2 it foows that π cosec π o [ cot 2 ] π π c ρ ψ δ. 2! 9 Takig ogs ad diffeetiatig the idetity π cosec π gives ψ ψ π cot π 2 ad equatio 8 foows fo equatios 9 ad 2. Cooay 4.. The euti poduct o eists ad cosec o π π ] ρ ψ δ 2! 2 fo ± ± 2 L ad 2 L. Poof. Equatio 2 foows o epacig by i equatio 8. We fiay ote that if we epace by i equatio 2 we get π cosec π δ 2! [ ] 2c ρ ψ o. ad we see that the poduct of the distibutios whe. ad is coutative oy Refeeces. J.G. va de Coput Itoductio to the euti cacuus J. Aayse Math. 7 959-6 29-398. 2.. Fishe The poduct of distibutios Quat. J. Math. Ofod 22 97 29-298. 3.. Fishe A o-coutative euti poduct of distibutios Math. Nach. 8 982 7-27. Keywods ad phases: distibutio deta-fuctio euti iit euti poduct. 99 AMS Subject Cassificatio: 46F