The Neutrix Product of the Distributions r. x λ

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Λαμία Μανιάκης
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1 ULLETIN u. Maaysia Math. Soc. Secod Seies 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 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 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 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 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

5 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

6 . 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 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 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 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!

10 . 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 Fishe The poduct of distibutios Quat. J. Math. Ofod Fishe A o-coutative euti poduct of distibutios Math. Nach Keywods ad phases: distibutio deta-fuctio euti iit euti poduct. 99 AMS Subject Cassificatio: 46F

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