On the Exchange Property for the Mehler-Fock Transform

Transcription

1 Avaabe at App. App. Math. SS: Vo. 11, ssue (December 016), pp Appcatos ad Apped Mathematcs: A teratoa oura (AAM) O the Echage Property for the Meher-Foc Trasform Abhshe Sgh Departmet of Mathematcs Amty sttute of Apped Sceces Amty Uversty Uttar Pradesh, oda mathdras@gma.com; asgh8@amty.edu Receved: December 15, 015; Accepted: August 8, 016 Abstract The theory of Schwartz Dstrbutos opeed up a ew area of mathematca research, whch tur has provded a mpetus the deveopmet of a umber of mathematca dscpes, such as ordary ad parta dffereta euatos, operatoa cacuus, trasformato theory ad fuctoa aayss. The tegra trasforms ad geerazed fuctos have aso show euvaet assocato of Boehmas ad the tegra trasforms. The theory of Boehmas, whch s a geerazato of Schwartz dstrbutos are dscussed ths paper. Further, echage property s defed to costruct Meher-Foc trasform of tempered Boehmas. We vestgate echage property for the Meher-Foc trasform by usg the theory of Meher-Foc trasform of dstrbutos. Agebrac propertes ad covergece s aso proved for ths reato o the tempered Boehmas whch s a atura eteso of tempered dstrbuto. eywords: Dstrbuto spaces; tempered Boehmas; Fourer trasform; Meher-Foc trasform MSC 010 o.: 46F10, 46F99, 44A10 1. troducto The cocept of Boehmas s motvated by the reguar operator troduced by Boehme (1973), whch forms a subagebra of the fed of Musńs operators ad thus they cude oy such fuctos whose support s bouded from the eft. The theory of Boehmas (uotet of seueces), ts propertes ad dfferet casses of Boehma spaces are studed by Musńs et a. (1981), Musńs (1983, 1995). 88

2 AAM: ter.., Vo. 11, ssue (December 016) 89 Tempered Boehmas s a atura eteso of tempered dstrbuto whch, therefore, maes t possbe to defe a eteso of the Fourer trasform for ths cass of Boehmas. The Fourer trasform of a tempered Boehma s a dstrbuto. A ftey dfferetabe fucto f : R C s caed rapdy decreasg f sup sup(1 m R 1... ) m D f ( ), (1) for every oegatve teger m, where = ( 1,... ), = ( 1,... ), 's are o-egatve teger, = , ad D. () The space of rapdy decreasg fuctos s deoted by S(R ) or smpy by S. f f ad S, the the covouto s we defed ad f. A seuece foowg codtos () ( ) d 1, for a, R ( f )( ) f ( u) ( u) du (3) R S s caed a deta seuece f t satsfes the () () ( ) d M, for some costat M ad for a, R m ( ) d 0, for every ad > 0. f S ad = 1, the the seuece of fuctos s a deta seuece. A cotuous fucto R such that f ( ) p( ) for a f : R C s caed sowy creasg f there s a poyoma p o R. The space of sowy creasg fucto w be deoted by (R ) or smpy by. Let f. { } s a deta seuece uder usua otato. The the space of euvaece casses of uotets of seuece w be deoted by ad ts eemets w be caed tempered Boehmas. For F [ / ] defe D f F = [( f D )/( )]. f F s a Boehma correspodg to dfferetabe fucto, the D F.

3 830 Abhshe Sgh f F [ / ] ad f S, for a, the F s caed a rapdy decreasg f Boehma, the space of whch s deoted by S. f the we defe the covouto F G F = [ / ] ad G [ / ], f = [( f g )/( )]. = g S what foows, we w deote by S' the space of tempered dstrbutos; that s, the space of cotuous ear fuctoa o S. The Meher-Foc trasform of a tempered dstrbuto f, deoted by Mf, s the fuctoa defed by Mf () = f (M), where M s the Meher-Foc trasform of defed by Baerj et a. (008). The Meher-Foc trasform o geerazed fuctos s studed by Patha (1997). The Meher-Foc trasform of Boehma spaces are vestgated by Looer ad Baerj (008, 009, 009). Secto we study Meher-Foc trasform ad ts propertes ad vestgate the echage property for the Meher-Foc trasform. Secto 3, agebrac propertes ad covergece s proved for ths reato o the tempered Boehmas.. The Meher-Foc Trasform ad the Echage Property The Meher-Foc trasformato s defed as [cf. Yaubovch ad Lucho (1994, p. 149)]: 1 M[ f ( )] F( P 1 ( ) f ( ) d, r 0, (4) r ad ts verso s gve by f ( ) r tah( P ( ) F( dr, 1. (5) 0 1 r The geerazato of the Meher-Foc trasformato s gve by [cf. Patha (1997, p. 343)] 0 m F(, f ( ) P (cosh ) sh d, (6) 1 r, where P m 1 (cosh ) s the geerazed Legedre fucto, defed for compe vaues of the r parameters, m ad by / m, ( z 1) m m 1 z P ( z) F1 1; ;1 m; / (1 m)( z 1), (7) m for compe z ot yg o the cross-cut aog the rea -as from 1 to The verso formua of (6) s. f ( ) m, ( P 0 r 1 (cosh ) F( dr, (8)

4 AAM: ter.., Vo. 11, ssue (December 016) 831 Where 1 m 1 m 1 m 1 m ( r r r r Whe m, (6) ad (8) ca be wrtte as m ( ) ( 1 r. (9) F( P ), (10) 0 r 1 (cosh )sh f ( d ad f ( ) r tah( r ) P (cosh ) ( ) 0 1 F r dr, (11) r whereas for m 0, (6) ad (8) reduce to (4) ad (5), respectvey. The Parseva reato for the Meher-Foc trasformato s defed as [cf. Seddo (1974, pp )] 0 r tah( F( G( dr f ( ) g( ) d, (1) 1 whose covouto s M[ f g] M[ f ] M[ g]. (13) The asymptotc behavor for (7) s defed by Patha (1997, p. 345) as ad Rem m, O( ), 0, 1/ r (cosh ) (1/) P O( e ),, (14) P O(1), r 0, (cosh ) (sh ) ( { e e O( r )}, r. m, 1/r 1/( m1) 1/ 1/ m1/ r ( m t) 1 (15) Smary, the fucto (, defed by (9), possesses the foowg asymptotc behavor [cf. Patha 1997, p. 345)] O r r m () r 1m ( 1 [1 O( r )], r. m ( ), 0 Re 1 Re, (16)

5 83 Abhshe Sgh The dstrbutoa geerazed Meher-Foc trasform f M ( R ), where R deotes the set of postve rea umbers ad Re(m), 1/, s defed as [cf. Patha (1997, p. 346)] F( : f ( ), P (cosh ), r 0, (17) m, 1 r where the space M R ) s the dua of the space M R ), whch s the coecto of a ( ( ftey dfferetabe compe vaued fucto defed o ope terva (0, ) deoted by R + such that for every o-egatve teger, ( ) sup ( ) ( ), (18) 0 where m D (coth ) D, (19) (1 cosh ) (1 cosh ) ad ( ), ( ) O( ), 0, O( ),. (0) The topoogy over M R ) s geerated by separatg coecto of semorms ( { } 0 ad s a seuetay compete ocay cove topoogca vector space. D(R + ), the space of ftey dfferetabe fuctos of compact support wth the usua topoogy, s a ear subspace of M R ). ( From the propertes of the hypergeometrc fuctos, the geerazed Legedre fucto [cf. Patha (1997, p. 346)], satsfes the foowg dffereta euato D m y (coth ) Dy r (1 cosh ) (1 cosh ) 1 4 y 0. (1) Therefore, () m, 1 m, P 1/ r (cosh ) r P 1/ r (cosh ). 4 Reatos (14) ad (15) prove the boudedess for the Legedre fucto [cf. Patha (1997, pp , Lemma , Euato (11.3.))] ( ) r 1/ m, 1 mr,0 P 1/ r (cosh ) C ( ) mr P 1/ (cosh ), (3) where C s a costat depedet of ad r ad, m r s Re( m ).

6 AAM: ter.., Vo. 11, ssue (December 016) 833 The dfferetabty of the Meher-Foc trasform s defed by [cf. Patha (1997, p. 347)] m, F ( : f ( ), P 1 (cosh ), (4) r r where f M R ), Re( m),re( m) 1/, 1/, r 0. ( Whe s a o-egatve teger depedg o f, the asymptotc behavor of the Meher-Foc trasform s O(1), r 0, Fr () O r ( ), r, (5) where F( Cma r 1 4. (6) * For the operator : M ( R ) M ( R ) uder aready stated symbos ad for f M R ), M ( R ), we defe the operator trasformato formua by ( * ( ) f ( ), ( ) f ( ), ( ), (7) ad for f beg the geerazed Meher-Foc trasformato, M * 1 ( ) f ( ) ( 1) r M f ( ) t. 4 f f M R ) ad M R ) we have by trasposto by Baerj et a. (008) ( ( M ( f ), = f, M ( ), (8) where fucto f s absoutey tegrabe ad s a testg fucto of rapd descet. For a famy {φ } = {φ }, where s a de set ad φ M R ) defe [Ataasu ad Musńs (005)]: ( S,, we Ψ({φ } ) = { R + : Mφ () = 0, }. (9) A famy of pars {(f, )}, where f M R ) S ad φ M R ) to have the echage property f ( ( S,, s sad

7 834 Abhshe Sgh f f,,. (30) We w deote by A the coecto of a fames of pars {(f, )}, where s a de set, f M R ) S ad φ M R ) S,, satsfyg the echage property such that ( ( ({ } ). f ( ) s a deta seuece, the ({ } ). Defto 1. f {(f, )} A, the the uue F D' (R) such that Mf = M F for a deoted by M{(f, )}. w be Let {( f, )},{( g, )} A. f f g for a ad, the we wrte {( f, )} ~ {( g, )}. Ths reato s ceary symmetrc ad refeve. We w show that t s aso trastve. Let f {( f, )},{( g, )},{( h, )} A. {( f )}, )} ~ {( g, ad {( g )} ~ {( h )} L L, the f g, g h, (31) for a,, L. Therefore, f g, g h, (3) for a,, L. Sce s commutatve, we have f h. (33) ow f ad L. Sce ({ } ) ad (3) hods for every, we cocude that f h for a ad L, whch meas that {( f, )} ~ {( h, )} L. Theorem 1. f a famy of par{( f, )} has the echage property ad {( ) } c (the compemet of {( ) } R + ), the there ests a uue F D' () such that M[f ] = FM[φ ],. (34)

8 AAM: ter.., Vo. 11, ssue (December 016) 835 Proof: For every there ests ad > 0 such that ( ) a ope eghborhood of. The we ca defe F Mf / M that eghborhood. Let for some > 0, we have M ( ) for a U ad M ( ) for a V, where U ad V are ope sets. Sce f f, we have M Mf M Mf M ad Mf M Mf M, (35) o U V Theorem.. Ths shows that F s a uue fucto. There ests {( f, )} A, for every F D (R) ad such that F M({( f, )} ). Proof: Sce D (R) deotes the space of smooth fucto wth compact support, there ests a tota seuece { such that M D (R) for a. The for every, there s f } M ( R ) S such that Mf = M F. Ceary {( f, )} A ad F M({( f, )} ). Ths competes the proof of the theorem. Defto. [Ataasu ad Musńs (005)] Let {U } be a ope coverg of R + ad et { be such that ( ) 0 for U. A } famy { such that ({ } ) w be caed tota. } Lemma 1. [Ataasu ad Musńs (005)] f { } ad { } are tota, the { * } s tota. Theorem 3. Let The, Proof: {( f, )}, {( g, )} A. M {( f, )} ~ {( g, )} f ad oy f M({( f, )} ) M({( g, )} ).

9 836 Abhshe Sgh Here, f F M({( f, ) }) ad G M({( g, )} ). {( f, )} ~ { g, )}, the F M M Mf M MgM FM M GM M,,. (37) Hece, F = G, by Lemma 1. ow assume F = G. The, Hece, Mf M F M M G M M Mg M,,. (38) Ths competes the proof of the theorem. Theorem 4. {( f, )} ~ {( g, )}. There ests a deta seuece ( ) such that for every T, T [{( f, )} ] for some f. Proof: Let ( ) be a deta seuece such that M D(R). The for ay T, we have TMψ M ( R ) S, sce MT D'(R). Coseuety, MTM Mg for some g M ( R ) S. t s easy to chec that T [{( g, )} ]. Sce f g ad ( ) ( ) s a deta seuece, where ( ) does ot deped o T, hece the theorem s proved. 3. Agebrac Propertes ad Covergece becomes a vector space wth the addto operato, defed by [{( f, )} ] [{( g, )} ] [{( f g, )} ]. (39) Moreover, mutpcato by a scaar ad the operato are defed by

10 AAM: ter.., Vo. 11, ssue (December 016) 837 f [{( f, )} ] [{( f, )} ], C. (40) [{( f, )} ], [{( g, )} ] ad g M R ) for a, ( the for the operato we ca defe [{( f, )} ] [{( g, )} ] [{( f g, )} ]. (41) Defto 3. [Ataasu ad Musńs (005)] Let T 0, T 1, T,.... The the seuece (T ) s sad to coverge to T 0, whch s wrtte as T T 0 f there ests a tota famy { } such that (a) there ests tempered dstrbuto f,, where ad such that T [{ f,, } ] for a = 0,1,,..., (b) f, f, 0 M ( R ) as for every. Theorem 5. The Meher-Foc trasform s a somorphsm from to D'(R). Proof: Sce T T 0 f ad oy f T T 0 0, t suffces to prove the cotuty at 0. Let T 0 s. The there ests tempered dstrbuto f, where ad such that T, } ] for a = 1,,... ad f, 0 M R ) S as for every. f [{ f, D(R), the there are 1,..., such that ( The, m MT supp suppm. m1 m m ( MT M ) m m1 M m m1 m1 m M M m m Mf, 0 m m1 M m, (4)

11 838 Abhshe Sgh because m Mf, 0 for, due to the cotuty of the Meher-Foc trasform M ( R ). Ths proves the cotuty of M : D (R), because m MT 0 M ( R ) for every D(R), mpes m MT 0 D' (R). ow, assume m MT 0 D (R). By Theorem 4, there ests a deta seuece ( ), such that for every, we have T [{( f,, )} ] for some f,. Let ( ), be a deta seuece such that M D(R) for every. The, Sce m MT M M 0 M R ) for every,. ( MT M f,,, m Mf, M 0 M R ) S, ( whch mpes m f, 0, M R ) S. ( But, T [{( f,, )} ] [{( f,, )} ], (43) for a = 0,1,,.... Thus, we have T 0. Ths proves the theorem. 3. Cocusos The preset paper focused o the echage property for the Meher-Foc trasform va tempered Boehmas whch s the atura eteso of tempered dstrbutos. Agebrac propertes ad covergece proved for ths reato are usefu ths area for deveopmet of the covouto propertes ad other operatos of Meher-Foc trasform of dstrbutos ad Boehmas [Patha et a. (016)]. The formua ad the property estabshed ths paper may aso be sutabe for a utraboehmas. The aforesad aayss ca be used to deveop the Cadero s formua for Meher-Foc trasform [Patha et a. (016)]. Acowedgemet: The author s thafu to the referees for frutfu crtcsm ad commets for the mprovemet.

12 AAM: ter.., Vo. 11, ssue (December 016) 839 REFERECES Ataasu, D. ad Musńs, P. (005). O the Fourer trasform ad the echage property, terat.. Math. Math. Sc. Vo. 16, pp Boehme, T.. (1973). The support of Muss operators, Tras. Amer. Math. Soc. Vo. 176, pp Looer, Desha ad Baerj, P.. (008). Meher-Foc trasform of tempered dstrbuto.. da Acad. Math. Vo. 30, o. 1, pp Looer, Desha ad Baerj, P.. (009). Meher-Foc trasform for tempered Boehmas. Mogoa Math.. Vo. 13, pp Looer, Desha ad Baerj, P.. (009). Meher-Foc trasformato of utradstrbuto. Appcatos ad Apped Mathematcs. Vo. 4, o. 1, pp Musńs,. ad Musńs, P. (1981). Quotets de sutes ecurs appcatos das aayse foctoee. C. R. Acad. Sc. Pars. Vo. 93 Seres, pp Musńs, P. (1995). Tempered Boehmas ad utradstrbutos. Proc. Amer. Math. Soc. Vo. 13, pp Musńs, P. (1983). Covergece of Boehmas. apa. Math. Vo. 9, pp Patha, R. S. (1997). tegra trasforms of Geerazed Fuctos ad ther Appcatos, Gordo ad Breach Scece Pub., Caada, Austraa, Frace. Patha, R. S. ad Sgh, Abhshe (016). Waveet trasform of geerazed fuctos {M p } spaces. Proc. da Acad. Sc. Math. Sc. Vo. 16, o., pp Patha, R. S. ad Sgh, Abhshe (016). Dstrbutoa Waveet Trasform. Proc. at. Acad. Sc., Sec. A, Vo. 86, o., pp Sgh, Abhshe (015), O echage property for the Hartey trasform. taa. Pure App. Math. Vo. 35, pp Sgh, Abhshe Looer, Desha ad Baerj, P.. (010). O the echage property for the Lapace trasform. da oura of Scece ad Techoogy Vo. 3, o., pp Seddo,.. (1974). The Use of tegra Trasforms, Tata McGraw-H Pub. Co. Ltd., ew Deh. Yaubovch, S. B. ad Lucho, Y. F. (1994). The Hypergeometrc Approach to tegra Trasforms ad Covoutos. uwer Academc Pub., Dordrecht, Bosto, Lodo. Zemaa, A.H. (1987). Dstrbuto Theory ad Trasform Aayss. Dover Pubcatos c. ew Yor. Frst pubshed by McGraw-H Boo Co., ew Yor, 1965.