I J Pure Al S Teol, 04, 64-77 Ierol Jourl o Pure d Aled Sees d Teoloy ISSN 9-607 Avlle ole wwwjos Reser Per O New Clss o rmo Uvle Fuos Deed y Fox-r Geerled yereomer Fuo Adul Rm S Jum d Zrr,* Derme o Mems, Uversy o Ar, Rmd, Ir Derme o Mems, Collee o See, Uversy o Sldd, Erl, Kurds, Ir * Corresod uor, e-ml: rr@yooom Reeved: 3-8-3; Aeed: --3 Asr: Us e Fox-r eerlo o e lssl yereomer uo, we rodue d sudy ew lss o rmo uvle uos e oe u ds e esls some resuls volv oee odo, dsoro ouds, exreme os, ovex omos, ovoluo d eoroods e lso dsuss lss reserv erl oeror Keywords: rmo uos, Fox-r eerled yereomer uo, Ierl oeror Iroduo A ouous omlex-vlued uo u v deed smly oeed dom D s sd o e rmo D o u d v re rel rmo D I y smly oeed dom we wre, were d re ly D e ll e ly r d e o-ly r o A eessry d sue odo or o e lolly uvle d sese reserv D s >, D Deoe y S e lss o uos re rmo uvle d sese reserv e oe u ds U { C: < } or w 0 0 0 Te or S we my exress e ly uos d s
I J Pure Al S Teol, 04, 64-77 65,, < I 984 Clue d Sel- Smll 3] vesed e lss oed some oee ouds, see lso 4], 9], 0] S s well s s eomer sulsses d Noe S redues o e lss S o ormled ly uvle uos e o-ly r o s memer s ero For s lss e uo my e exressed s A uo o e orm s rmo srle o order, 0 < or r < θ r re R{ } > θ * Te lss o ll rmo srle uos o order s deoed y S d exesvely suded y Jr 7] Te se 0 d 0 were suded y Slverm d Slv 3] d Slverm ], see lso ] I 7] Jr roved e oee odo, s sue odo or uos o e rmo srle o order I we u 0 ove eules e we o sue odo or uo elo o e lss * S o rmo srle uos Furer, we deoe y V e sulss o S oss o uos o e orm,, < 3, were Te dmrd rodu or ovoluo o wo ower seres d s deed y φ λ ϕ φ * ϕ φ * ϕ λ 4,5 For osve rel rmeers, A,,, A d B,,, B, N N {0}, 0 ssy e odo B A 0 U, e Fox-r eerlo
I J Pure Al S Teol, 04, 64-77 66 ψ,, A,,, A ;, B,,, B ; ] ψ, A,, B ; ] o e yereomer uo F,,,,,, s deed y 4]; see lso ] ψ, A,, B, ; ] { Γ A }{ Γ B } U! 0 I A,, d,,, e we ve e ollow ovous relos: B Θ ψ,,,, ; ] F,,,,,, 6 0! were F,, ;,,, s e eerled yereomer uo see or dels 4], s e mlr Pommer symol, d Θ s ve y Θ 0 Γ Γ 7 By us e eerled yereomer uo, Do d Srvsv 6] rodued ler oeror w ws suseuely exeded y Do d R 5] y us e Fox-r eerled yereomer uo 0 Le,,, A, B ]: S S e ler oeror deed y e o us 4 d 7, we e,, A,, B, ] φ { Θ ψ, A, ;, B ; ]}* φ,, A,, B, ] φ Θ λ were Θ s deed y 7, d s ve y,8 ΘΓ A Γ A! Γ B Γ B For oveee se, we do e ored oo ] o rerese e ollow: ] φ, A,,, A;, B,,, B ] φ,0 w we use e seuel rouou Te ler oeror ] os e Do-Srvsv oeror see 6], d s s vrous sel ses o su ler oerors s e olov oeror, Crlso-Ser oeror, Rusewey dervve oeror, eerled Berrd-Ler-Lvso oeror d rol dervve oeror Dels d reerees ou ese oerors e oud 5] d 6]
I J Pure Al S Teol, 04, 64-77 67 I vew o e relos 6 d e ler oeror 8 or e rmo uo y, we dee e oeror ve ] ] ], d rodue elow ew sulss L ], o S erms o e oeror deed y Deo : For <, U d ] s deed y, we le L ], deoe e sulss o S oss o rmo uos S o e orm ssy e odo R { ] ] } < e urer le L ], L ], V e deem rore o meo ere some o e useul sulsses w sem rom e lss L ], deed ove y redues o e lsses w we llusre elow I we u A,, d B,,, e e mly L ], deed y redues o e lss deoed y L ], w sses e euly: ] R { were ] s e Do-Srvsv oeror 6] ] } <, Nex, vew o e relos,; ] L,, we o lss GL ], ssy e euly: L, L, R { } <, were L, s e Crlso-Ser oeror ] 3 Also, y o e relos δ λ,;] D, we rrve e lss RL, w sses e euly: δ δ D D R { } <, δ were D δ > s e Rusewey dervve oeror ]lso see 8] 4 Lsly, vew o e relos,; ] Ω, we o oer lss FL, ssy e odo Ω ] R Ω ] } < {,
I J Pure Al S Teol, 04, 64-77 68 were Ω s e Srvsv-Ow rol dervve oeror 5] ve y < 0 Γ Ω D I s er, we sudy oee ouds,dsoro ouds, exreme os, ovex omos, ovoluo odo, eorood rolems d dsuss lss reserv erl oeror M Resuls I our rs eorem, we rodue sue oee or uos o e ], L Teorem : Le e su d ve y Furermore, le 3 Te s sese-reserv, rmo uvle U d ], L Proo: I, e > 0 ee, s uvle U Now, s sese-reserv U Ts s euse
I J Pure Al S Teol, 04, 64-77 69 > > I rems o sow ], L Us e w < R, d oly, < w w, sues o sow U <, ] ] ] ] e ve
I J Pure Al S Teol, 04, 64-77 70, y ee e roo s omlee Te rmo uvle uos o e orm, y x were < d y x, sow e oee oud ve y 3 s sr I e ollow eorem we sow e odo 3 s lso eessry or e uo were d re o e orm 3 Teorem : Le e deed y 3 Te ], L d oly 4 Proo: Se ], ], L L, we oly eed o rove e "oly " r r o e eorem For uos o e orm 3, we oe e odo } < ] ] { R s euvle o } { R < Te ove reured odo mus old or ll vlues o U Uo oos e vlues o o e rel xs d le, we ve, w s e reured odo Sress o 4 e see y e rmo uvle uo, y x
I J Pure Al S Teol, 04, 64-77 7 were <, x 0, y 0 d x y elos o e lss L ], Te ollow eorem ves e dsoro ouds or uos L ], Teorem 3: Le ], Te or r < we ve L d r r r r,5,6 Proo: Le ], Te L r r r r r r r r r By Teorem, we o: r ] r r ] r d r r r r, r r r r
I J Pure Al S Teol, 04, 64-77 7 r r r ee e roo s omlee Te ollow over resul ollows rom euly 6 Corollry : Le ], Te L r w : w < } U { Nex we deerme e exreme os o losed ovex ulls o L ], deoed y lo ], L Teorem 4: Le e ve y 3 Te lol ], d oly were d λ were λ 0, 0 d λ,,,3,,,, I rulr, e exreme os o L ], re } d } { { Proo: Suose Te λ λ λ Tus lol ], λ λ
I J Pure Al S Teol, 04, 64-77 73 Coversely, suose lol ], Le λ,,3,,,,3, Also ee, we o Ts omlees e roo o e eorem λ λ γ λ Teorem 5: L * ], S were < Proo: I ],, e y Teorem, we ve L Bu Tereore * S Teorem 6: Te lss L ], s losed uder ovex omo Proo: Suose ],, or,,, were L To rove e eorem s eou o sow were 0 λ d λ λ L ],, Now, y Teorem, we ve
I J Pure Al S Teol, 04, 64-77 74 Te λ λ λ By Teorem, we ve λ λ λ λ Tereore λ L ], Ts omlees e roo o e eorem Now we sow e lss L ], s losed uder ovoluo Teorem 7: For < γ, le ], d F ], γ L Te * F L ], L ], γ Proo: Le ], L, L d Te F A B L ], γ * F A B Se F ], γ, we oe A d B L Now y Teorem, we e: A B
I J Pure Al S Teol, 04, 64-77 75 Tereore * F L ], L ], γ ee e roo s omlee Te δ - eorood o s e se N δ { F : F δ d A B } Teorem 8: Le L ], I F Nδ, e F s rmo srle uo, were δ Proo: Suose F N δ Te d Tus F A A B A B B δ, A B B δ Tereore F s rmo srle uo 3 A Fmly o Clss Preserv Ierl Oeror e exme e losure roeres o e lss L ], uder e eerled Berrd-Ler-Lvso erl oeror L w s deed y L 0 d, > Teorem 3: Le ], e o e orm 3 Te L ], L Proo: Le ], e o e orm 3 Te L L
I J Pure Al S Teol, 04, 64-77 76 Now >, 0 0 d d L 0 0 d d By Teorem, we ve Tereore ee ], L L Ts omlees e roo o e eorem Reerees ] Y Av d E Zloew, O rmo uvle ms, A Uv Mre Cure- Slodows Se, A 44990, -7 ] BC Crlso d SB Ser, Srle d resrle yereomer uos, SIAM, J M Al, 500, 737-745 3] J Clue d T Sel-Smll, rmo uvle uos, A Ad S Fe Seres AI M, 93 984, 3-5 4] P Dure, rmo Ms e Ple, Cmrde Trs Mems Vol 56, Cmrde Uversy Press, Cmrde, 004 5] J Do d R, Fmles o ly uos ssoed w e r eerled yereomer uo, Demosro M, 373 004, 533-54 6] J Do d M Srvsv, Cer sulsses o ly uos ssoed w e eerled yereomer uo, Ierl Trsorm Se Fu, 4003, 7-8 7] JM Jr, rmo uos srle e u ds, J M Al Al, 35999, 470-477 8] G Muruusudrmoory, A lss o Rusewey-Tye rmo uvle uos w vry rumes, Souwes J Pue Al M, 003, 90-95 9] S Pousmy d A Rsl, Pler rmo ms, RMS Mems Newsleer, 7
I J Pure Al S Teol, 04, 64-77 77 007, 40-57 0] S Pousmy d A Rsl, Pler rmo d usoorml ms, RMS Mems Newsleer, 73 007, 85-0 ] S Rusewey, New rer or uvle uos, Pro Amer M So, 49975, 09-5 ] Slverm, rmo uvle uos w eve oees, J M Al Al, 0998, 83-89 3] Slverm d EM Slv, Sulsses o rmo uvle uos, New Zeld J M, 8999, 75-84 4] M Srvsv d P Krlsso, Mulle Guss yereomer Seres, lsed Press Ells- orwood Lmed, Ceser, Jo ley d Sos, New Yors, Ceser, Brse d Toroo, 985 5] Srvsv d S Ow, Some rero d dsoro eorems volv rol lulus, eerled yereomer uos, dmrd rodus, ler oerors d er sulsses o ly uos, Noy M J, 06987, -8