ON CONVERGENCE ALMOST EVERYWHERE OF MULTIPLE FOURIER INTEGRALS (Mengenai Penumpuan Hampir di Mana-Mana bagi Kamiran Fourier Berganda)

Transcription

1 Joua of Qua Measuee ad Aass JQMA Jua Pegukua Kua da Aass O OVERGEE AMOST EVERYWHERE OF MUTPE FOURER TEGRAS Megea Peupua Hap d Maa-Maa bag Kaa Foue Begada AVARJO AHMEDOV ORASHK ABDU AZZ & MOHD ORZA MOHTAR ABSTRAT hs pape we vesgae he ppe of he geeased oasao fo spea expasos of he pohao opeao whh odes wh he upe Foue egas sued ove he doas oespodg o he sufae eves of he pohao pooas s poved ha he paa sus of he upe Foue egas of a fuo f R ovege o zeo aos-evewhee o R \ supp f Kewods: Foue egas; spea expasos; axa opeaos ABSTRAK Daa akaah dka psp peepaa eak uuk kebaga speku bag pegopeas pohaok ag bedh dega kaa Foue begada ag duahka ke aas doa ag sepada dega aas peukaa pooa pohaok Dbukka bahawa uah sepaa kaa Foue begada bag fugs f R beupu kepada sfa hap d aa-aa pada R \ supp f Kaa ku: Kaa Foue; kebaga speku; pegopeas aksu oduo The we kow assa oasao Theoe of Rea saes ha f π peod fuo f s egabe ove he eva [ π π ] he he ovegee o dvegee of a oe desoa Foue sees a a gve po depeds o o he behavou of he fuo f a aba sa eghbohood of hs po The ew oep of oasao ppe has bee odued 968 ad s aed geeased ppe of oasao: he aos evewhee ovegee of he spea expasos of he gve fuo o o he se whee expadg fuo vashes s eesg o kow whehe he suffe odos fo oasao a be weakeed f we osde he geea oasao pobe hs wok we vesgae he pobes of he geeased oasao fo he upe Foue egas oespodg o he pohao opeao Fo a posve ege ube we deoe b a pohao opeao wh he doa of defo R whee s he apae opeao As a se x opeao he pohao opeao has a eas oe sef ado exeso R e A deoe a sef ado exeso of he opeao R We osde he oespodg paa egas of he upe Foue egas:

2 Avao Ahedov oashk Abdu Azz & Mohd oza Moha E f x π x fˆ e d < whee ˆ f s he Foue asfo of a fuo f p R p e A deoes a uque sef ado exeso of he pohao opeao R The he fa { E } - be a deoposo of he de of A ad he oespodg spea expaso of a fuo f R odes wh fo oe deas we efe he eade o Aov e a 99 Fo he ase of he apae opeao he geeased oasao ppe he aos evewhee ovegee of he spea expasos o R \ supp f asses p R was vesgaed b a auhos Bass 983; aeso 966; abe & Soa 988; 997; Sö 983 The pobe of geeased oasao fo spea expasos of he apae opeao s opee soved asses p R whe p : f p he we have he geeased oasao ad f p < he we do o The eesg ase of u sphee was vesgaed Avao 9 shoud be oed ha he pobe of he geeased oasao fo Resz eas of he Foue-apae sees o he a e fs vesgaed hs wok hs pape we pove he foowg: Theoe f f R ad f x o a ope se ovege o aos evewhee o Ω E f x π < fˆ e Ω R x d he paa egas ohe wods he heoe asses ha fo he upe Foue egas he ppe of geeased oasao hods he ass R We oe ha ase of apae opeao a sa saee has bee esabshed abe ad Soa 988 We ogaes fo he ae pape whh he auhos obaed he eeva equaes fo axa opeao of he spea deoposo of he apae opeao The axa opeao s defed b E* f x sup E f x > Theoe s based o he foowg esaes of he axa opeao Theoe f f R ad f s suppoed { x 3} he fo a < 3 hee exss a osa suh ha

3 O ovegee aos evewhee of upe Foue egas x { E f x } dx f x dx * 3 x 3 The esao of axa opeao a be apped poofs oeg he aos evewhee ovegee of spea expasos The aos evewhee ovegee of spea expasos wee suded b a auhos Ashuov 983; Avao 9; aeso 966; Sö 983; Se 958; see fo a evew Aov 97 ad Zhzhashv ad Topua 979 The Poof of Theoes e f R ad supp f { x R : x 3} foowg equa hods: We pove ha fo eve < 3 he sup E x > τ f x dx x 3 f x dx 4 τ x wh τ : < τ < hee E f x π fˆ e d shoud < be oed hee ha he saee of he Theoe s paua ase of he ae equa whe τ e χ b be he haaes fuo of he sege [b] ad φ be a sooh fuo defed fo suh ha χ φ χ The we defe τ 3/3 3 /3 x x φ x φ x ad x fo We oba φ x x τ τ e Θ Θ f supp f { x 3} he fo a x : x we have E τ f x Θ τ f Θ τ f x beause Θ τ φ * f x f x < < 3 s ea ha o pove he equa 4 s suffes o pove τ sup Θ g x dx g x dx g x R > R R

4 Avao Ahedov oashk Abdu Azz & Mohd oza Moha B dua we a pove ha he ae equa a be esabshed b vesgag he τ τ τ Foue asfo of he "spea fuo" Θ x e ˆ Θ Whe τ we use oao e Fo we have ea Fo a > R we have < ˆ d Ths ea s poved abe ad Soa 988 ea Fo a posve ege hee exss a osa suh ha fo a > R we have he foowg esae Poof We osde he foowg fuo: d The ega we dvde o wo pas as foows / / d d Esao of Usg he foowg foua d d we oba : d d < d The fo we have < d

5 O ovegee aos evewhee of upe Foue egas 3 dd < / / dd B hagg he ode of egao ad akg o aou he eao } : { } / : { d dd ow we osde Usg he egao b pas we oba / / / d d Usg he equa d we ge / / dd Sa as pevous ase we have dd dd / / ε

6 Avao Ahedov oashk Abdu Azz & Mohd oza Moha oseque fo we have Fa we oba The ea s poved Fo he devave of he fuo we oba ea 3 Fo a posve ege hee exss a osa suh ha fo a > ad R we have he foowg esaes d d ε oe fo 4 b pug we have he saee of Theoe Poof of Theoe e f p R ad f x o a ope se Ω R We have o pove ha E f x aos evewhee o Ω o f x Ω s a aba po he o he ba B x Ω Theefoe whou oss of geea we assue ha supp f R \ B x Due o he vaae of E f x wh espe o he asao ad dao we a edue he pobe o he osdeao of he fuos wh supp f { x 3} Thus we have o pove ha fo a fuo f R wh supp f { x 3} oe has E f x aos evewhee o { x < } < 3 Bu hs s a osequee of equa fo Theoe beause he ae equa aows us o sae ha he se { x : E f xdoes o ovege o f x} has a easue fo oe deas see Se ad Wess 97 Theoe s opee pove Akowedgee Ths eseah has bee suppoed b Uves Pua Maasa ude Reseah Uves Ga RUGS: RU The fs auho wshes o hak Pof Ashuov RR fo hs kd adve ad suag dsussos Refeees Aov ShA Ashuov RR & Puaov AK 99 Mupe Foue sees ad Foue egas : Khav VP & ko'sk KEds ouave Hao Aass V ew Yok: Spge-Veag 4

7 O ovegee aos evewhee of upe Foue egas Aov ShA 97 Suab aos evewhee of Foue sees p wh espe o egefuos Joua of Dffeea Equaos 6: 64-7 Avao Ahedov 9 The ppe of geea oazao o u sphee Joua of Maheaa Aass ad Appaos 356: 3-3 Ashuov RR983 Suab aos evewhee of Foue sees p wh espe o egefuos Maheaa oes of he Aade of Sees of he USSR 346: Bass AJ 983 Geeazed oazao of Foue sees wh espe o he egefuos of he apae opeao he asses p ovsk Maeahesk Sbok 33: aeso 966 O ovegee ad gowh of paa sus of Foue sees Aa Maheaa 6: abe A & Soa F 988 Aos evewhee ovegee of Foue egas fo fuos Soboev spaes ad a -oazao ppe Revsa Ma beoaeaa 4: abe A & Soa F 997 Powse Foue veso ad oasao R Joua of Foue Aass ad Appaos 3 Spea ssue: ' VA 968 oazao ad ovegee pobes fo Foue sees b fudaea fuo sses of he apae opeao Russa Mah Suves 3: 59-6 Sö P 983 Regua ad egab of sphea eas Moash Mah 964: 77-9 Se EM & Wess G 97 oduo o Foue Aass o Eudea Spaes Peo J: Peo Uv Pess Se EM 958 oazao ad suab of upe Foue sees Aa Maheaa : Zhzhashv V & Topua SB 979 Foue-apae sees o a sphee Joua of Sove Maheas 6: Depae of Poess ad Food Egeeg Fau of Egeeg Uves Pua Maasa 434 UPM Sedag Seago DE MAAYSA E-a: ava@egupedu 5