α ]0,1[ of Trigonometric Fourier Series and its Conjugate

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1 aqartvelo mecierebata erovuli aademii moambe 3 # 9 BULLETIN OF THE GEORGIN NTIONL CDEMY OF SCIENCES vol 3 o 9 Mahemaic Some pproimae Properie o he Cezàro Mea o Order ][ o Trigoomeric Fourier Serie ad i Cojugae Dali Maharadze Shoa Ruaveli Sae Uiveriy 35 Niohvili Sr Baumi 6 Georgia Preeed by cademy Member G haraihvili BSTRCT Some approimae properie o he Cezàro mea o rigoomeric Fourier erie ad i cojugae are eablihed 9 Bull Georg Nal cad Sci ey word: rigoomeric Fourier erie cojugae rigoomeric erie Cezàro mea ume ha T [ ] ad : R R are ucio wih period L T he [ ] ad [ ] R ] [ I a ucio deoe repecively a rigoomeric Fourier erie ad i cojugae o ie [ ] a co b i [ ] b co a i a b a b a T i d T co d We deoe by he cojugae ucio o ie ε ε Iroduce he ollowig deigaio: lim [ ] cg d ψ ψ cg d N Deoe by Φ ee [] he cla o all ucio [ ] R i coiuou o [ ]; : wih he properie: 9 Bull Georg Nal cad Sci

2 Some pproimae Properie o he Cezàro Mea o Order ][ o 4 i o-decreaig; 3 ; 4 > < The ymbol ad o order amely: ad Bull Georg Nal cad Sci vol 3 o 9 D deoe repecively he Cezàro mea o he erie [ ] [ ] ψ τ d τ D D d co D i ] ] L Remar Below he ymbol ad correpodig parameer ad [ ] N >! are poiive iie coa depedig o he I hi paper we eablih ome local approimaio properie o Cezàro mea ad order ] [ The obaied reul geeralize Obreho [] reul he The ollowig aeme are rue L T Φ Theorem Le ] [ Proo I i ow ha j o D ad we coclude ha The we ca wrie ad T I d < d j d [ ] d ad hereore aig io accou he deiiio d d d d i 3 i o

3 4 Dali Maharadze Bull Georg Nal cad Sci vol 3 o 9 Sice c by he codiio o he heorem equaliy implie d d I i ow ha ee [3] ~ r i i r Thereore rom iequaliy we ca wrie d 3 pplyig he ormula o parial iegraio we obai d d d d d d d d Hece by he codiio o he heorem we ca coclude ha d d d The by virue o eimae 3 we ca wrie d 4 I ca be eaily oed ha or he epreio 3 coaied i he ollowig eimaio i valid: 3 5 Thu by relaio 4 ad 5 we have ha d

4 Some pproimae Properie o he Cezàro Mea o Order ][ o 43 he The Theorem i proved From he proved heorem i ollow ha i < < he l Thi eimaio wa proved by Obreho [] L T Φ ad T I Theorem ume ha Proo I i ow ha We have ha Uig he iequaliy Bull Georg Nal cad Sci vol 3 o 9 ψ ] [ d < d ψ τ d τ cg ψ γ ] ] 6 ψ τ d ψ ψ d 4 [ ] d Y ψ γ d ψ ψ γ i 7 τ by he codiio o he heorem we coclude ha c Y 8 For he epreio ψ ad γ i equaliy 6 we ow ha ee [3] co ψ i ad γ Thereore by virue o hee relaio ad aig io accou ha ~ equaliy 7 implie ψ Y Y3 d i

5 44 Dali Maharadze Hece applyig he ormula o parial iegraio rom he codiio o he Theorem ee [4] we obai Y Y alogouly or he epreio 3 d 9 4 ccordig o ad we obai: The Theorem i proved Y coaied i 7 he ollowig iequaliy i valid: Y 4 Α Α d matemaia urie rigoomeriuli da mii SeuRlebuli mwrivebi ][ rigi Cezaro asualoebi zogierti aproqimaciuli Tvieba d maaraze SoTa rutaveli aelmwio uiveriei batumi warmodgeilia aademio g araisvili mier aiasi dadgeilia urie rigoomeriuli mwrivi da mii SeuRlebuli ] [ Cezaro asualoebi zogierti loaluri aproqimaciuli Tvieba rigi REFERENCES N Bari SB Sechi 956 Trudy Moov Ma Obhch 5: i Ruia N Obrecho 934 Bull Soc Mah De Frace 6: LVZhizhiahvili 993 Neoorye voproy eorii rigoomericheih ryadov Fourier i ih opryazheyh Tbilii i Ruia Received November 8 Bull Georg Nal cad Sci vol 3 o 9

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