Certain sufficient conditions on N, p n, q n k summability of orthogonal series

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1 Avilble olie t J. Nolier Sci. Appl , 7 77 Reserch Article Certi sufficiet coditios o N, p, k summbility of orthogol series Xhevt Z. Krsiqi Deprtmet of Mthemtics d Iformtics, Fculty of Eductio, Uiversity of Prishti Hs Prishti, Aveue Mother Theres 5, Prishtië, Kosovë. Commuicted by Mohd Slmi Md Noori Abstrct I this pper we obti some sufficiet coditios o N, p, k summbility of orthogol series. These coditios re expressed i terms of the coefficiets of the orthogol series. Also, severl kow d ew results re deduced s corollries of the mi results. c 014 All rights reserved. Keywords: Orthogol series, geerlized Nörlud summbility. 010 MSC: 4C15, 40F05, 40G Itroductio Let be give ifiite series with its prtil sums s }. The, let p deotes the sequece p }. For two give sequeces p d q, the covolutio p is defied by p = p m m = p m q m. d We write := p, R j := p m q m m=j +1 = 0, R 0 =. Emil ddress: xhevt.krsiqi@ui-pr.edu Xhevt Z. Krsiqi Received

2 Xh. Z. Krsiqi, J. Nolier Sci. Appl , Also we put P := p 1 = p m d Q := 1 = q m. Whe 0 for ll, the geerlized Nörlud trsform of the sequece s } is the sequece t p,q } obtied by puttig t p,q = 1 p m q m s m. The ifiite series is sid to be bsolutely summble N, p, k with idex k, if for k 1 the series coverges [8], d we write i brief k 1 t p,q t p,q k N, p, k. We ote tht for k = 1, N, p, k summbility is the sme s N, p, N, p, 1 summbility itroduced by Tk [7]. Let ϕ x} be orthoorml system defied i the itervl, b. We ssume tht fx belogs to L, b d fx ϕ x, 1.1 where = b fxϕ xdx, = 0, 1,,.... Our mi purpose of the preset pper is to study the N, p, k summbility of the orthogol series 1.1, for 1 k, d to deduce s corollries ll results of Y. Okuym [6]. Throughout this pper K deotes positive costt tht it my depeds oly o k, d be differet i differet reltios. The followig lemm due to Beppo Levi see, for exmple [3] is ofte used i the theory of fuctios i which re ivolved the series d itegrls d which re ivolved i [1] d [] too. It will eed us to prove mi results. Lemm 1.1. If f t LE re o-egtive fuctios d the the series E f tdt <, 1. f t coverges lmost everywhere o E to fuctio ft LE. Moreover, the series 1. is lso coverget to f i the orm of LE.. Mi results We prove the followig theorem.

3 Xh. Z. Krsiqi, J. Nolier Sci. Appl , Theorem.1. If for 1 k the series coverges, the the orthogol series is summble N, p, k lmost everywhere. k j } k Rj j R ϕ x Proof. For the geerlized Nörlud trsform t p,q x of the prtil sums of the orthogol series ϕ x we hve tht t p,q x = 1 = 1 = 1 p m q m j ϕ j x j=0 m j=0 j ϕ j x p m q m m=j j j ϕ j x j=0 where m j=0 jϕ j x re prtil sums of order k of the series 1.1. As i [6] pge 163 oe c fid tht t p,q x := t p,q x t p,q x = j Rj j ϕ j x. R Usig the Hölder s iequlity d orthogolity to the ltter equlity, we hve tht b Hece, the series x k dx b 1 k = b 1 k k 1 b b [ x k dx K t p,q x t p,q x dx k j ] k Rj j. R k 1 [ j ] k Rj j R.1 coverges by the ssumptio. From this fct d sice the fuctios x re o-egtive, the by the Lemm 1.1 the series k 1 x k coverges lmost everywhere. This completes the proof of the theorem. For k = 1 i Theorem.1 we hve the followig result.

4 Xh. Z. Krsiqi, J. Nolier Sci. Appl , Corollry. [6]. If the series j } 1 Rj j R coverges, the the orthogol series is summble N, p, lmost everywhere. ϕ x Let us prove ow other two corollries of the Theorem.1. Corollry.3. If for 1 k the series coverges, the the orthogol series p P 1/k P k p j ϕ x is summble N, p k N, p, 1 k lmost everywhere. Proof. After some elemetry clcultios oe c show tht R j Rj R = p P P P P } k j j p p j P P j p j p p j for ll = 1, d the proof follows immeditely from Theorem.1. Remrk.4. We ote tht: 1. If p = 1 for ll vlues of the N, p k summbility reduces to C, 1 k summbility. If k = 1 d p = 1/ + 1 the N, p k is equivlet to R, log, 1 summbility. These fcts show us tht Theorem.1 icludes lso sufficiet coditios uder which the series 1.1 is C, 1 k summble, respectively R, log, 1 summble. Corollry.5. If for 1 k the series coverges, the the orthogol series q 1/k Q 1/k Q k ϕ x is summble N, k N, 1, k lmost everywhere. Proof. Sice Q j 1 j } k R j Rj R = Q j 1 Q Q for ll p = 1, the the proof follows directly from Theorem.1.

5 Xh. Z. Krsiqi, J. Nolier Sci. Appl , Also, puttig k = 1 i Corollries.3 d.5 we obti Corollry.6 [4]. If the series coverges, the the orthogol series p P P is summble N, p lmost everywhere. Corollry.7 [5]. If the series coverges, the the orthogol series is summble N, lmost everywhere. p j P P } 1 j j p p j ϕ x Q Q ϕ x Q j 1 j } 1 Now we shll prove geerl theorem cocerig N, p, k summbility of orthogol series which ivolves positive sequece with certi dditiol coditios. For this reso first we put Λ k j := 1 4 k k R j Rj. R the the followig theorem holds true. j k 1 =j Theorem.8. Let 1 k d Ω} be positive sequece such tht Ω/} is o-icresig sequece d the series 1 Ω coverges. Let p } d } be o-egtive. If the series Ω k 1 Λ k coverges, the the orthogol series ϕ x N, p, k lmost everywhere, where Λ k is defied by.. Proof. Applyig Hölder s iequlity to the iequlity.1 we get tht K = K K k 1 b x k dx k 1 [ 1 Ω k 1 Ω [ k j ] k Rj j R Ω k 1 [ k Ω k 1 k j ] k Rj j R j ] k Rj j R

6 Xh. Z. Krsiqi, J. Nolier Sci. Appl , K j Ω k 1 k R j } k Rj R K =j Ωj j j k 1 =j = K j Ω k 1 jλ k j 4 k k R j } k Rj R } k which is fiite by ssumptio, d this completes the proof. A direct cosequece of the theorem.8 is the followig k = 1., Corollry.9 [6]. Let Ω} be positive sequece such tht Ω/} is o-icresig sequece d the series 1 Ω coverges. Let p } d } be o-egtive. If the series ΩΛ 1 coverges, the the orthogol series ϕ x N, p, lmost everywhere, where Λ 1 is defied by Λ 1 j := 1 j =j j Rj R. Refereces [1] M. H. Froughi, M. Rdi, Some properties of L p,w, J. Nolier Sci. Appl., 009, [] J. Kur, S. S. Bhti, Covergece of ew modified trigoometric sums i the metric spce L, J. Nolier Sci. Appl., 1 008, [3] I. P. Ntso, Theory of fuctios of rel vrible vols, Frederick Ugr, New York, [4] Y. Okuym, O the bsolute Nörlud summbility of orthogol series, Proc. Jp Acd., , [5] Y. Okuym d T. Tsuchikur, O the bsolute Riesz summbility of orthogol series, Al. Mth., 7, 1981, [6] Y. Okuym, O the bsolute geerlized Nörlud summbility of orthogol series, Tmkg J. Mth., 33 00, ,,.,.9 [7] M. Tk, O geerlized Nörlud methods of summbility, Bull. Austrl. Mth. Soc., , [8] M. A. Srigol, O some bsolute summbility methods, Bull. Clcutt Mth. Soc., ,