Approximation of Functions Belonging to. Lipschitz Class by Triangular Matrix Method. of Fourier Series

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1 I Jorl of Mh Alysis, Vol 4, 2, o 2, 4-47 Approximio of Fcios Blogig o Lipschiz Clss by Triglr Mrix Mhod of Forir Sris Shym Ll Dprm of Mhmics Brs Hid Uivrsiy, Brs, Idi shym _ll@rdiffmilcom Biod Prsd Dhl Bwl Mlipl Cmps Tribhv Uivrsiy, Npl Absrc I his ppr, w horm o h dgr of pproximio of h fcio f lipα by lowr riglr mrix smmbiliy mhod of Forir sris hs b sblishd Mhmics Sbjc Clssificio: 42B5, 42B8 Kywords: Lipschiz Clss of fcios, Forir sris, Dgr of pproximio, Mrix smmbiliy mhod Irodcio Th dgr of pproximio of fcios f blogig o Lip α clss by (C,), (C, δ ), δ >, Nörld ms (N, p ), Risz ms ( N, p ) d grlizd Nörld ms (N,p,q) hs b drmid by svrl ivsigors li Brsi [4], Alxis [], Chdr [5], Shy d Gol [7], Alxis d Krli [3], Alxis ~

2 42 S Ll d B P Dhl d Lidlr[2] Th dgr of pproximio of fcios f lipα sig lowr riglr mrix smmbiliy hs o b sdid so fr Lowr riglr mrix smmbiliy mhod iclds Csáro ms, Nörld ms d grlizd Nörld ms s priclr css I his ppr, h dgr of pproximio of fcio f Lip α, < α, by lowr riglr mrix mhod is Forir sris hs b drmid 2 Dfiiios d Noios α Lip if ( ) A fcio f α f (x + ) f (x) O for < α L f b priodic wih 2 d igrbl ovr (-,) i h Lbsg ss d f Lipα L is Forir sris b giv by f (x) + ( cos x + b si x) () 2 Th dgr of pproximio of fcio f : R R by rigoomric polyomil T of ordr is giv by T f sp{ T (x) f (x) : x R} (Zygmd,[] p4) L T(, ) b ifii riglr mrix sisfyig h Töpliz[9] codiio of rglriy, i, s,, cos L s m b, for d M, > ifii sris sch h whos, fii h pril sm Th sqc-o-sqc rsformio, s dfis h sqc { } of lowr riglr mrix ms of h sqc { s } grd by h sqc of cofficis (, ) Th sris is sid o b smmbl o h sm s by mrix mhod if lim xiss d is ql o s (Zygmd () p 74) d i is dod by s(), s φ f x + + f x 2f x, W wri () ( ) ( ) ( ), τ, τ A, τ [ / ] h lrgs igr i ( / )

3 Approximio of fcios 43 3 Thorm W prov followig horm: Thorm L T (, ) b ifii lowr riglr mrix sch h h lms (, ) b o-giv, o-dcrsig wih, τ, τ A d A, L f L [, 2] b 2-priodic fcio blogig o Lip α ( < α ), h h dgr of pproximio of f by lowr riglr mrix ms of is Forir sris () is giv by f 4 Lmms ( ) O ( + ), < α <, O( log( + ) /( + ) ), α, for,,2,3 For h proof of or horm followig lmms r rqird Lmm M () O (+), if < /( + ) Proof For < /( + ), si( + ) ( + ), M () 2, ( 2 + ) ( / 2) (2 ) (2 ) 4, 4 A O( + ) Lmm 2 If (, ) is o-giv, o-dcrsig wih, h, O(, A, τ ), iformly for < (2) Proof L τ [ / ], h, τ, +, τ

4 44 S Ll d B P Dhl d B by Abl`s lmm, τ,, τ,, τ +, τ+ τ τ 2 A ;,, τ i(p+ ), mx τ i p τ 4, τ 2, τ 2, τ, i / 2 i / 2 i / 2 si ( / 2), τ +, τ + + ( τ + ), τ+, τ hrfor, 2A Also τ,, τ, τ,,,, A, τ τ τ τ,, τ, τ Ths (2 + )A O(A ) Lmm 3 () A, τ M O, if /( + ) < Proof For /( + ) <, si( / 2) ( / ), w hv M () Im 2, i(+ ) 2 i / 2 2, A τ O,, sig lmm 2

5 Approximio of fcios 45 5 Proof of h horm Followig Tichmrsh [8], s (x) of Forir () is giv by, w hv s (x) f (x) 2 2 si( + / 2) φ() d si( / 2) si + / 2 si( / 2) h, ( ) ( ), s, (x) f (x) φ(), d or Usig Lmm d Nx, sig lmm 3, I 2 Q O A, ( ( ) + ) is moooic ( x) f( x) φ( ) M( ) d /(+ ) + φ φ ( ) M ()d () M () d /(+ ) I +, sy (3) I 2 f Lipα φ Lip α [[ 6] ; Lmm 527], w hv O /(+ ) + / I α A, O α+ O ( + ) /(+ ) α d O (( + ) ) (4) A, τ d d + A O, d +, α / + + {( + ) (/ ) } ( ) log( + ), α, < α < O (( + ) ), < α < O(log( + ) ), α Combiig (3) o (5) d wriig log, w hv (5)

6 46 S Ll d B P Dhl f x2 ( ) O ( + ), < α <, sp ( x) f( x) O( log( + ) /( + ) ), α 6 Corollris Followig corollris c b drivd from h mi horm Cor If /( + ) h dgr of pproximios f Lipα by Csáro, σ + ms s of Forir sris (2) is giv by σ f Cor 2 If ms N O (( + ) ), P p, O < α <, ( log( + ) /( + ) ), α p h dgr of pproximios f Lipα by Nörld P s is giv by N O (( + ) ), < α <, f O( log( + ) /( + ) ), α 7 Rmrs Rmr() Thr r svrl simios of h fcio f Lipα by (C,), (C, δ ), δ >, (N, p ),(N, p, q) mhods of is Forir sris, for xmpl, Brsi [4], Alxis [], Chdr [5], Shy d Gol [7], Alxis d Krli [3], Alxis d Lidlr[2] b mos of hs rsls r o sisfis for, or α Thrfor his dficicy hs b moivd o ivsig h grlizio of hs rsls by mos grl lowr riglr mrix cosidrig css () < α < (2) α sprly Or simio is shrpr d br h ll prviosly ow simios of his dircio (2) By or horm, f s h his sim is corrc d h bs simio [Zygmd [9], p 5] (3) Or corollris provid simplifid d corrcd forms of w rsls

7 Approximio of fcios 47 Rfrcs [] Alxis, G: Übr di Aährg ir sig Fio drch di Csàrosch Mil ihrr Forirrih,Mh A, (928), [2] Alxis, G d Lidlr, L: Übr di Approximio im sr Si (Grm), Ac Mh Acd Sci Hgr, 6(965), [3] Alxis, G d Krlic : Übr di Approximio mi sr d l vll-possich mil, Ac Mh Acd Sci Hgr 6(965), [4] Brsi S, Srl Ordr d l Millr pproximio dsfcios cois prds polyoms d dgr, do`s Mmoris AcdRoyBlyiq (2),4(92),-4 [5] Chdr Prm, O h dgr of pproximio of fcios blogig o Lipschiz clss, N Mh 8(975) o,88-9 [6] McFdd, Lord: Absol Nörld smmbiliy, D Mh J 9 (942), [7] Shy B N d Gol DS,O dgr of pproximio of coios fcios, Rchi Uiv Mh J 4(973), 5-53 [8] Tichmrsh EC,Th hory of fcios, Prss (939),42-43 d 2 Ediio, Oxford Uivrsiy [9] Töpliz O,Ubrllgmi lir Milbdg, PMF, 22(93), 3-9 [] Zygmd A,Trigomic sris, scod diio Volm, Cmbridg Uivrsiy Prss, Cmbridg (959),74, 4-5 Rcivd: Spmbr, 29

International Journal of Mathematical Archive-3(1), 2012, Page: Available online through

International Journal of Mathematical Archive-3(1), 2012, Page: Available online through eril Jrl f Mhemicl rchive-3 Pge: 33-39 vilble lie hrgh wwwijmif NTE N UNFRM MTRX SUMMBLTY Shym Ll Mrdl Veer Sigh d Srbh Prwl 3* Deprme f Mhemics Fcly f Sciece Brs Hid Uiversiy Vrsi UP - ND E-mil: shym_ll@rediffmilcm