APPROXIMATION OF CONTIONUOUS FUNCTIONS BY VALLEE-POUSSIN S SUMS

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1 italia joural of ure ad alied mathematics APPROXIMATION OF ONTIONUOUS FUNTIONS BY VALLEE-POUSSIN S SUMS Rateb Al-Btoush Deartmet of Mathematics Faculty of Sciece Mutah Uiversity Mutah Jorda btoush@mutahedujo Al-Oushoush Nizar KhKh Deartmet of Mathematics Faculty of Sciece Balqa Alied Uiversity Al-Salt Jorda aloushoush@bauedujo Abstract Let V,m α,β f; x m+ S α,β f; x be the Vallee-Poussi s artial sums of Fourier-Jacobi series I this aer, we study the deviatios of V,m α,β f; x o [, ] for cotiuous fuctio fx Itroductio Let P α,β x,,, deote the Jacobi orthoormal system of olyomials with weight fuctio Furthermore, let x α + x β, α >, β > o [, ] fp α,β x, be the Fourier-Jacobi series of the fuctio fx, where f t α + t β ftp α,β tdt

2 54 rateb al-btoush, al-oushoush izar hh Deote the Fourier-Jacobi series of artial sums, S α,β f; x as 3 S α,β f; x Deote the Fejer sum of fx, σ α,β f; x as 4 σ α,β f; x + fp α,β x S f; x Defie the Vallee-Poussi s artial sums of Fourier-Jacobi series,v,m α,β f; x, as 5 V,m α,β f; x m + S α,β f; x A estimatio for the deviatio of the cotiuous fuctio fx, with eriod from its Fourier sum S f; x is give i [], whe fxhas a bouded variatio ad sureme modulus of cotiuity This aer geeralizes ad imroves some results i theory of aroximatio of cotiuous fuctios, as those reorted i [] [] The efficiet study for aroximatio by Vallee-Poussi sums has bee carried out for several decades Recetly, several studies dealig with the Vallee- Poussi sums have bee itroduced, see [8] [] The results reseted i this aer geeralize ad imrove may results of [] [] ad may others i theory of aroximatio of cotiuous fuctios Our roblem, here, is to study the deviatios of S α,β cotiuous fuctio of oe variables fx I this regard, three theorems have bee itroduced f; x o [, ] for Jacso s theorem We state oe of most imortat theorem i the aroximatio theory, amely Jacso s theorem, ad which is used i our study Let fx be cotiuous fuctio o closed iterval [a, b] Deote by E f the best uiform aroximatio elemet of a fuctio fx by algebraic olyomials of order ot exceeded o [a, b], ie, { } E f if max x [a,b] fx x Jacso s Theorem If ωf; is a modulus of cotiuity of a fuctio fx, the the iequality E f c ω f; b a,

3 aroximatio of cotiouous fuctios by vallee-oussi s sums 543 holds, where c a absolute, ad ωf; t is defied as 3 ωf; t su fx fx x x x,x [a,b] I case of fx [, ], the 4 E f c ω f; 3 The mai result Let P α,β x,,, deote the Jacobi orthoormal system of olyomials with weight fuctio x α + x β, α >, β > o [, ] We start with the followig secial cases of the Jacobi olyomial: α, β, the 3 P, x cos cos x, where P, x, which are called the Tschebyscheff olyomials of the first id α, β, the 3 P, x si + cos x, si cos x which are called the Tschebyscheff olyomials of the secod id 3 α, β, the 33 P, si + cos x x si cos x, 4 α, β, the 34 P, cos + cos x x cos cos x, I this wor, we will rove the followig three theorems:

4 544 rateb al-btoush, al-oushoush izar hh Theorem If M max fx, the the followig iequalities hold x 35 V,,m f; x M + l + m + m + t V,,m f; x V,,m f; x V,,m f; x M + l x M x + l M + x + l + m + m + + m + m + + m + m + Proof First, we cosider the case α β equatio 36 Defie where P x P, x, S f; x S, f; x, σ f; x σ, f; x, V,m f; x V,,m f; x, S f; x fp x, f t ftp tdt Usig the defiitio of P x, give i 3, we have 39 The f S f; x fp x si y t ft si + cos t dt sicos t ft si + cos t dt fcos t si + t si tdt fcos t si t si + t si + ydt,

5 aroximatio of cotiouous fuctios by vallee-oussi s sums 545 where y cos x ad, sice si α si β cosα β cosα + β, we obtai 3 S f; x fp x si y fcos t si t cos+t y cos+t+ydt Usig the well-ow quality + cos x si + x si x, equatio 3 ca be writte as S f; x si + + t y fcos t si t si y si t y si + + t + y si dt, t + y that is, 3 S f; x fcost + y sit + y si y fcost y sit y si + + t si t dt From equatio 5, we have 3 V,m f; x S f; x m + m + S f; x S f; x To obtai a estimatio for V,m f; x, first, from the itegral reresetatio of V,m f; x, let M,m m + + m + si t si m + t si t dt, ad, if m +, r +m +, for, r, we obtai M,m m + si rt si t si t dt

6 546 rateb al-btoush, al-oushoush izar hh Sice the fuctio si t t 4 is bouded o [, ] ad, the Sice ad the M,m si rt si t t si rt si t dt t M,m ad agai, sice the fuctio si t Next, we must show that M,m si rt si t dt + O t dt si rt si t dt, t si rt si t dt t si rt si t dt + O, t t 4 dt t, is bouded o [, ] ad, the si rt dt + O t si rt dt l r + O, t where r < +, for For this, let Ad, sice for t r i si rt dt t t + i r i i r i r ad from r i si rt i + r i r si rt t + i dt + O r i si rt dt + O t t + i r dt + O si + x si + x si x i + O r l + O, we have:

7 aroximatio of cotiouous fuctios by vallee-oussi s sums 547 The So si rt dt t r r si rt l + O dt l + O l r + O M,m 4 l r + O 4 l + + m + m + + O 4 l + m + m + Therefore, usig the itegral form of V,m, we obtai our result for, ie, V,,m f; x M + m + + l x m + I a similar way, we ca rove the other three cases Theorem Suose that fx is cotiuous fuctio o [, ] The 33 fx V, + m +,m f; x E f + l m + + O fx V,,m f; x fx V,,m f; x fx V,,m f; x E f + l x E f + l x E f + l + x + m + m + + m + m + + m + m + Proof First, we cosider the case α β Let Q x be the best uiform aroximatio of algebraic olyomial for fuctio fx of order ot exceeded o [, ], the fx V,m f; x fx Q x + Q x V,m f; x fx Q x + V,m f Q ; x + m + E f + E f + l x m + + m + E f + l x m + I a similar way, we ca rove the other three cases

8 548 rateb al-btoush, al-oushoush izar hh Theorem 3 Suose that fx is cotiuous fuctio o [, ], the 37 fx V,,m f; x m + E f + + l + 38 fx V,,m f; x m + E f x + l fx V,,m f; x m + x E f + + l + 3 fx V,,m f; x m + + x E f Proof We start the case α β hoose the iteger such that m + < +, the l + fx V,m f; x fx S f; x m + m + fx S f; x fx S f; x } + i+ fx S i f; x Sice we get which yields ad S f; x m + V,m f; x, + S f; x V, f; x, i+ S i f; x V +, f; x, S i f; x m + V +,m f; x, i+

9 aroximatio of cotiouous fuctios by vallee-oussi s sums 549 So fx V,m f; x m + {fx V,f; x + fx V +, f; x +m + fx V +,m f; x} Alyig Theorem, we obtai 3 Thus 3 fx V, f; x fx V +, f; x fx V +,m f; x fx V,m f; x x + l + E f + l x + l x + + m+ E + f m + x { + l + E f + + l + E + f +m + + m + + l m + E + f } E + f Note that u, v >, we have the iequality u + v + u + v ad the lu + v l + u + l + v Settig u x, v y, we obtai z z So 33 I l x + y z + E + f l l + x + l + y z z E + f l + + I + I For I, as metioed above E + f l I E f

10 55 rateb al-btoush, al-oushoush izar hh For I, ote that I E + f l + 35 E + f l + + E + f l + + E + f l i+4 + { + i+ E + f l + + i+ E + f l + E + f l + E + 3f l E f l i+ i+ E + f l E + f l + } + ombiig the estimates give i 35 ad 34 for I ad I, we obtai 36 + E + f l { + E f + + E f l Thus, from the choice of such that m +, we have: m + E + f } + E f I additio, ote that for ay atural umbers α, β such that α β, also we have therefore, α l β α α l β α α l β α β l α+ + l β α α + l β α β α β + α + + l β α α + + l β α α

11 aroximatio of cotiouous fuctios by vallee-oussi s sums 55 m + l + m + m + +m+ + + l + m + m l + m + l + m + + l l + m+ + where α m +, β m +, but where we used ad so l + m + + m l + x l x + O m + l + m + m + Sice m +, the therefore, m+ l m +, l + + m +, x, x m l l m m! l em m, [ + l + + ] + E f + E + f + + E f m + E + f E + f,

12 55 rateb al-btoush, al-oushoush izar hh 37 m + E + f l + m+ [ E + f + [ E f + [ E f + E f l E + f l E + f l ] + ] + ] + ombiig all of the above estimates 3 37, we get the desired result This eds of the roof of the theorem Refereces [] Stechi, SB, O aroximatio of cotiuous fuctios by Fourier series, Usehi Mat Nau, , 39-4 [] Al-Btoush, R, O Aroximatio of bouded variatio fuctios by Fourier Legedre sums, J of MU TAH Lil-Buhuth Wad-Dirasat, 5, [3] Abelov, VA, Aroximatio of fuctios by Fourier-Lagerre sums, Mat Zameti, , 63-7 [4] Niolsi, SM, Aroximatio of fuctios of may variables ad embeddig theorems, Naua, Moscow, 977 [5] Abelov, VA, Koorbaova, MM, Aroximatio of fuctios by Vallee Poussi sums, Viiti, Moscow, 987 [6] Al-Btoush, R, Aroximatio of cotiuous fuctios by algebraic Polyomials, Italia Joural of Pure ad Alied Mathematics, 4 8, 9-8 [7] Al-Btoush, R, Aroximatio of eriodic fuctios by Vallee Poussi sums, Hoaido Mathematical Joural, 3, 69-8 [8] Kholshcheviova, NN, O the de la Vallee-Poussi theorem o the uiqueess of the trigoometric series reoresetig a fuctio, Sb Math, , [9] Noviov, OA, Ruaasov, VI, Aroximatio of classes of cotiuous fuctios by geeralized de la Vallee-Poussi sums, Ur Mat Zh, , [] Nowaowsi, K, Tabersi, R, Estimates for the strog de la Vallee- Poussi meas i case of eriodic cotiuous fuctios of two variables, Fuct Arox ommet Math, 4 996, 83- [] Nahmwoo, Hshm, Hog, BI, Degree of aroximatio by eriodic eural etwor, Korea Joural Math, 4, Acceted: 986