ON APPROXIMATION BY SPHERICAL ZONAL TRANSLATION NETWORKS BASED ON BOCHNER-RIESZ MEANS

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1 Vo 8 o 3 Joura of Systems Sciece ad Compexity Ju, 2005 O APPOXIMATIO BY SPHEICAL ZOAL TASLATIO ETWOKS BASED O BOCHE-IESZ MEAS SHEG Baohuai Departmet of Mathematics, Shaoxig Coege of Arts ad Scieces, Shaoxig 32000, Chia Emai: bhsheg@zscaseduc LI Hogtao Departmet of Mathematics, Baoji Coege of Arts ad Scieces, Baoji 72007, Chia Abstract A sequece of spherica zoa trasatio etwors based o the Bocher-iesz meas of spherica harmoics ad the iesz meas of Jacobi poyomias is itroduced, ad its degree of approximatio is achieved The resuts obtaied i the preset paper actuay impy that the approximatio of zoa trasatio etwors is coverget if the actio fuctios have certai smoothess Key words Spherica harmoics, zoa trasatio etwors, approximatio Itroductio I recet years there has bee growig iterest i the probems of eura etwors ad their reated stabiity 2] ad approximatios 3], ad may importat resuts o the quatitative estimate of the degree of approximatio have bee obtaied The cocept of Sigmoida fuctio of order is defied i 4] ad it was proved by H Mhasar ad C A Micchei 4] that whe the actio fuctio φ defied o d is ot a poyomia fuctio ad K d is a compact set, the fuctio cass φ x = {φλx t : λ, t d }, x K is dese i L p K Moreover, a sequece of eura etwors was costructed by B-spie fuctios ad a Jacso type theorem of approximatio i C0, ] was estabished Let s d be itegers, φ : d be a 2π periodic fuctio ad φ L p π, π] d, p +, J = J d,s be the cass of a d s matrices of ra d with iteger etries, ad et φx = {φ Ax + t : A J, t π, π] d } {}, x π, π] s The, H Mhasar ad C A Micchei 5] gave the ecessary ad sufficiet coditios for φ x to be dese i Lp π, π] s space, ad costructed with the de a Vaée Poussi meas of Fourier series a very importat sequece of etwors, ad obtaied estimates of the degree of approximatio i L p π, π] s I a recet paper 6], the authors cosidered the probem of approximatio of o-periodic fuctios by the fuctio cass c φx = {φcosa arccos x + t] : A J, t, ] } {}, x, ] s, eceived December 23, 2003 *This research is supported i part by the atioa atura Sciece Foudatio of Chia o 04730, of Chia, the atura Sciece Foudatio Y of Zhejiag Provice ad the Doctor Foudatio 2004A62007 of igbo city

2 362 SHEG BAOHUAI LI HOGTAO Vo 8 ad gave the ecessary ad sufficiet coditios such that c φ x is dese i weighted space L p W,] where W x = x 2 s 2 x x 2 s 2, ad costructed a sequece of etwors with the de a Vaée Poussi meas of Chebyshev poyomias of first id ad obtaied a estimate of the degree of approximatio i weighted space L p W,] Let q 2 be s a iteger, deote the surface of the uit sphere i the Eucidea space q+ The H Mhasar et a 7] showed that whe fuctio φ :, ] satisfies some coditios, the zoa trasatio etwor fuctio cass S φx = {φx y : y } {}, x where x y = xy deotes the ier product of vectors x ad y is dese i L p ad furthermore gave a Jacso estimate o the degree of approximatio i space L p with the hep of deayed meas of spherica harmoics I preset paper, we sha aso ivestigate this probem ad, with the hep of both the Bocher-iesz meas of the spherica harmoics ad the Bocher iesz meas of Jacobi poyomias, we give a sequece of zoa trasatio etwors which has a more expicit form The paper is orgaized as foows I Sectio 2, we sha review for the coveiece of the readers some basic facts cocerig spherica harmoics ad their approximatios which wi be eeded i the paper I Sectio 3, we sha give some estimates of approximatio by the iesz meas of Jacobi orthogoa poyomias which wi aso be eeded i the paper I Sectio 4, we sha costruct the sequece of spherica trasatio etwors I the ed, we sha give the order of approximatio I the foowig we sha write A B if there exists a costat C > 0 such that A CB, ad we sha write A B if A B ad B A 2 Some Cocepts ad esuts of Spherica Harmoics Let q be a iteger which wi be fixed throughout the rest of this paper, ad et be the uit sphere i the Eucidea space q+, with dµ q beig its usua voume eemet The voume of is = dµ q = 2π q+ 2 Γ q+ 2 Correspodig to dµ q, we defie the orm of space L p by fx p p dµ q x, p < +, f p, = ess sup fx, p = + x The cass of a measurabe fuctios f : Chere, C is the set of a compex umbers with f p, < + wi be deoted by L p, with the usua uderstadig that fuctios that are equa amost everywhere are cosidered equa eemets of L p For a iteger 0, the restrictio to of a homogeeous harmoic poyomia of degree is caed a spherica harmoic of degree The cass of a spherica harmoics of degree wi be deoted by H q, ad the cass of a spherica harmoics of degree wi be deoted by Π q Of course, Π q = H q, ad it comprises the restrictio to Sq of a agebraic poyomias =0

3 o 3 APPOXIMATIO BY ZOAL TASLATIO ETWOKS 363 i q + variabes of tota degree ot exceedig The dimesioa of H q 8, p65] is give by 2 + q + q,, d q = + q q dimhq =, = 0, ad that of Π q is =0 dq By 9] we ow { L 2 = cosure Hece, if we choose a orthoorma basis {Y, : =, 2,, d q } for each H q, the the set {Y, : = 0,, 2, ; =, 2,, d q } forms a orthoorma basis for L2 Oe has the we-ow additio formua 0] : d q = H q } Y, xy, y = dq p q+ x y, = 0,,, where p q+ x is the degree- Legedre poyomia The Legedre poyomias are ormaized so that p q+ =, ad satisfy the orthogoa reatio p q+ xp q+ xw q xdx = d q δ,, where W q x = x 2 q 2 From the fact that Π q = ow that for ay p Π q ad x Sq px = d q =0 pyp q+ H q =0 x ydµ q y ad the additio formua, we I additio to the ier product ad orms defied above o, we sha eed the foowig reated orms for, ] with geeraized Jacobi weight fuctios W α,β x = x α + x β α >, β > fx p p W α,β xdx, p < + ; f = ess sup fx, p = + x,] We aso ote that the Fu-Hece formua 8, Chapter3] impies the foowig usefu coectio betwee itegras over ad itegras over, ] with respect to the weight fuctio W q x = W q 2, q 2 For ay φ L W, x q,] Sq, ad ay Y H q we have φxzy zdµ q z = φy S d q x, q where φ = d q φxp q+ xw q xdx

4 364 SHEG BAOHUAI LI HOGTAO Vo 8 8, Theorem 34] Moreover, we have foowig reatio φx ydµ q x = φxw q xdx The orthogoa projectio Y f, x of a fuctio f L o H q Y f, x = dq x yfydµ q y, p q+ is defied by9] correspodig to which we have the foowig Fourier-Lapace expasio of f fx Y f, x, x =0 Let C be a fiite set of distict poits o The mesh orm of C is defied to be δ C = max distx, C = max x Sq mi x y C distx, y, where distx, y = arccosx y is the geodesic distace betwee x ad y With these otios i mid, we ow rewrite some ow resuts which wi be used i Sectio 4 Lemma 2 3] There exist costats α q ad q with the foowig property Let p +, C be a fiite set of distict poits o ad be a iteger with q α q δ C The, there exist oegative weights {A ξ } ad {a ξ } with a ξ A ξ C, such that for every p Π q, pxdµ q x = a ξ pξ, ad where p C,p = p C,p p p,, p pξ p Aξ, sup{ pξ }, p = + Further, {ξ : aξ 0} q dimπ q Let r > 0 be a iteger ad S r,δ f be the geeraized Bocher-iesz meas of spherica harmoics, S r,δ f, x = 0 < ] r δ Y f, x, x

5 o 3 APPOXIMATIO BY ZOAL TASLATIO ETWOKS 365 The, Li gave i 4] a importat estimate Lemma 22 4] Let f L p, p ad δ satisfy: i q = 2, δ > max{2 p 2, 0}, p + ; or ii q 3, δ p := max{0, q p 2 2 }, p 2 q+, δ > δ p, p + The for ay positive iteger r we have where the K-fuctioa K f, t p is defied by S r,δ f f p, K f, r p, 2 K f, t p = if g D p,r f g p, + t g D p,r, D p,r = {f L p : r f L p }, D p,r = r p,, the mutipier operator r is a operator for which r Y = λ r Y x, Y H q 3 The iesz Meas of Jacobi Poyomias I this sectio, we sha give the estimate of the degree of approximatio by iesz meas of Jacobi poyomias To defie K-fuctioa, we first defie the Jacobi differetia operator P α,β D = W α,β x d dx W α,βx x 2 d dx, ad from which we defie P α,β D r = P α,β DP α,β D r, P α,β D 0 = P α,β D The eigefuctios of P α,β D are the Jacobi poyomias p α,β x ad P α,β D r p α,β x = + α + β + r p α,β x, where the Jacobi poyomias p α,β x is ormaized by p α,β xp α,β m xw α,β xdx = δ,m The forma expasio of f L W α,β correspodig to p α,β x is fx + =0 a fp α,β x, x, ], where a f = fxp α,β xw α,β xdx

6 366 SHEG BAOHUAI LI HOGTAO Vo 8 For itegers b > 0, r, the iesz mea f is f, x = 0 < + α + β + + α + β + r b a fp α,β x A K-fuctioa K α,β f, t r p correspodig to differetia operator P α,β D is defied i 5] by K α,β f, t r p = if g C r,p α,β D r g L p W α,β f g + t r P α,β D r g, where P α,β D j gx is give i such a way that P α,β D j gxp α,β xw α,β xdx = gxp α,β D j p α,β xw α,β xdx hods for p α,β x, = 0,, 2, The, Z Ditzia 6] showed that,0,0, f f p,w0,0 K 0,0 f, 2 p, f L p, ] I 7], Che showed that r,0,0, f f p,w0,0 K 0,0 f, 2r p, f L p, ] For the eed of costructig zoa trasatio etwors i ext paragraph we give here a upper estimate of covergece rate of f i case of α, β 0, 0 Lemma 3 Let f L p W α,β, p +, b > maxα + 2, β + 2, α, β >, ad α + β The f Cr, b, α, β f Proof The case of r = ca be foud from 5] We ow give the proof for r > Let where A b m = m+b b where σf b = σf, b x = A b A b P f, x, x, ], =0, P f, x = a fp α,β x The by 5, 58] we ow b+ b + + b j P f, x = j σ j b jf, b x j=0 = b+ + b σ b b f, x, g = g g, m g = m g, ad σ b f, x 0, =, 2, 3,

7 o 3 APPOXIMATIO BY ZOAL TASLATIO ETWOKS 367 Let a, be a sequece of rea umbers i The, by the way of Theorem 5 i 5], usig Abe trasformatio b + times, we have where =0 b a, P f, x = b+ + =0 + b b σ b f, x b+ a, b j j b j j + b a j, σ b b jf, x, j=0 a, = a +, a,, m a, = m a, Let The, for 0 j b we have a j, = a, = + α + β + ] r b + α + β + j j + α + β + ] r b + α + β + rb Sice, for 0 j b, j+b b b, ad b is a fiite umber, we have b j b j j + b j a j, b j=0 Sice for g, a poyomia i of degree, b+ g is a poyomia of degree b i ecaig that + α + β + ] r b a, = + α + β + is a poyomia of degree br i, we ow that b+ a, is a poyomia of degree br b i It foows that ecaig that +b b b, therefore, b+ br b a, br b =0 b+ + b b+ a, b By Theorem A i 5] we ow that for b > maxα + 2, β + 2, α, β >, ad α + β Lemma 3 is therefore proved σ b f C f, p +

8 368 SHEG BAOHUAI LI HOGTAO Vo 8 Lemma 32 Let r be a iteger, g C 2r, ], p +, b > maxα + 2, β + 2, α, β >, ad α + β The we have Sice Proof oe has = 0 < = 0 < = = 0 < i= g g 2r P α,β D r g g gx + α + β + + α + β + + α + β + + α + β + r ] b a ] r b a gp α,β x + α + β + ] r b + α + β + + α + β + + α + β + b b + α + β + i i + α + β + i= ] r b a gp α,β x ri g gp α,β b b ir+ ri a g + α + β + ri p α,β x i + α + β + 0 < P α,β D r p α,β + α + β + ] r b + α + β + g gx = x = + α + β + r p α,β x, x b b ir+ i + α + β + i= By the defiitio of operators P α,β D ri we have Sice P α,β D ri m P α,β D ri g, x g, x = P α,β D ri g, x g, x = m g, x, ri

9 o 3 APPOXIMATIO BY ZOAL TASLATIO ETWOKS 369 we have m m g, x m+ m g, x = m m g, x m g, x + m g, x m+ m g, x b b = ir+ P α,β D ri m g, x i i= ri ri mm + α + β + m + m + α + β + 2 = b i= b ir+ P α,β D ri m g, xo i m 2ri+ 8, Theorem42] ecaig the Berstei iequaity P α,β Dp Cα, β 2 p, p P, we have by Lemma 3 that Pα,β D ri m g = Pα,β Dm P α,β D ri g Therefore, m 2 m P α,β D ri g m 2ri m P α,β D r g m 2ri Pα,β D r g g, x g, x b b ir+ ri i + α + β + i= Pα,β D ri g Pα,β D r g, 2r m m g, x m+ m g, x Pα,β m 2r+ D r g I a simiar way, oe has m+ m+ g, x m m+ g, x Pα,β m 2r+ D r g Hece, im m m g m+ m+ g + m= Pα,β im + m 2r+ D r g m= Pα,β 2r D r g

10 370 SHEG BAOHUAI LI HOGTAO Vo 8 It foows that g g + + m= m g g g g g g m g g g + m m= m+ g g m g g g m+ By Lemma 3 ad the desity of poyomias i L p W α,β, we have Therefore, + + g g m+ g m+ g + + g + g + + g g 0 + g g + + m= m g g m Pα,β D r g 2r g m+ m+ g Lemma 33 Let f L p W α,β, p +, b > maxα + 2, β + 2, α, β >, ad α + β The, f f Cr, b, α, βk α,β f, 2r p 3 Proof By Lemma 3 ad Lemma 32, we ow that whe g satisfies P α,β D r g L p W α,β, f f f g + g g + f g f Pα,β g + D r g 2r

11 o 3 APPOXIMATIO BY ZOAL TASLATIO ETWOKS 37 Hece, f f K α,β f, 2r p 4 A Sequece of Zoa Trasatio etwors ad Its Degree of Approximatio Choosig a orthoorma basis {Y, : =, 2,, d q } for each H q φ, x, = ] +, the we have d q Y, x = s,q,b s,q,bφ φ, xξy,ξdµ q ξ s, q 2, q 2,b Let The, we have for x that S r,δ f, x = 0 < = = ˆf, = fuy, udµ q u d q f, Y,x S r,δ d q S r,δ 0 < = = s,q,b 0 < s,q,b = s,q,b d f, q φ s,q,b φ, xξy,ξdµ q ξ d q 0 < s,q,b = S r,δ 0 < = which ad Lemma 2 aows S r,δ f, x = φ d q = φ, xξy,ξdµ q ξ S r,δ f, u dq 0 < s,q,b p q+ S r,δ f, uy,udµ q u d q d q s,q,b φ φ, xξ = f, uy,ξy, udµ q u dµ q ξ d q s,q,b s,q,b φ φ, xξ ξudµ q u dµ q ξ d q φ d q 0 < s,q,b φ s,q,b, ad taig s,q,b φ, x = φ, xξy S r,δ f, ξ dµ q ξ, a ξ s,q,b φ, xξy S r,δ f, ξ 4

12 372 SHEG BAOHUAI LI HOGTAO Vo 8 This fact remids us of the itroductio of foowig zoa trasatio etwor operators M,φ f, x = d q 0 < s,q,b φ a ξ φxξy S r,δ f, ξ 5 Theorem 4 Let q 2, b > q 2, f Lp, p +, φ L p W q such that ˆφ 0 = 0,, 2,, δ satisfy the coditios of Lemma 22, itegers s > r > 0, ad α φ,q = 0 < d q 2 p q+ p,w q +q ] s b, ]+]+q ˆφ p + p = The, there exists a costat Cp, r, s > 0 such that f M,φ f p, Cp, r, s K f, r + α φ,q fp,k q 2, q 2 φ, 2s p ] Proof By Höder iequaity ad 4 5, oe has M,φ f, x S r,δ f, x a ξ By we ow a ξ A ξ Sice C Therefore, 0 < s,q,b s,q,b d q Y S r,δ φ f, ξ φ, xξ φxξ a ξ s,q,b φ, xξ φxξ p p a ξ 0 < s,q,b d q p p Y S r,δ φ f, ξ d q p p A = a ξ Y S r,δ 0 < s,q,b φ f, ξ d q p p C A ξ Y S r,δ 0 < s,q,b φ f, ξ d q C Y S r,δ 0 < s,q,b φ f, x p, Y S r,δ f, x = dq = dq S r,δ S r,δ f pq+ x f, ηpq+ xηdµ q η,

13 o 3 APPOXIMATIO BY ZOAL TASLATIO ETWOKS 373 the Höder iequaity maes oticig that we have which aows Cosequety, Y S r,δ f, x d q S r,δ ω f p, q C dq fp, p q+ xη p dµ q η p q+ C dq ωq p p q+ fp,s ω p,w q q q p q 2, q 2 ωq d q x = 2 p q+ p x p p W q xdx x, s,q,b + q ] s b φ, x = ˆφp q+ x, + q 0 < φ + q ] s b = ] + ] + q ˆφ s,q,b M,φ f, x S r,δ f, x p, C a ξ s,q,b p p q+ 0 < C 0 < = C s,q,b φ, xξ φxξ p dµq x p,w q p d q 2 ωq +q ] fp,s s b q ]+]+q ˆφ a ξ ω p q s,q,b φ, x φx p Wq xdx d q 2 p q+ p,w q +q ] s b f p, ]+]+q ˆφ φ φ p,wq α φ,q fp, It foows by Lemma 2 that a ξ = Therefore, a ξ p ω p q ad M,φ f, x S r,δ f, x p, M,φ f f p, Cα φ,q fp,k q 2, q 2 φ, 2s p, S r,δ f, x fx p, + M,φ f, x S r,δ f, x p, C K f, r p + α φ,q fp,k q 2, q 2 φ, 2s p

14 374 SHEG BAOHUAI LI HOGTAO Vo 8 Coroary 4 Let q 2, b > q 2, f Lp, p +, φ L p W q such that ˆφ 0 = 0,, 2,, δ satisfy the coditios of Lemma 22 If iteger s > 0 big eough, ad φ smooth eough such that im + αφ,q K q 2, q 2 φ, 2s p = 0, the im + f M,φ f p, = 0 efereces ] Y P Li, J B Li ad X H Zhao, Loca stabiity ad bifurcatio i a three uit deayed eura etwor, Joura of Systems Sciece ad Compexity, 2003, 6: ] Z H Gua ad G Che, O the equiibria stabiity ad istabiity of Hopfied eura etwors, IEEE Tra eura etwors, 2000, 2: ] T P Che, H Che ad W Li, Approximatio capabiity i C by mutiayer feedforward etwors ad reated probems, IEEE Tra eura etwors, 995, 6: ] H Mhasar ad C A Micchei, Approximatio by superpositio of sigmoida ad radia basis fuctios, Advaced i Appied Mathematics, 992, 3: ] H Mhasar ad C A Micchei, Degree of approximatio by eura ad trasatio etwors with sige hidde ayer, Advaced i Appied Math, 995, 6: ] J L Wag, B H Sheg ad S P Zhou, O approximatio by o-periodic eura ad trasatio etwors i L p w spaces, Acta Mathematica Siicai Chiese, 2003, 46: ] H Mhasar, F J arcowich ad J D Ward, Approximatio properties of zoa fuctio etwors usig scattered data o the sphere, Advaces i Computatioa Mathematics, 999, : ] H Groemer, Geometric Appicatios of Fourier Series ad Spherica Harmoics, Cambridge Uiversity Press, 996 9] K Y Wag ad L Q Li, Harmoic Aaysis ad Approximatio o the Uit Sphere, Sciece Press, ew Yor, ] C Muer, Spherica harmoic, Lecture otes i Mathematics, Vo7, Spriger-Verag, Beri, 966 ] H Mhasar, F J arcowich ad J D Ward, Zoa fuctio etwor frames o the sphere, eura etwors, 2003, 6: ] H Mhasar, F J arcowich ad J D Ward, Spherica Marciiewicz-Zygmud iequaities ad positive quadrature, Mathematics of Computatio, 2000, 70: ] H Mhasar, F J arcowich ad J D Ward, Corrigedum to Spherica Marciiewicz- Zygmud iequaities ad positive quadrature, Mathematics of Computatio, 200, 7: ] L Q Li, The iesz meas o the compact symmetric spaces, Math achr, 994, 68: ] W Che ad Z Ditzia, Best approximatio ad K-fuctioas, Acta Math Hugar, 997, 75: ] Z Ditzia, A K-fuctioa ad the rate of covergece of some iear poyomia operators, Proc Amer Math Soc, 996, 24: ] S Y Che, The Peetre K-fuctioas ad the iesz summabiity operators, J of MathPC, 998, 84: ] A S Dzafarov, Berstei iequaity for differetia operators, Aaysis Math, 986, 2: