Probabilistic and Average Linear Widths in L -Norm with Respect to r-fold Wiener Measure

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1 joural of approximatio theory 84, 3140 (1996) Article No Probabilistic ad Average Liear Widths i L -Norm with Respect to r-fold Wieer Measure V. E. Maiorov Departmet of Mathematics, Techio, Haifa, Israel ad G. W. Wasilkowski* Departmet of Computer Sciece, Uiversity of Ketucky, Lexigto, Ketucky Commuicated by Alla Pikus Received February 9, 1994; accepted i revised form February 6, 1995 We show that for r-fold Wieer measure, the probabilistic ad average liear widths i the L -orm are proportioal to &(r+12) - l $ ad &(r+12) -l, respectively Academic Press, Ic. 1. Itroductio We study probabilistic liear (, $)-widths ad average liear -widths for L -approximatio of fuctios that are distributed accordig to the r-fold Wieer measure. As the clasical -widths (see, e.g., [9]); probabilistic ad average widths quatify the error of best approximatig operators. However, i the classical approach, the errors are defied by their worst case with respect to a give class (typically a uit ball of the uderlyig space). I the probabilistic approach, the errors are defied by the worst case performace o a subset of measure at least 1&$, ad i the average case approach, they are defied by their expectatios, both with respect to a give probability measure. The study of probabilistic ad average widths has bee suggested oly recetly (see, e.g., [8, 13]) ad relatively few results have bee obtaied so far (see, e.g., [1, 2, 47, 10, 12, 13]). These iclude results o probabilistic ad average Kolmogorov widths i L q -orm for ay q ad o probabilistic ad average liear widths i L q -orm for fiite q. I both cases, the uderlyig space of fuctio is the C r [0, 1] space equipped with * Partially supported by the Natioal Sciece Foudatio uder Grat CCR Copyright 1996 by Academic Press, Ic. All rights of reproductio i ay form reserved.

2 32 maiorov ad wasilkowski the r-fold Wieer measure. More specifically, the upper bouds o the average widths follow from [11] for q< ad [10] for q=. The asymptotic lower bouds o average Kolmogorov widths with arbitrary q ad o average liear widths with fiite q are maily due to [46]. The results cocerig probabilistic Kolmogorov ad liear widths are also due to [46]. Our result cocerig the probabilistic ad average liear widths for q= provides the last missig piece as far as the probabilistic d average liear widths with r-fold Wieer measures are cocered. Thus, deotig probabilistic Kolmogorov ad liear (, $)-widths by d (p), $(C r, L q, r ) ad -widths by d (a) that, $(C r, L q, r ), ad average Kolmogorov ad liear (C r, L q, r ), respectively, we coclude (C r, L q, r ) ad * (a) d (p),$ (C r, L q, r ) &(r+12) - 1+ &1 l(1$),,$ (Cr,L q, r ) { &(r+12) -1+ &mi[1,2q] l(1$), &(r+12) -l($), d (a) (C r,l q, r ) &(r+12), * (a) (C r,l q, r ) { &(r+12), &(r+12) -l, 1q, 1q<, q=, 1q, 1q<, q=. It is iterestig to ote that for fiite q, the average Kolmogorov ad average liear -widths are equal modulo multiplicative costats. We have also equality betwee probabilistic Kolmogorov ad liear (, $)-widths for q2. For such values of q, liear approximatio opertors are (modulo a costat) as good as oliear operators. The differece is oly for q= (for average widths) ad for q>2 (for probabilistic widths); however, the liear operators lose to optimal oliear operators oly by a factor of - l ad (12&1q) +, respectively. The paper is orgaized as follows. Basic defiitios ad the mai result are provided i Sectio 2. The proof of the result is i Sectio Mai Result For a oegative iteger r, let C r be the space of r times cotiuously differetiable fuctios defied o [0, 1]. Recall that the correspodig Kolmogorov ad liear -widths are defied respectively by d (C r, L q )= if T # 4 * (C r, L q )= if T # L sup & f &T( f )& q, (1) f # B(C r ) sup & f &T( f )& q, (2) f # B(C r )

3 liear-widths i L -orm 33 where B(C r ) is the uit ball i C r, 4 is the class of all (ot ecessarily liear) operators T : B(C r ) L q whose rage is cotaied i a -dimesioal subspace of L q, ad L is the class of all liear operators from 4. Let + be a probability measure defied o the Borel _-field of C r. Give $ # [0, 1], the correspodig probabilistic Kolmogorov ad probabilistic liear (, $)-widths are defied by d (p), $ (C r, L q, +)=if G, $ (Cr, L q, +)=if G if T # 4 if T # L sup & f &T( f )& q, (3) f # G sup & f &T( f )& q. (4) f # G The first ifima are take with respect to all measurable sets G/C r with +(G)1&$. The average Kolmogorov ad average liear -widths are defied by d (a) (C r, L q, +)= if E + (& f &T( f )& q ), (5) T # 4 * (a) (Cr, L q, +)= if E + (& f &T( f )& q ). (6) T # L Here E + deotes the expectatio with respect to +, i.e., E + (& f &T( f )& q )= &f&t( f )& q +(df). C r Obviously, d (a) (C r, L q, +)= 1 * (a) (C r, L q, +)= d (p), $ (Cr, L q, +) d$,, $(C r, L q, +) d$. (7) I what follows we assume that + equals the r-fold Wieer measure r. For basic properties of r, see, e.g., [3]. Here we oly metio that r is a zero mea Gaussia measure with the covariace fuctio E r ( f(x) f( y))= 1 0 (x&t) r + (y&t)r + (r!) 2 dt ad that r (A)= 0 (D r A), where 0 is the classical Wieer measure o the space C 0 ad D r is the differetial operator, D r f = f (r). As metioed i Itroductio, the probabilistic ad average Kolmogorov widths have bee foud for ay q, ad the probabilistic ad

4 34 maiorov ad wasilkowski average liear widths have bee foud oly for fiite q. The followig theorem deals with probabilistic ad average liear widths for q=. Theorem 1. For every r ad $ #(0, 1 2 ),, $ (Cr, L, r ) &(r+12) - l($), * (a) (Cr, L, r ) &(r+12) - l. Actually, the proof of Theorem 1 provides aother result cocerig liear widths for fiite dimesioal spaces. Let l m deote the space R m equipped with the maximum orm, ad let # deote zero mea ormal distributio with the idetity covariace matrix, #=N(0, I). (8) Theorem 2. Let m>2 ad $ #(0, 1 2 ). The, $ (Rm, l m, #) - l((m&)$), * (a) (Rm, l m, #) - l(m&). (9) 3. Proof Due to (7), we oly eed to show the equality cocerig the probabilistic widths. We begi with few auxiliary lemmas. Lemma 1. Let m>2. Let T be ay operator i R m whose rage is cotaied i a -dimesioal subspace. For i=1,..., m, let g i =e i &T*(e i ), where e i is the ith uit vector. The there are distict idices i 1,..., i m&2 such that for all s=0,..., m&2&1. dist[g is+1, li[g i1,..., g is ]]1-2 Proof. It is kow (see, e.g., [9]) that the followig Kolmogorov -widths equal Therefore, there is i 1 such that d (cov(e 1,..., e m ), l m 2 )=-(m&)m. (10) dist[g i1, [0]]=&e i1 &T*(e i1 )& 2 - (m&)m. Assume by iductio that i 1,..., i s (sm&2&1) exist. Cosider the idex sets I=[i 1,..., i s ] ad I$=[1,..., m]"i, ad the followig operator P: l m l m&s 2 2, Px=(x i ) i # I$. From (10) it follows that d (cov[pe i : i # I$], l m& 2 )-(m&s&)(m&s).

5 liear-widths i L -orm 35 Therefore, there is i s+1 # I$ for which dist[pe is+1, PL]- (m&s&)(m&s), where L=Im T*. Sice Pg i =&PT*e i # PL for ay i, we have dist[g is+1, li[g i1,..., g is ]]dist[pg is+1, li[pg i1,..., Pg is ]] dist[pe is+1, PL]- (m&s&)(m&s). Sice sm&2, we have (m&s&)(m&) 1 2. This completes the proof. K Let k=m&2, ad let r 1,..., r k # R m. By G a (r 1,..., r k ) we deote the followig polytop i R m : G a (r 1,..., r k )=[x # R m : max (r i, x) a]. 1ik Lemma 2. Let h s =g is for s=1,..., k. The Proof. #(x : &x&tx& a)1&#(g a (h 1,..., h k )). Sice &x&tx& = max (e i, x&tx) 1im = max (e i &T*e i, x) 1im max (h i, x), 1ik we have #(x : &x & Tx& a)#(x: max 1ik (h i, x) a)=1& #(G a (h 1,..., h k )), which completes the proof. K Lemma 3. we have For a= 1 2 l m&2 2$ 2$ #(G a (h 1,..., h k ))\1& m&2+ #(G a(h 1,..., h k&1 )). Proof. Due to rotatioal ivariace of # we ca assume without loss of geerality that h 1,..., h k&1 # li[e 1,..., e k&1 ], h k # li[e 1,..., e k ].

6 36 maiorov ad wasilkowski For x # R m, let x$=(x 1,..., x k&1 ). The #(G a (h 1,..., h k ))=(2?) &k2 Ga (h 1,..., h k&1 ) exp(&&x$& 2 2 2) _ (x$, h $ k )+x k h kk a exp(&x 2 k 2) dx k dx$ (2?) &(k&1)2 Ga (h 1,..., h k&1 ) exp(&&x$& 2 22) dx$(2?)&12 _ thkk a exp(&t 2 2) dt =#(G a (h 1,..., k k&1 ))(2?) &12 thkk a exp(&t 2 2) dt. From Lemma 1 it follows that h kk =dist[h k, li[h 1,..., h k&1 ]]1-2. Therefore, #(G a (h 1,..., h k ))#(G a (h 1,..., k k&1 ))(2?) &12 t a -2 exp(&t 2 2) dt. (11) To complete the proof we eed oly to show that (2?) &12 t a-2 exp(&t 2 2) dt1& 2$ m&2 (12) for sufficietly large (m&2)(2$). For this ed, we use the fact that 2? x e &t2 2 dt c x e&x2 2 \c #(0,-2?), \x c - 2?&c, (13) which follows from the fact that f(x)=-2? 2 x dt&ce e&t2 &x22 x has a egative derivative for such values of c ad x ad the fact that f()=0. Usig c=1-? ad x=a - 2, we have x- c(- 2?&c) wheever (m&2)(2$)exp(1(- 2&1)). Moreover, cx- 2$(m&2) wheever (m&2)(2$)? l((m&)(2$)). Thus (13) holds whe m&2 2$ max m&2? l { 1(-2&1)=, e 2$. This completes the proof of Lemma 3. K

7 liear-widths i L -orm 37 We are ready to prove Theorem 2. Proof of Theorem 2. Applyig Lemma 3 k=m&2 times, we get k #(G a (h 1,..., h k )) k+ \1&2$. Hece, Lemma 2 yields k #(x : &x&tx& a)1& k+ \1&2$ $. Sice T is arbitrary, this proves that $ (Rm, l m, #)a -l((m&)$). To prove equality observe that #(Q)1&$ for Q=[x # R m : max x i -l((m&)$)]. 1im& This meas that for the orthogoal projectio operator T o li[e m&+1,..., e m ], sup &x&tx& - l((m&)$). x # Q This proves that, $ - l((m&)$), as claimed, ad completes the proof of Theorem 2. We are ready to prove Theorem 1. Proof of Theorem 1. We begi with the lower boud:, $ (C r, L, r )c r &(r+12) - l($) (14) for a positive costat c r. For this ed, we cosider the iverse fuctio of probabilistic widths. That is, give, let e (=; C r, L, r ) := if T # L r ( f # C r : & f &T( f )& =) for =0. Obviously, e (=; C r, L, r )=$ for ==, $ (C r, L, r ). Take ow m=2 ad a i =im for i=1,..., m. Let + r,0 = r (} N(f)=0) be the coditioal measure with N( f )=[ f ( j) (a i )=0 : 0jr,1im]. Sice r is Gaussia, e (=; C r, L, r )e (=; C r, L, + r,0 ) if + r,0 (f#c r : max f(t i )&T( f )(t i ) =), T # L 1im

8 38 maiorov ad wasilkowski where t i =(a i&1 +a i )2=(2i&1)(2m). Let M( f )=[ f(t 1 ),..., f(t m )], let +* r,0 =+ r,0 M &1 be the iduced probability o R m, ad let &=+ r,0 (} M(f)=0) be the coditioal probability with M( f )=0. Lettig s y (x)= m i=1 y is j (x) be the mea elemet of + r,0 (} M(f)=y) (y=(y 1,..., y m )#R m ), we have that ay f ca be represeted as f =s y +h, where y is distributed accordig to +* r,0,his distributed accordig to &, ad y ad h are idepedet. Sice s y (t i )=y i, we have that for a arbitrary T # L + r,0 (f#c r : max f(t i )&T( f )(t i ) =) 1im =(+* r,0 _&)((y, h)#r m _C r : max y i &T(s y )(t i )+h(t i )&T(h)(t i ) =) 1im +* r,0 (y#r m : max y i &A(y) =) 1im with the matrix A=(a i, j ) give by a i, j =T(s j )(t i ). (The iequality above follows from the fact that & is zero mea Gaussia.) Sice the rak of T does ot exceed, so does the rak of A. This implies that e (=; C r, L, r )e (=; R m, l m, +* r,0), or equivaletly, that Fially, sice, $ (C r, L, r ), $ (Rm, l m, +* r,0). (15) +* r,0 =N(0, _I) with _c 1 m &(r+12) for some costat c 1 (see [14]), this ad (15) imply that, $ (C r, L, r ), $ (Rm, l m, +* r,0)c 1 m &(r+12), $ (Rm, l m, #). Hece, Lemma 1 with m=2 completes the proof of the lower boud (14). We ow prove the upper boud,, $ (C r, L, r )c &(r+12) - l($) (16) for a positive costat c. For this ed, we use the followig geeral result; see [5]. Let [ k ] k be a sequece of oegative itegers ad [$ k ] k be a sequece of reals from [0, 1]. If the k 2 k, : k=0 (C r, $, L, r ) : k, : k=0 k=0 $ k $, \k0, (17) 2 &(r+12)k k, $ k (R 2k, l 2k, #). (18)

9 liear-widths i L -orm 39 Without loss of geerality, we ca assume that =2 k$. Cosider k = {2k, w2 k$&k x, k<k$&1, kk$&1, $ k = {0, $2 k$&k, k<k$&1, kk$&1. Obviously, [ k ] k ad [$ k ] k satisfy (17), ad : k=0 2 &(r+12)k k, $ k (R 2k, l 2k, #) c 1 : k=k$&1 c 2 2 &(r+12)k$ -l(2k$$) 2 &(r+12)k - l((2 k & k )($2 k$&k )) for some positive costats c 1 ad c 2. This proves (16) ad, hece, completes the proof of Theorem 1. K Ackowledgmets We thak Klaus Ritter ad Heryk Woz iakowski for their valuable commets. Refereces 1. A. P. Buslaev, O best approximatio of the radom fuctios ad fuctioals, Third Saratov Witer School 2 (1988), S. Heirich, Lower bouds for the complexity of Mote-Carlo approximatio, J. Complexity 8 (1992), H. H. Kuo, ``Gaussia Measures i Baach Spaces,'' Lecture Notes i Math., Vol. 463, Spriger-Verlag, Berli, V. E. Maiorov, Average -widths of the Wieer space i L -orm, J. Complexity 9 (1993), V. E. Maiorov, Kolmogorov (, $)-widths of the spaces of the smooth fuctios, Math. Sb. 184 (1993), V. E. Maiorov, About -widths of Wieer space i L q -orm, J. Complexity, to appear. 7. P. Mathe, Radom approximatio of Sobolev embeddigs, J. Complexity 7 (1991), C. A. Micchelli, Orthogoal projectios are optimal algorithms, J. Approx. Theory 40 (1984), A. Pikus, ``-Widths i Approximatio Theory,'' Spriger-Verlag, New York, K. Ritter, Approximatio ad optimizatio o the Wieer space, J. Complexity 6 (1990), P. Speckma, ``L p Approximatio of Autoregressive Gaussia Processes,'' Techical Report, Departmet of Statistics, Uiversity of Orego, Eugee, OR, 1979.

10 40 maiorov ad wasilkowski 12. J. F. Traub, G. W. Wasilkowski, ad H. Woz iakowski, ``Iformatio-Based Complexity,'' Academic Press, New York, S. M. Voroi ad N. Temirgaliey, O some applicatios of Baach measure, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. 5 (1984), G. W. Wasilkowski, O average complexity of global optimizatio problems, Math. Programmig 57 (1992),