On regularization in superreflexive Banach spaces by infimal convolution formulas

a Xi :ma h/9706214 1 [ma h.FA] 12 Jun 1997 ON REGULARIZATION IN SUPERREFLEXIVE BANACH SPACES BY INFIMAL CONVOLUTION FORMULAS Manuel Cepedello Boiso May 28, 1997 Abs ac . We p esen he e a new me hod o app oxima ing unc ions de ined on supe e lexi e Banach spaces by di e en iable unc ions wi h α-H¨olde de i a i es ( o some 0 < α ≤1). The smoo h app oxima ion is gi en by means o an explici o mula enjoying good p ope ies om he minimiza ion poin o iew. Fo ins ance, o any unc ion which is bounded below and uni o mly con inuous on bounded se s his o mula gi es a sequence o ∆-con ex C1,α unc ions con e ging uni o mly on bounded se s o and p ese ing he in imum and he se o minimize s o . The echniques we de elop a e based on he use o ex ended in -con olu ion o mulas and con exi y p ope ies such as he p ese a ion o smoo hness o he con ex en elope o ce ain di e en iable unc ions. 0. In oduc ion and P elimina ies This pape in oduces an explici egula iza ion p ocedu e o unc ions de ined on supe e lexi e Banach spaces. Fo any bounded below l.s.c. ( esp. uni o mly con inuous on bounded se s) unc ion on a supe e lexi e Banach space Xwe gi e by means o a “s anda d” o mula a sequence o C1,α-smoo h unc ions con e ging poin wise ( esp. uni o mly on bounded se s) o (whe e 0 < α ≤1 only depends on X). Unde some addi ional condi ions, he con e gence o he sequence o app oxima e unc ions is uni o m on he whole space X. Mo eo e , he app oxima e unc ions p ese e he in imum and he se o minimize s o . We ema k ha hese ea u es al oge he canno be easily ob ained om egula iza ion me hods like he smoo h pa i ions o he uni y echniques ( o a de ailed s udy o his opic we e e o Chap e VIII.3 o [DGZ], he e e ences he ein and [F ]) o o he esul s ha only ensu e he exis ence o smoo h app oxima es ( o ins ance, see [DFH]). In Hilbe spaces, ou wo k is closely linked wi h he Las y-Lions app oxima ion me hod (in oduced in [LL] and subsequen ly s udied by se e al au ho s, such as [AA]) and i s mo e gene al e sion gi en by T. S ¨ombe g in [S 2]. Ac ually, we imp o e he esul s o [S 2] in he supe e lexi e case by p o iding he bes uni o mly smoo h app oxima ion possible o his se ing. None heless, we wan o ema k ha he app oxima e unc ions explained he ein canno be educed o hose o S ¨ombe g (o Las y-Lions app oxima es in Hilbe spaces); we e e o he ema k a e P oposi ion 8 o a mo e p ecise explana ion. Ou app oach o smoo h 1991 Ma hema ics Subjec Classi ica ion. P ima y 46B20; Seconda y 46B10. Key wo ds and ph ases. Regula iza ion in Banach spaces, con ex unc ions.. The au ho was suppo ed by a FPU G an o he Spanish Minis e io de Educaci´on y Ciencia. Typese by A M S-T EX egula iza ion in non-Hilbe spaces comes om wo main ac s: he densi y o he linea span o he con ex unc ions (s udied in [C]) and he smoo hness o he con ex en elope o a “somehow” smoo h unc ion. In his di ec ion, we also p esen mo e gene al e sions o ce ain esul s in [GR] o in ini e dimensional Banach spaces. This pape is o ganized in he ollowing way. Ou main esul o his pape , Theo em 1, and se e al co olla ies a e explained in Sec ion 1. The p oo o Theo em 1 is showed in Sec ion 4 wi h he ools p o ided by sec ions 2 and 3. Sec ion 2 deals wi h he exis ence o app oxima es o a gi en unc ion using some esul s on ex ended in -con olu ion o mulas. Sec ion 3 de elops a p ocedu e o egula iza ing ce ain ∆-con ex app oxima es. This p ocedu e is based on he smoo hness o he con ex en elope o ce ain “somehow” smoo h unc ions. No a ion: In wha ollows, Xdeno es a Banach space and k · k an equi alen no m on X. By BXwe mean he uni closed ball o Xunde he no m k · k and by BX( ) he closed ball o adius > 0. A unc ion :X→R∪ {+∞} is called p ope i 6≡ +∞and SIn ( ) is he (possibly emp y) se {x∈X: (x) = in }. We will deal wi h he poin wise, compac , uni o m on bounded se s and uni o m on Xcon e gence in he se o lowe semi-con inuous (in sho , l.s.c. ) unc ions on X, abb e ia ed espec i ely by τp,τK,τband τu. A unc ion de ined on Xis called ∆-con ex i i can be exp essed as he di e ence o wo con inuous con ex unc ions. The con ex en elope co o a unc ion :X→ R∪ {+∞} is de ined as he g ea es p ope con ex l.s.c. unc ion below (i he e exis s a con ex mino an o ). The explici alue o he con ex en elope o a a poin x∈Xis gi en by he o mula (co )(x)= in n∈N(n X i=1 λi (xi) : x= n X i=1 λixi, n X i=1 λi=1,xi, λin i=1 ⊂(X×R+) ).(1) Unless s a ed o he wise, di e en iabili y will be unde s ood in he F ´eche sense. The ollowing no a ion is used h oughou his wo k. By C1,u(X) ( espec i ely C1,u B(X)) we unde s and he se o di e en iable unc ions de ined on Xwi h uni- o mly con inuous ( esp. uni o mly con inuous on bounded se s) de i a i e. Simi- la ly, C1,α(X) ( esp. C1,α B(X)) s ands o he class o unc ions on Xha ing α-H¨olde con inuous ( esp. α-H¨olde con inuous on bounded se s) de i a i e (0 < α ≤1). 1. The main esul We begin by s a ing he main esul o his wo k. Theo em 1. Le p > 1,Xbe a Banach space and k · k be an equi alen no m on Xwhich is locally uni o mly con ex and uni o mly smoo h. Fo any p ope lowe semi-con inuous bounded below unc ion :X→R∪ {+∞}, conside he sequence o ∆-con ex unc ions gi en by he o mula ∆p n := co gp n−2p−1nk · kp(n∈N), whe e gp na a poin x∈Xis de ined as gp n(x) := in y∈Xn (y) + 2p−1nkxkp+ 2p−1nkykp−nkx+ykpo+ 2p−1nkxkp. 2 Then he ollowing asse ions a e sa is ied: (i) Fo all n,in ≤∆p n ≤ and SIn (∆p n ) = SIn ( ). (ii) (∆p n )n∈N⊂ C1,u B(X)and (∆p n )n∈N⊂ C1,α B(X)p o ided ha he modulus o smoo hness o he no m k · k is o powe ype 1 + α; ac ually, we ha e ha (∆1+α n )n∈N⊂ C1,α(X). (iii) ∆p n τp −→ n poin wise and ∆p n τK −−→ n i :X→Ris con inuous. I mo eo e he no m k · k is uni o mly con ex hen (i ) ∆p n τb −→ n whene e is uni o mly con inuous on bounded se s. ( ) ∆p n τu −→ n p o ided ha is uni o mly con inuous on X(no necessa ily bounded below) and he modulus o con exi y o he no m k·k is o powe ype p(p≥2). Rema k. I is well-known ha he exis ence o a uni o mly smoo h no m k · k on a Banach space Ximplies he supe e lexi i y o X(and ecip ocally, he a icles [E] o P. En lo and [Pi] o G. Pisie ell us ha any supe e lexi e Banach space admi s an equi alen uni o mly smoo h no m). Simila ly, we wan o poin ou ha he conclusions o Theo em 1 canno be expec ed ou side he supe e lexi e se ing. Fi s , he τb-densi y o he se o ∆-con ex unc ions de ined on Xin he se o unc ions on X ha a e uni o mly con inuous on bounded se s is equi alen o he supe e lexi i y o he Banach space X(as i was p o ed in [C]). On he o he hand, he exis ence o C1,α bump unc ions ( o some 0 < α ≤1) on Ximplies he exis ence o an equi alen no m k · k on Xwi h modulus o smoo hness o powe ype 1 + α(see Theo em V.3.1. o [DGZ]). Rema k. The op imal applica ion o Theo em 1 is achie ed when we conside a Hilbe ian no m k · k. In his case, aking p= 2 in Theo em 1 we ob ain simila app oxima ion esul s as hose gi en by he Las y-Lions app oxima ion me hod (see [LL]). Ne e heless, he di e en sequences o app oxima es a e no he same e en in his se ing (see ema k a e P oposi ion 8). We p oceed o s a e some co olla ies o Theo em 1. They a e ela ed wi h ce - ain esul s known on a supe e lexi e Banach spaces om he exis ence o smoo h pa i ions o he uni y (see Theo em VIII.3.2 in [DGZ]). Thei p oo is easily ob- ained appealing o Theo em 1 and Pisie ’s eno ming Theo em ( he o iginal p oo can be ound in [P]; we e e o [L] o a simple and mo e geome ical p oo ). The i s co olla y imp o es Co olla y 1 o [S 2] o supe e lexi e Banach spaces. Co olla y 2. Le Xbe a supe e lexi e Banach space. Then he e exis s some 0< α ≤1such ha any non-emp y closed se Fo Xis he se o ze os o a ∆- con ex C1,α-di e en iable unc ion on X. Mo eo e , Fis he limi o he Hausdo dis ance o a sequence o se s Sn={x∈X: n(x)< σn∈R}(n∈N) whe e he unc ions ( n)na e ∆-con ex and in C1,α B(X). P oo o Co olla y 2. Fo a supe e lexi e Banach space X, Pisie ’s eno ming The- o em ensu es he exis ence o an equi alen no m k·k on Xwi h modulus o smoo h- ness o powe ype q(1 < q ≤2). Gi en a closed se Fin X, conside he p ope unc ion dde ined a a poin x∈Xas d(x) := dis (x, F ) = in y∈Fkx−yk(dis p ope because Fis no emp y). By Theo em 1(i)–(ii) we ha e ha he unc ion ∆q 1dis ∆-con ex, C1,q−1-di e en iable and sa is ies ha SIn (∆q 1(d)) = SIn (d) = F. 3 Mo eo e , using Asplund a e aging echnique (see P oposi ion IV.5.2 o [DGZ]), we can assume ha he modulus o con exi y o he no m k·k is in addi ion o powe ype p( o some p≥2). Since dis Lipschi z con inuous on X, om Theo em 1(i ) i ollows o e e y n ha F⊆ {∆p nd(x)<1 n:x∈X}:= Sn, whe e ∆p ndis a ∆-con ex C1,q−1 B-di e en iable unc ion and (Sn)ncon e ges o F o he Hausdo dis ance.  The nex co olla y gi es a sligh ly s onge e sion o some o he s app oxima- ion esul s ob ained by using pa i ion o he uni y echniques ( o ins ance, see Theo em 1 o [NS]). Co olla y 3. Fo any supe e lexi e Banach space X he e is 0< α ≤1so ha o e e y uni o mly con inuous on bounded se s ( esp. uni o mly con inuous) unc- ion on Xone has he ollowing: is he uni o m limi on any ixed bounded se Bo X( esp. on X) o a sequence o ∆-con ex C1,α-di e en iable ( esp. C1,α B- di e en iable) unc ions ha ing he same in imum and se o minimize s on Bas . P oo o he Co olla y 3. Appealing again o Pisie ’s eno ming Theo em o su- pe e lexi e Banach spaces, we can suppose ha he e is an equi alen no m k · k on Xwi h modulus o smoo hness o powe ype q(1 < q ≤2). Fix some bounded se Bo Xand de ine ˜ := max{ , in B }. Since is uni o mly con inuous on B, we ha e ha in B > −∞. The e o e, ˜ is uni o mly con inuous on bounded se s and bounded below. No e ha i ially ˜ (x) = (x) o all x∈Band hen he in imum and se o minimize s on Bo and ˜ a e he same. Hence, Theo em 1(ii) and ( i) ell us ha he sequence (∆q n˜ )nsa is ies he equi ed condi ions o he claim o α=q−1. I is uni o m con inuous on X, he p oo o Co olla y 3 ollows he same lines, using he exis ence on Xo an equi alen no m k · k wi h non- i ial moduli o con exi y and smoo hness and Theo em 1( ).  The las co olla y is an ex ension o Rema k ( iii) in [LL]. I deals wi h he p ope y o ex ending and egula izing unc ions de ined on subse s o supe e lexi e Banach spaces o he whole space. Co olla y 4. Le Xbe a supe e lexi e Banach space. The ollowing holds ue o some 0< α ≤1depending only on X: Le Sbe a subse o Xand :S→Rbe a unc ion ha is uni o mly con inuous on bounded se s o S. Then o e e y > 0and ε > 0 he e exis s a ∆-con ex unc ion F ,ε :X→Rsa is ying he ollowing condi ions: (i) in S = in XF ,ε and SIn ( ) = SIn (F ,ε), (ii) F ,ε ∈ C1,α(X), o some 0< α ≤1, and (iii) (x)−ε≤F ,ε(x)≤ (x) o e e y x∈S∩BX( ). P oo o he Co olla y 4. By he same a gumen as abo e, le k · k be an equi alen no m on Xwi h modulus o smoo hness 1 + α( o some 0 < α ≤1). Conside he ollowing simple ex ension o : F(x) :=  (x) o x∈S +∞o he wise. No ice ha SIn (F) = SIn ( )⊂S. I is no ha d o see using P oposi ion 8(i) and he p oo o P oposi ion 6( ) ha he sequence (∆1+α nF)n∈N, which sa is ies (i) and (ii) o Theo em 1, also con e ges uni o mly on bounded se s o S o . 4 The p oo o Theo em 1 will be done in a gene al scheme in ol ing wo main s eps. Fi s , we explain an ex ended in -con olu ion o mula ha gi es us a s anda d way o app oxima e unc ions on X. Then, we de elop some con exi y echniques in o de o ge smoo h ∆-con ex unc ions be ween he unc ions gi en by he ex ended in -con olu ion o mula. 2. The ex ended in -con olu ion In his sec ion we explain he con e gence esul s we need in he p oo o The- o em 1. Fi s , we in oduce he de ini ion o ex ended in -con olu ion. This de i- ni ion gene alizes he classical one o in -con olu ion (see [S 1] o a gene al su ey o he subjec ) and will be an impo an ool in ou wo k. De ini ion. Fo any applica ion K:X×X→R∪ {+∞} and any unc ion :X→R∪ {+∞} we de ine he ex ended in -con olu ion o by Kas he unc ion  K(x) := in y∈Xn (y) + K(x, y)o, x ∈X. Kwill be called he ke nel o he ex ended in -con olu ion. Example. I o g:X→R∪ {+∞} we conside he ke nel Kg(x, y) := g(x−y), hen he ex ended in -con olu ion  Kgis no hing else bu he classical in - con olu ion  g. Be o e he s a emen o he main esul o his sec ion, we need o de ine some na u al p ope ies o ke nels. De ini ion. A ke nel Kis poin wise sepa a ing i o e e y x0∈Xand e e y δ > 0 he e exis s Cx0,δ >0 such ha K(x0, y)≥Cx0,δ whene e kx0−yk ≥ δ. A ke nel Kis called uni o mly sepa a ing on bounded se s i o all > 0 and δ > 0 he e exis s C ,δ >0 so ha K(x, y)≥C ,δ p o ided kxk ≤ and kx−yk ≥ δ. A ke nel Kis uni o mly sepa a ing i o e e y δ > 0 he e is some βδ>0 in such a way ha K(x, y)≥βδkx−ykwhene e kx−yk ≥ δ. De ini ion. Gi en a unc ion :X→R∪ {+∞} and a ke nel K, we de ine he ollowing sequences o unc ions: IK,n :=  nKand SK,n := −− nK(n∈N). Rema k. Fo any Hilbe no m k · k conside he ke nel KL(x, y) = kx−yk2. Then, wi h ou no a ion he sequence SKL,mIKL,n m>n deno es he Las y- Lions app oxima es o ela ed o he no m k · k. Rema k. No e ha he Las y-Lions app oxima es commu es wi h ansla ions in he same way as he classical in -con olu ion also does. This is a consequence o he ollowing p ope y o he ke nel: KL(x−a, y) = KL(x, y +a) ( o all x,y and a). Howe e , he p oblem o egula izing (no necessa ily con ex) unc ions in a non-Hilbe space leads na u ally o mo e gene al ke nels which do no yield ansla ion-in a ian app oxima es. The nex ac s a e easy o check. 5 Fac s 5. Le :X→R∪ {+∞} be a unc ion. 1Fo x∈X, IK,n = in y∈Xn (y) + nK(x, y)o, SK,n (x) = −IK,n(− )(x) = sup y∈Xn (y)−nK(x, y)o. 2Le Cbe a cons an . Then IK,n( +C) = IK,n +C, o any n. 3Suppose ha he ke nel Kis posi i e (i.e., K(x, y)≥0 o all x, y ∈X) hen (i) IK,n n∈Nis an inc easing sequence o unc ions bounded below by in . (ii) I ≤g, hen IK,n ≤IK,ng o any n. (iii) IK,mIK,n ≤IK,mIK,m , o any m > n. We now p oceed o s a e and p o e a echnical p oposi ion which is he main esul o his sec ion. P oposi ion 6. Le K:X×X→Ra ke nel sa is ying he ollowing condi ions: (1) Kis posi i e and K(x, x) = 0 o all x∈X, (2) Kis symme ic (i.e., K(x, y) = K(y, x) o all x, y ∈X), (3) K(x, y)−−−→ y→∞ +∞uni o mly on bounded se s, (4) Kis uni o mly con inuous ( esp. Lipschi z con inuous) on bounded se s and (5) Kis poin wise sepa a ing. Then o e e y p ope l.s.c. bounded below unc ion :X→R∪ {+∞} he ollowing s a emen s hold: (i) IK,n ≤SK,nIK,n ≤ . (ii) in IK,n = in and SIn (IK,n ) = SIn ( ). (iii) IK,n is uni o mly con inuous ( esp. Lipschi z con inuous) on bounded se s. (i ) IK,nIK,n τp −−−→ n→∞ and IK,nIK,n τK −−−→ n→∞ when is con inuous. I in addi ion Kis uni o mly sepa a ing on bounded se s hen ( ) IK,nIK,n τb −−−→ n→∞ whene e is uni o mly con inuous on bounded se s. Finally, when Kis uni o mly sepa a ing one has ( i) IK,nIK,n τu −−−→ n→∞ p o ided is uni o mly con inuous on X(no necessa - ily bounded below). Rema k. The sequence o unc ions IK,nIK,n plays an impo an auxilia y ˆole in his wo k; namely, i p o ides a lowe bound o he sequence (∆K,n )n∈Nin P oposi ion 8(i). P oo o he P oposi ion 6. (i) Since K(x, x) = 0 we ge ha IK,n ≤ ( ake y=xin he in imal de ini ion o IK,n a any poin x∈X). The e o e we deduce ha SK,n(IK,n) =−IK,n(−IK,n )≥IK,n . To see he o he inequali y, no ice ha om Fac 5-1 we ob ain o x∈X he exp ession SK,n(IK,n )(x) = sup y∈X in z∈Xn (z) + nK(y, z)−K(x, y)o.(2) 6 Fo some ixed x, i we ake z=xin (2) we conclude om he symme y o K ha SK,n(IK,n )(x)≤ (x). (ii) F om (i) and Fac 5-1(i) we ha e in IK,n = in and SIn ( )⊆SIn (IK,n ). Conside any minimum x0∈Xo IK,n . Then, he e exis s a sequence (yk)k∈N⊂X so ha in =IK,n (x0)≤ (yk) + nK(x0, yk)−−−→ k→∞ in . (3) Hence, since Kis posi i e i ollows om (3) ha lim k→∞ (yk) = in and lim k→∞ K(x0, yk) = 0.(4) Bu Kis poin wise sepa a ing, so he second pa o (4) implies ha yk−→ x0. Using he lowe -semicon inui y o and he i s pa o (4) we conclude ha in ≤ (x0)≤lim k→∞ (yk) = in . and his p o es asse ion (ii). Be o e p oceeding wi h he es o he p oo , we se up he ollowing use ul de ini ion: Ωn(x) := y∈X: (y) + nK(x, y)≤IK,n (x) + 1(x∈X, n ∈N) (5) Wi h hese no a ions, we ema k ha o n∈Nand x∈X IK,n (x) = in y∈Ωn(x) (y) + nK(x, y)≥in Ωn(x) (6) ( he las inequali y coming om he posi i i y o K). I is clea om (6) ha he beha iou o IK,n is di ec ly linked wi h he size o he se s Ωn(x)x∈X. We shall see ha he g ow h condi ion (3) ensu es ha he se s Ωn(x) a e no a bi a ily big when x uns on bounded se s o X. Mo e p ecisely, we claim he ollowing. Claim 6.1. Fo any > 0, he se Ω := Sn∈NSkxk≤ Ωn(x)is bounded. The p oo o his claim is based on he nex simple ac . Fac 6.2. Fo any > 0,sup nIK,n (x) n:x∈BX( ), n ∈No:= M <+∞. P oo o he Fac 6.2. Since is p ope , ake y0such ha (y0)≤in +1 <+∞. Then by de ini ion o IK,n i ollows ha o any x∈X IK,n (x) n≤ (y0) n+K(x, y0)≤in + 1 + sup K(x, y0) : x∈BX( ), and his exp ession is bounded abo e on bounded se s because Kis uni o mly con inuous (o Lipschi z con inuous) on bounded se s. The p oo o Fac 6.2 is inished. P oo o he Claim 6.1. Fo 0>0, le M 0>0 be he uppe bound de ined in Fac 6.2. Thus, o any x∈BX( 0) and n∈Ni y∈Ωn(x) i ollows om he de ini ion o Ωn(x), gi en in (5), ha K(x, y)≤1 nIK,n (x) + 1 − (y)≤M 0+ 1 −in . (7) 7 Bu he g ow h condi ion on Kgi en by (3) implies ha he se o ysa is ying (7) is uni o mly bounded o x∈BX( 0). The p oo o Claim 6.1 is done. We can now con inue wi h he p oo o P oposi ion 6. (iii) Suppose he ke nel Kis Lipschi z con inuous on bounded se s ( he p oo o he uni o mly con inuous case is p ac ically he same). Fo 0>0 ake x, x′∈BX( 0) and le LK, 0be he Lipschi z cons an o Kon BX( 0)×Ω 0(Ω 0being bounded by Claim 6.1). Using he equali y o (6) we can cons uc a sequence (yk)k∈N⊂Ω 0 in such a way ha o e e y k∈None has (yk) + nK(x′, yk)≤IK,n (x′) + 1 k. The e o e, we ob ain IK,n (x′)−IK,n (x)≤ (yk) + nK(x′, yk)− (yk)−nK(x, yk) + 1 k ≤nLK, 0kx′−xk+1 k−−−→ k→∞ nLK, 0kx′−xk. This concludes he p oo o (iii). We i s p o e (i ), ( ) and ( i) o (IK,n )nins ead o IK,n(IK,n )n. We will comple e he p oo a e wa ds. (i ’) Fix x0∈X. I limn→∞ IK,n (x0) = supnIK,n (x0) = +∞ hen by (i) one has (x0) = +∞and he esul holds. Thus, suppose ha Ix0:= limnIK,n (x0)<+∞. By he in imal de ini ion o IK,n a x0, we can choose a sequence (yn)n∈N⊂X such ha IK,n (x0)≤ (yn) + nK(x0, yn)≤IK,n (x0) + 1 n−−−→ n→∞ Ix0(8) Hence, om (8) i ollows o n∈N ha K(x0, yn)≤1 nIK,n (x0)− (yn)+1 n2≤1 nIx0−in +1 n2−−−→ n→∞ 0.(9) Bu Kis poin wise sepa a ing, so we ha e om (9) ha (yn)nis no m con e ging o x0. Using he lowe -semicon inui y o , he posi i i y o Kin (8) and (i), we ge ha (x0)≤lim in n→∞ (yn)≤Ix0≤ (x0). I is con inuous, since by Fac 5-3(i) and (iii) (IK,n )nis an inc easing se- quence o con inuous unc ions, Dini’s Theo em ell us ha he poin wise con e - gence o (IK,n )n o is ac ually uni o m on compac se s. ( ’) Le be an uni o mly con inuous unc ion on bounded se s and O 0be he oscilla ion o on he se BX( 0)∪Ω 0, o some ixed 0>0. Then, o any n∈N, x∈BX( 0) and y∈Ωn(x) a e he i s inequali y o (7) and (i) we ha e ha K(x, y)≤1 nIK,n (x)+1− (y)≤1 n (x)− (y)+1≤1 n(O 0+1) −−−→ n→∞ 0.(10) Suppose ha Kis uni o mly sepa a ing on bounded se s . Then, a di ec con- sequence o (10) is ha limndiam(Ωn(x)) = 0 uni o mly on BX( 0). The e o e, i ollows om (i),(6) and he uni o m con inui y o on BX( 0) ha (x)≥lim n→∞ IK,n (x)≥lim n→∞ in Ωn(x) −−−→ n→∞ (x) (11) 8 uni o mly on x∈BX( 0). ( i’) Suppose ha is uni o mly con inuous on X. Then sa is y he ollowing ac (whose simple p oo is le as an exe cise o he eade ): he e exis s α > 0 such ha (x)− (y)≤max{1, αkx−yk} o all x, y ∈X. (12) Then, in he same way as in (10) be o e, using his ime (12), we deduce ha o n∈N,x∈Xand any y∈Ωn(x) K(x, y)≤1 n( (x)− (y) + 1) ≤max n1 n,α nkx−yko+1 n.(13) Fo 1 > δ > 0, since Kis uni o mly sepa a ing he e is some βδ>0 so ha om (13) we deduce o x∈Xand y∈Ωn(x) ha kx−yk ≤ max n1 nβδ ,α nβδ kx−yko+1 nβδ whene e kx−yk> δ. (14) Hence, aking nbig so ha max 2 nβδ,2α nβδ≤δ < 1, (14) shows o e e y x∈X ha diamΩn(x)≤2δ. Tha is, we ha e shown ha diamΩn(x)→0 uni o mly on x∈X. The e o e, as is uni o mly con inuous on Xwe can epea he same easonings o (11) o conclude ha (IK,n )ncon e ges o uni o mly on X. (i ) and ( ) a e s aigh o wa d co olla ies o (i ’) and ( ’) i we ema k he ollowing. Suppose ha o ε > 0 he e exis s n0∈Nso ha −ε 2≤IK,n0 on some se S (Sbeing a single on, o a compac se o a bounded se o X). By Fac 5-3(i) and (iii), we can hen apply (i ’) (o ( ’)) o he bounded below, uni o mly con inuous unc ion IK,n0 o ob ain m > n0such ha IK,n0 −ε 2≤IK,mIK,n0 on he same S. Thus, by Fac 5-3(iii) and (i) i ollows ha −ε≤IK,n0 −ε 2≤IK,mIK,n0 ≤IK,mIK,m ≤ on S. ( i) is also easily deduced om ( i’) h ough he ollowing a gumen . I −ε≤IK,n ≤ , o some ε > 0 and n∈N, hen applying Fac s 5-2 and 5-3(ii) we ge ha −2ε≤IK,n −ε=IK,n( −ε)≤IK,nIK,n ≤IK,n ≤ .  Rema k. Wi h he abo e echniques i is no di icul o check ha IK,n(IK,n )n con e ges o o he epig aphical dis ance (see [AW] o he de ini ion). We e e o he p oo o Lemma 3( ) in [S 2] o de ails. 3. Con exi y echniques and smoo hness esul s In his sec ion we shall show a p ocedu e o ob ain smoo h unc ions om he ope a o s IK,n(·) and SK,n(·). We will need o impose some addi ional condi ions o con exi y and smoo hness on he ke nel K o achie e he smoo h egula iza ion. The in e es ing ea u e o hese con exi y a gumen s is he p ese a ion o he app oxima ing p ope ies ob ained in he p e ious sec ion. The main ool we shall use o ge smoo h egula iza ion is explained in he nex heo em. I deals wi h he smoo h p ope ies inhe i ed by he con ex en elop o a “somehow” smoo h unc ion. 9 Lemma 1.3. I k · k ∈ C1,α(X) hen k · k1+α∈ C1,α(X). P oo o he Lemma 1.3. This ac elies s ongly in he con exi y and homogenei y o a no m. Since i is clea ha k · k1+α∈ C1,α B(X), le C > 0 be he α-H¨olde con inui y cons an o he de i a i e o he no m k · k in BX. We shall show ha he condi ion (15) holds ue o k · k1+α. Take any x, y ∈Xand deno e by ω he maximum o kxkand kyk. The lemma is p o ed by he nex compu a ion. kx+yk1+α+kx−yk1+α−2kxk1+α= ω1+α  x ω+y ω  1+α−  x ω  1+α+  x ω−y ω  1+α−  x ω  1+α≤ ω1+α2αC  y ω  1+α= 2αCkyk1+α. By he abo e, his concludes he p oo o Theo em 1. Acknowledgmen s. The au ho wishes o hank Gilles Gode oy o his con- s an suppo and many ui ul con e sa ions. The au ho also wan s o exp ess his g a i ude o he Depa men o Ma hema ics o he Uni e si y o Missou i- Columbia, whe e his wo k was de eloped. Re e ences [AA] H. A ouch-D.Az´e, App oxima ion and egula iza ion o a bi a y unc ions in Hilbe spaces by he Las y-Lions me hod, Ann. Ins . H. Poinca ´e Anal. Non Lin´eai e 10 (1993), no. 3, 289–312. [AW] H. A ouch-R. We s, Quan i a i e s abili y o a ia ional sys ems: I. The epig aphical dis ance, T ans. Ame . Ma h. Soc. 328 (1991), no. 2, 695–729. [C] M. Cepedello Boiso, App oxima ion o Lipschi z unc ions by ∆-con ex unc ions in Ba- nach spaces, p ep in . [DFH] R. De ille-V. Fon -P. H´ajek, Analy ic and Ckapp oxima ions o no ms in sepa able Ba- nach spaces, S udia Ma h. 120 (1996), no. 1, 61–74. [DGZ] R. De ille-G. Gode oy-V. Zizle , Smoo hness and eno mings in Banach spaces, Pi man Mono. and Su . in Pu e and App. Ma h., ol. 64, Longman, Bos on, 1993. 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