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.
[E] P. En lo, Banach spaces which can be gi en an equi alen uni o mly con ex no m, P ocee-
dings o he In e na ional Symposium on Pa ial Di e en ial Equa ions and he Geome y
o No med Linea Spaces II(Je usalem, 1972), Is ael J. Ma h. 13 (1973), 281–288.
[Fa] M. Fabi´an, Lipschi z smoo h poin s o con ex unc ions and isomo phic cha ac e iza ions
o Hilbe spaces, P oc. London Ma h. Soc. 51 (1985), no. 1, 113-126.
[F ] J. F on isi, Smoo h pa i ions o uni y in Banach spaces, Rocky Moun ain J. Ma h. 25
(1995), no. 4, 1295–1304.
[GR] A. G iewank-P.J. Rabie , On he smoo hness o con ex en elopes, T ans. Ame . Ma h.
Soc. 322 (1990), no. 2, 691–709.
[H] J. Ho man-Jø gensen, On he Modulus o Smoo hness and he G∗-Condi ions in B-spaces,
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[L] G. Lancien, On uni o mly con ex and uni o mly Kadec-Klee eno mings, Se dica Ma h.
J. 21 (1995), no. 1, 1–18.
[LL] J.M. Las y-P.L. Lions, A ema k on egula iza ion in Hilbe spaces, Is ael J. Ma h. 55
(1986), 257–266.
[NS] A.S. Nemi o ski˘ı-S.M. Semeno , The polynomial app oxima ion o unc ions on Hilbe
space, Ma . Sb. (N.S.) 92 (134) (1973), 257–281, 344.
[Ph] R.R. Phelps, Con ex unc ions, Mono one ope a o s and di e en iabili y, Lec u e No es
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16

[S 1] T. S ¨ombe g, The ope a ion o in imal con olu ion, Disse a iones Ma h. (Rozp awy
Ma .) 352 (1996), 58 pp.
[S 2] T. S ¨ombe g, On egula iza ion in Banach spaces, A k. Ma . 34 (1996), 383–406.
Equipe d’Analyse, Uni e si ´
e Pie e e Ma ie Cu ie–Pa is 6, Pa is.
Depa amen o de An´
alisis Ma em´
a ico, Uni e sidad de Se illa, Se illa.
Cu en add ess: Depa men o Ma hema ics, Uni e si y o Missou i-Columbia, Columbia.
E-mail add ess:[email protected] h.missou i.edu, cepedel@cc .jussieu. , [email protected]
17