C 1 -FINE APPROXIMATION OF FUNCTIONS ON BANACH SPACES WITH UNCONDITIONAL BASIS

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1 C 1 -FINE APPROXIMATION OF FUNCTIONS ON BANACH SPACES WITH UNCONDITIONAL BASIS D. AZAGRA, R. FRY, J. GÓMEZ GIL, J.A. JARAMILLO, M. LOVO Abstract. We show that if X is a Baach space havig a ucoditioal basis ad a C p -smooth Lipschitz bump fuctio, the for every C 1 -smooth fuctio f from X ito a Baach space Y, ad for every cotiuous fuctio ε : X (0, ), there exists a C p -smooth fuctio g : X Y such that f g ε ad f g ε. 1. Itroductio Give a Fréchet smooth fuctio f betwee Baach spaces, we cosider i this ote the problem of uiformly approximatig both f ad its derivative by fuctios with a higher order of differetiability. More geerally, if f : X Y is a C k -smooth fuctio betwee Baach spaces, ad ε : X (0, ) a cotiuous map, the we say that f is C k -fie approximated by a C p -smooth fuctio g : X Y, where p > k, if f (i) (x) g (i) (x) < ε (x) holds for i = 0, 1,..., k o X. The fiite dimesioal case was completely solved i the classical paper of H. Whitey [W]. The ifiite dimesioal settig has prove to be more difficult, ad heceforth i this paper all spaces are take to be ifiite dimesioal. The questio of C 0 -fie approximatio, i.e., uiform approximatio of cotiuous fuctios by smooth fuctios, has bee well ivestigated over the last several decades ad usually relies o the use of smooth partitios of uity. For a survey of some results i this directio see Chapter VIII [DGZ] ad [FM]. The problem of C k - fie approximatio whe k > 0 is much less uderstood ad ot geerally ameable to a solutio by partitios of uity. The most fudametal work i this directio has bee by N. Moulis [M]. Variatios o Moulis results ca be foud i [H2], although there is a gap i the proof of the geeralizatio of Theorem 2 [M] claimed i [H3] ad aouced i [H1]. I fact, to our kowledge the oly complete results o C k -fie approximatio i ifiite dimesioal Baach spaces X whe k > 0 is the work of Moulis, which cosiders the case where X = l p for p (1, ), or X = c 0. The mai result of our ote is to exted Theorem 1 [M] o C 1 -fie approximatio by C α -smooth fuctios i l p or c 0, to ay Baach space which admits a Date: September 10, Mathematics Subject Classificatio. 46B20. Key words ad phrases. Smooth approximatio, Baach spaces. The secod amed author is partly supported by NSERC (Caada). 1

2 2 AZAGRA, FRY, GÓMEZ-GIL, JARAMILLO, LOVO ucoditioal Schauder basis ad a Lipschitz, C α -smooth bump fuctio. This geeralizatio is sufficiet to allow for a characterizatio of Baach spaces i which C 1 -fie approximatio by smoother fuctios is possible withi the class of Baach spaces with ucoditioal bases which admit a C 1 -smooth bump fuctio. The otatio we employ is stadard, with X, Y, etc. deotig Baach spaces, ad X, Y, etc. their (cotiuous) duals. The collectio of all cotiuous, liear maps betwee Baach spaces X ad Y is deoted by L (X, Y ). Smoothess i this ote is meat i the Fréchet sese. A C p -smooth bump fuctio o X is a C p -smooth, realvalued fuctio o X with bouded, o-empty support. Most additioal otatio is explaied as it is itroduced i the sequel. For ay uexplaied terms we refer the reader to [DGZ] ad [FHHSPZ]. The mai result of this ote was idepedetly proved by two ad three of the authors respectively. I order to avoid uecessary duplicity we all agreed to make a joit paper. 2. Mai Results Theorem 1. Let X be a Baach space with ucoditioal basis, ad Y be a arbitrary Baach space. Assume that X has a C p -smooth, Lipschitz bump fuctio. Let G be a ope subset of X. The, for every C 1 -smooth fuctio f : G Y ad every cotiuous fuctio ε : G (0, ), there exists a C p -smooth fuctio g : G Y such that f(x) g(x) Y ε(x) ad f (x) g (x) L(X,Y ) ε(x) for x G. Here, as throughout the paper, p N { }, p 1. We will say that the map g is a C 1 -fie approximatio of f. As oted i the itroductio, this result provides a characterizatio, withi the class of Baach spaces possessig ucoditioal bases ad C 1 -smooth bump fuctios, of those spaces i which C 1 -fie approximatio by smoother fuctios occurs. Specifically we have the followig. Corollary 2. Let X be a Baach space with a ucoditioal basis ad a C 1 -smooth bump fuctio, G X a ope set, ad Y a Baach space. The followig statemets are equivalet: (1) X has a C p -smooth Lipschitz bump fuctio. (2) Every C 1 -smooth fuctio f : G Y ca be C 1 -fiely approximated by C p -smooth fuctios g : G Y. Proof. (1) = (2) is the precedig Theorem. Let us prove (2) = (1). It is well kow that every separable Baach space with a C 1 -smooth bump fuctio has a C 1 -smooth equivalet orm as well (see e.g., Theorem II.5.3 [DGZ]). Hece we may assume that the orm of the space X is C 1 -smooth away from the origi. Take a C smooth fuctio θ : R [0, 1] such that θ 1 (1) = (, 1/3], θ 1 (0) = [1, ), ad θ (R) [ 2, 0]. Cosider the fuctios f, ε : X R defied by f(x) = θ( x ), ε(x) = 1/3. By assumptio there is a C p smooth fuctio g : X R such that f g 1/3 ad f g 1/3. The we have that g(0) 2/3; g(x) 1/3 if x 1; ad g (x) f (x) + 1/ /3 = 7/3 for all x X. By composig

3 C 1 -FINE APPROXIMATION IN BANACH SPACES 3 g with a C smooth fuctio ϕ : R [0, 1] such that ϕ 1 (0) = (, 1/3], we get a C p smooth Lipschitz bump fuctio o X. Remark 3. We do ot kow whether a Baach space X with o C 1 -smooth bump fuctio (for istace, X = l 1 ) might have the property that every C 1 -smooth fuctio f : X R ca be C 1 -fiely approximated by C p -smooth fuctios, with p 2. Some results o approximatio i Baach spaces with o C 1 -smooth bump fuctios ca be foud i [F]. Proof of Theorem 1 We will eed to use the followig result, which is implicitly proved i Propositio II.5.1 [DGZ]; see also [L]. Propositio 4. Let Z be a Baach space. The followig assertios are equivalet. (1) Z admits a C p -smooth Lipschitz bump fuctio. (2) There exist umbers a, b > 0 ad a Lipschitz fuctio ψ : Z [0, ) which is C p -smooth o Z \ {0}, homogeeous (that is ψ(tx) = t ψ(x) for all t R, x Z), ad such that a ψ b. For such a fuctio ψ, the set A = {z Z : ψ(z) 1} is what we call a C p smooth Lipschitz starlike body, ad the Mikowski fuctioal of this body, µ A (z) = if{t > 0 : (1/t)z A}, is precisely the fuctio ψ (see [AD] ad the refereces therei for further iformatio o starlike bodies ad their Mikowski fuctioals). We will deote the ope (resp. closed) ball of ceter x ad radius r, with respect to the orm of X, by B(x, r) (resp. B(x, r)). If A is a bouded starlike body of X, we defie the ope A-pseudoball of ceter x ad radius r as B A (x, r) := B(x, r; µ A ) := {y X : µ A (y x) < r}, ad we defie B A (x, r) to be the closure of B A (x, r). Accordig to Propositio 4 ad the precedig remarks, because X has a C p - smooth Lipschitz bump fuctio, there is a bouded starlike body A X whose Mikowski fuctioal µ A = ψ is Lipschitz ad C p -smooth o X \ {0}, ad there is a umber M 1 such that 1 M x µ A(x) M x for all x X, ad µ A (x) M for all x X \ {0}. Notice that i this case we have that B(x, r M ) B A(x, r) B(x, Mr) for every x X, r > 0. We ext itroduce some other otatio used throughout the proof. Let {e j, e j } be a ucoditioal Schauder ( basis o X, ad P : X X the caoical projectios give by P (x) = P j=1 x je ) j = j=1 x je j. Let the ucoditioal basis costat be C 1 1. Followig Moulis [M], we put E = P (X), ad E = E, otig that dim E = ad E = X.

4 4 AZAGRA, FRY, GÓMEZ-GIL, JARAMILLO, LOVO I the sequel the symbol stads for ay of the differet orms of the spaces X, X, ad L(X, Y ). The followig lemma gives us the key to provig Theorem 1. Lemma 5. Let X, Y, G be as i the statemet of Theorem 1. There exists a costat C > 0, depedig oly o the space X ad the basis costat, such that: for every ope ball B 0 = B(z 0, r 0 ) with B(z 0, 2r 0 ) G, ad for every C 1 fuctio f : G Y ad umbers ε, η > 0 with sup x B(z0,2r 0 ) f (x) < η, there exists a C p -smooth map g : G Y such that sup f(x) g(x) < Cε, ad sup g (x) < Cη. x B 0 x B 0 Proof. We may assume that z 0 = 0 ad 2r 0 < 1. Choose r > 0 with r < mi{ε/c 1 Mη, r 0 /C 1 M}. Let ϕ : R [0, 1] be a C -smooth fuctio such that ϕ (t) = 1 if t < 1/2, ϕ (t) = 0 if t > 1, ϕ (R) [ 3, 0]. We ow costruct C 1 -fie smooth approximatios to f o the fiite dimesioal subspaces E. This classical itegral-covolutio method already appears i Whitey [W], but we follow Moulis [M] for cosistecy. Cosider the map ˆf : G Y, defied by ˆf (x) = (a ) f(x y)ϕ(a µ A (y))dy, c E where c = ϕ (µ A (y)) dy, E ad we have chose the costats a > 0 large eough so that sup ˆf (x) f (x) < ε 2, x B 0 E sup x B 0 E ˆf (x) f (x) < η 2. Oe ca check that ˆf is C 1 -smooth o G, ad restrictig ˆf to E gives rise to a C -smooth map. We ext defie a sequece of fuctios f : G Y as follows. Put f 0 = f (e 0 ), ad supposig that f 0,..., f 1 have bee defied, we set f (x) = ˆf (x) + f 1 (P 1 (x)) ˆf (P 1 (x)). Oe ca verify by iductio that: (i) The restrictio of f to E is C -smooth ad f is a extesio of f 1. (ii) sup x E B 0 f (x) f (x) < 2ε ( ).

5 C 1 -FINE APPROXIMATION IN BANACH SPACES 5 (iii) sup x E B 0 f (x) f (x) < 2η ( ) We ow defie the map f : E Y by f (x) = lim f (x). Fact 6. The fuctio f has the followig properties: (i) The restrictio of f to every subspace E is C -smooth (ii) sup x E B 0 f (x) f (x) < 2ε (iii) sup x E B 0 f (x) f (x) < 2η This is easily checked by usig properties (i) (iii) above. Next, let us write x = x e X, ad defie the map [ ] µa (x P 1 (x)) χ (x) = 1 ϕ, r (here we use the covetio that P 0 = 0), ad ow set Ψ (x) = χ (x) x e. Fact 7. The mappig Ψ : X E is well defied, C p -smooth o X, ad has the followig properties: (1) Ψ (x) 4M 2 C 1 (1 + C 1 ) for all x X; (2) x Ψ(x) C 1 Mr for all x X; (3) Ψ(B 0 ) 2B 0. Proof. For ay x 0, because P (x 0 ) x 0, there exist a eighbourhood N 0 of x 0 ad a 0 so that χ (x) = 0 for all x N 0 ad 0, ad so Ψ (N 0 ) E 0. Thus, Ψ : X E is a well-defied C p -smooth map. We ext estimate its derivative. We have that (χ (x) x ) = χ (x) x + χ (x) e. Now, sice ϕ (t) 3, µ A (x) M ad (I P 1) (x) 1 + C 1 for all x, t, we get that, for ay, χ (x) ( ) µa (x P 1 (x)) ϕ r 1 µ A (x P 1 (x)) (I P 1 ) (x) r 3M(1 + C 1 )r 1.

6 6 AZAGRA, FRY, GÓMEZ-GIL, JARAMILLO, LOVO Cosider ow the derivative of the map Ψ. We have Ψ (x) ( ) = χ (x) ( )x e + χ (x) e ( ). For a fixed x, defie 0 = 0 (x) to be the smallest iteger with µ A (x P 0 1 (x)) r. The for all m < 0, χ m (x) = 1, χ m (x) = 0, ad so, usig Lemma 6.33 [FHHSPZ] sice {e } is ucoditioal with basis costat C 1, ad our estimate above, we have that for every h B X, Ψ (x) (h) = χ (x) (h) x e + χ (x) h e χ (x) (h) x e 0 + χ (x) h e C 1 sup χ (x) (h) x e 0 + C 1 sup χ (x) h e 3C 1 M(1 + C 1 )r 1 h x e 0 + C 1 h = 3C 1 M(1 + C 1 )r 1 h x P 0 1(x) + C 1 h 3C 1 M(1 + C 1 )r 1 h Mµ A (x P 0 1(x)) + C 1 h 3C 1 M(1 + C 1 )r 1 h Mr + C 1 h 4M 2 C 1 (1 + C 1 ) h, which yields (1). We ext estimate x Ψ (x). We have, agai usig Lemma 6.33 [FHHSPZ] sice {e } is ucoditioal with basis costat C 1, ad with 0 = 0 (x) as above, x Ψ (x) = x (1 χ (x)) e 0 C 1 sup 1 χ (x) x e 0 C 1 x e 0 C 1Mµ A (x P 0 1(x)) C 1 Mr, which proves (2). Lastly, property (3) is immediate from (2) ad the choice of r.

7 C 1 -FINE APPROXIMATION IN BANACH SPACES 7 To ed the proof of the lemma, we defie g (x) = f (Ψ (x)). Note that g is C p -smooth o G, beig the compositio of C p -smooth maps. Also we have, accordig to Facts 6, 7, ad the choice of r, f (x) g (x) f (x) f (Ψ (x)) + f (Ψ (x)) f (Ψ (x)) η x Ψ (x) + f (Ψ (x)) f (Ψ (x)) ηc 1 Mr + 2ε < 3ε. Lastly, we have, agai usig Facts 6 ad 7, that g (x) f (Ψ (x)) Ψ (x) ( f (Ψ (x)) + 2η ) 4M 2 C 1 (1 + C 1 ) 12M 2 C 1 (1 + C 1 )η. This establishes the Lemma with C = 12M 2 C 1 (1 + C 1 ). Now we fiish the proof of Theorem 1. Usig separability ad opeess of G, as well as cotiuity of the fuctios f ad ε, we let {B(x j, r j /M)} j=1 be a coverig of G by ope balls B(x j, r j /M) G with ceters x j ad radii r j /M so that T j (x) f (x) < ε j /8C ad ε(x) ε j /2 for all x B(x j, 2Mr j ), where T j is the first order Taylor Polyomial to f at x j, ad ε j = ε(x j ). Sice B(x, r M ) B A(x, r) B(x, Mr) for every x, r, we have that G = B A (x j, r j ), ad T j (x) f (x) ε j < 8C o B A(x j, 2r j ). j=1 Next, let ϕ j C p (X, [0, 1]) with bouded derivative so that ϕ j = 1 o B A (x j, r j ) ad ϕ j = 0 outside B A (x j, 2r j ) (such a fuctio ca be easily defied as ϕ j (x) = θ j (µ A (x x j )), where θ j is a suitable smooth real fuctio). Now, via Lemma 5, we may choose C p -smooth maps δ j : G Y so that o each ball B(x j, 2Mr j ) we have both T j (x) f (x) δ j (x) < 2 j 2 ε j Mj 1, ad δ j (x) < εj /8, where M j = j M k=1 k ad M k = sup x B(xk,2Mr k ) ϕ k (x). The we also have T j (x) f (x) δ j (x) T j (x) f (x) + δ j (x) < ε j /8C + ε j /8 ε j /4. Next, we defie h j = ϕ j (1 ϕ k ), k<j

8 8 AZAGRA, FRY, GÓMEZ-GIL, JARAMILLO, LOVO ad g (x) = j h j (x) (T j (x) δ j (x)) Note that for each x, if := (x) := mi {m : x B A (x m, r m )} the, because 1 ϕ (x) = 0 ad B A (x, r ) is ope, it follows from the defiitio of the h j that there is a eighbourhood N of x so that for y N, g (y) = j h j (y) (T j (y) δ j (y)), ad j h j(y) = j h j(y). Also, by a straightforward calculatio, agai usig the fact that ϕ = 1 o B A (x, r ), we have that j h j (y) = 1 for y B A (x, r ) (hece for every y G). Now, fix ay x 0 G, ad let 0 = (x 0 ) ad a eighbourhood N 0 of x 0 be as above. The for ay x N 0, sice support (h j ) B A (x j, 2r j ) B(x j, 2Mr j ), g (x) f (x) = h j (x) (T j (x) δ j (x)) f (x) j 0 = h j (x) (T j (x) δ j (x)) h j (x) f (x) j 0 j 0 h j (x) (T j (x) f (x) δ j (x)) j 0 < h j (x) ε j 4 ε(x). j 0 A straightforward calculatio shows that g (x) f (x) h j (x) Mj, ad so we have = < h j (x) (T j (x) f (x) δ j (x)) + h j (x) (T j (x) f (x) δ j (x)) j 0 h j (x) T j (x) f (x) δ j (x) + h j (x) (T j (x) f (x) δ j (x)) j 0 j 0 ( ) M j 2 j 2 ε j Mj 1 + h j (x) ε j 4 j 0 j 0, x B A (x j,2r j ) j 0, x B(x j,2mr j ) 2 j ε j 4 + ε(x) ε(x). 2

9 C 1 -FINE APPROXIMATION IN BANACH SPACES 9 Remark 8. By usig Moulis ideas [M] ad some refiemets of the techiques deployed above, oe ca also show the followig result: if X is a Baach space with a ucoditioal basis ad a C smooth bump fuctio with bouded derivatives, the every C 2k 1 smooth fuctio ca be C k -fiely approximated by C smooth fuctios. We do ot feel that this statemet justifies the iclusio of its (ecessarily techically ivolved) proof i this ote. Of course, the atural problem as to whether C k fuctios ca be C k -fiely approximated by C fuctios o such spaces X remais ope, eve i the case where X = l 2. Ackowledgmets The secod amed author wishes to thak P. Reyolds for his valuable research assistace. [AD] [DGZ] [F] [FM] Refereces D. Azagra ad T. Dobrowolski, O the topological classificatio of starlike bodies i Baach spaces, Topology ad Its Applicatios 132 (2003), R. Deville, G. Godefroy, ad V. Zizler, Smoothess ad reormigs i Baach spaces, vol. 64, Pitma Moographs ad Surveys i Pure ad Applied Mathematics, R. Fry, Approximatio i o-asplud spaces, Rocky Mt. J. Math., to appear. R. Fry ad S. McMaus, Smooth Bump Fuctios ad the Geometry of Baach Spaces A Brief Survey, Expos. Math., 20 (2002), [FHHSPZ] M. Fabia, P. Habala, P. Hájek, V.M. Satalucía, J. Pelat, ad V. Zizler, Fuctioal aalysis ad ifiite-dimesioal geometry, CMS books i mathematics 8, Spriger- Verlag, [H1] M.P. Heble, Approximatio of differetiable fuctios o a Hilbert space, C.R. Math. Rep. Acad. Sci. (Ca.) V (1983), [H2] M.P. Heble, Approximatio of differetiable fuctios o a Hilbert space II, Cotemp. Math. 54 (1986), [H3] M.P. Heble, Approximatio of differetiable fuctios o a Hilbert space III, Math. A. 282 (1988), [L] M. Leduc, Desité de certais familles d hyperplas tagets, C.R. Acad. Sci. (Paris) Ser: A 270 (1970), [M] N. Moulis, Approximatio de foctios différetiables sur certais espaces de Baach, A. Ist. Fourier, Greoble 21, 4 (1971), [W] H. Whitey, Aalytic extesios of differetial fuctios i closed sets, Tras. Amer. Math. Soc. 36 (1934), Daiel Azagra, Javier Gómez Gil, Jesús A. Jaramillo, Mauricio Lovo. Departameto de Aálisis Matemático. Facultad de Ciecias Matemáticas. Uiversidad Complutese Madrid, SPAIN. Robb Fry. Departmet of Mathematics ad Computer Sciece. St Fracis Xavier Uiversity. P.O. Box Atigoish,NS B2G 2W5, CANADA. addresses: daiel azagra@mat.ucm.es, rfry@stfx.ca, Javier Gomez@mat.ucm.es, jaramil@mat.ucm.es.