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A NOTE ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS OF UNIFORMLY CONTINUOUS FUNCTIONS ON M. JIM ENEZ-SEVILLA Abstract. A real Banach space satisﬁes property (K) (deﬁned in [7]) if there exists a real-valued function on which is uniformly (real) analytic and separat- ing. We obtain that every uniformly continuous function , where is an open subset of a separable Banach space with property (K) and containing (thus for some Banach space ) can be uniformly approximated by (real) analytic functions such that ∂g ∂c p> (where ∂f ∂c is the set of partial derivatives ∂f ∂x x,y ) : ( x,y ). Similar statements are obtained for uniformly continuous functions with values in a (ﬁnite or inﬁnite dimensional) Banach space . Some consequences of these results are studied. 1. Introduction Several properties related to the size and shape of the set of derivatives of smooth functions between Banach spaces have been studied in [6], [2], [12], [8], [9] and [3] among others. Azagra and Deville [2] constructed smooth bump functions , in any Banach space with a Lipschitzian and smooth bump function, such that the set of derivatives ﬁlls the dual space, that is ) = (a bump function is a function with non-empty and bounded support). Some generalizations of this result are given in [5]. If has the Radon-Nikod´ym property it follows from Stegall’s variational principle that the cone generated by the set of derivatives of any continuous and Gˆateaux diﬀerentiable bump function is residual. In turn, P. H´ajek [13] proved that every Fr´echet diﬀerentiable function with locally uniformly continuous derivative from into a Banach space with non trivial type has locally compact derivative. Azagra and Cepedello [1] proved that every continuous function from into can be uniformly approximated by smooth functions with no critical points. P. H´ajek and M. Johanis [14] stated a related result for separable Banach spaces with a smooth bump function and containing : every real- valued continuous function deﬁned on can be uniformly approximated by smooth functions whose range of derivatives is of ﬁrst category and avoids a pre- ﬁxed subset. Recently, it was proved in [4] that every real valued continuous function deﬁned on a separable Banach space with separable dual (with a LUR and smooth equivalent norm, respectively) can be approximated by smooth ( smooth, respectively) functions with no critical points. Date : July, 2008. 2000 Mathematics Subject Classiﬁcation. Primary 46B20, 46T30. Secondary 58E05, 58C25. Key words and phrases. Approximation by analytic functions, range of the derivatives. The author has been supported by a fellowship of the Ministerio de Educacion y Ciencia, Spain.

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2 M. JIM ENEZ-SEVILLA Let us focus now on the (real) analytic case. J. Kurzweil [15] proved that every continuous function from into a Banach space can be uniformly approximated by analytic functions. R. Fry [11] proved the same assertion for uniformly continuous functions deﬁned on . Independently, M. Cepedello and P. H´ajek [7] proved the same result for and for the class of separable Banach spaces with property (K). A real separable Banach space satisﬁes property (K) if there is a function which is uniformly analytic and separating: that is, is real analytic at every point with radius of convergence uniformly bounded below by some constant M > and there is some such that the set < } and is not empty. In this note, we will prove that every uniformly continuous function , where is an open subset of a separable Banach space with property (K) and containing (thus for some Banach space ), can be uniformly approximated by (real) analytic functions such that ∂g ∂c p> (where ∂f ∂c ) is the set of partial derivatives ∂f ∂x x,y ) : ( x,y )). Our proof in based on the construction of real analytic approximations for a uniformly continuous function given by M. Cepedello and P. H´ajek [7]. We shall prove that, under additional conditions, we can obtain that p> . This result allows us to obtain real analytic approximations such that ∂h ∂c for any pre-ﬁxed 0 < p < q < 1 (or even ∂h ∂c span p> }\ p> , for any pre-ﬁxed \ p> ). A similar estimation of the set of derivatives is given for vectorial functions , where is a (ﬁnite or inﬁnite) dimensional Banach space. Let us point out that from the results of P. H´ajek [13] we can obtain an approx- imate Morse-Sard theorem, namely the existence of real analytic approximations with no critical points for uniformly continuous functions with open subset of (separable real Banach space with property (K) containing ) and ﬁnite dimensional. The results given in this note provide an alternative proof of this assertion, but with the advantage that it gives an additional description of the range of the derivatives. Finally, let us remark that, whenever has the property that every uniformly continuous function can be approximated by real analytic functions with no critical points, a real analytic separation result can be applied to any pair of closed subsets at positive distance in (via 1-codimensional real analytic submanifolds of ). This can be viewed as a real analytic approximation result for closed subsets in 2. Main results Let us ﬁrst state the following result for . Then we will state an analoguous result for (real) separable Banach spaces containing with property ( ). Theorem 2.1. Every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated by analytic functions whose set of deriva- tives is included in Proof. From now on, let us follow the notation given in [7]. Let us consider a uniformly continuous function . Let us ﬁx := 1 and take ρ > satisfying | whenever || || . We shall denote the complex Banach space by . Recall that ) := for and the complex

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN function ) = deﬁned in are real-analytic and analytic respectively, with radius of convergence one at every point [16, Example 5.5]. We will consider the real Banach space , the complex Banach space (which we shall denote by ) and the real quasi-Banach space with the quasi-norm ||·|| deﬁned as || || for all . Notice, that for every , we have that || || and thus ( . Let us consider, following the proof of [7], (1) the real-analytic function ) := ρx ) and the analytic function ) := ρz ) deﬁned in and respectively, with radius of convergence at every point . Let us denote by x,r ) = (2) a dense sequence in (3) a strictly decreasing sequence of real numbers converging to 0 with < (4) the two locally ﬁnite coverings of formed by the open sets and deﬁned as 1) 1 + 2 i 1) ) ) for n> i 1 + 2 ) ) for n> 1; (5) the -dimensional intervals deﬁned as =[ 1 + =[1 1) ··· × [1 1 + for n> 1; (6) the holomorphic functions deﬁned on as ) := + 1) exp =1 dτ, where = ( ··· , and are sequence of positive numbers and is the norming factor exp =1 d n/ n/ ··· In order to simplify the notation, let us write the following := and := M. Cepedello and P. H´ajek proved in [7] that whenever the sequence of positive numbers is taken small enough so that n>m m,n for every where m,n = sup {| ) + e is the closed unit ball of ) and the sequence is taken large enough so that (1) , whenever (2) + 1) , whenever (3) + 1) exp(

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4 M. JIM ENEZ-SEVILLA then the function + 1) ) locally uniformly converges in and the functions ) := ), ) := ) and ) := are well deﬁned and holomorphic on an open subset of containing . Therefore, the restrictions of the above functions to are real-analytic on . Moreover, they prove that 1 for every , that is 1-approximates in An additional condition we set for is that it is a decreasing sequence (oth- erwise, we change min ··· ,a by ). Once we ﬁx the sequence , we shall prove in the next that, under additional conditions on , the derivatives ) = ) = and , for every First, let us compute the derivative of ), for . Then, ) = + 1) =1 ) exp =1 d The derivative ) is a ﬁnite linear combination of ) : = 1 ,...,n . Since ) : = 1 ,...,n } , we obtain that . Moreover, , p> for every If a sequence of holomorphic functions locally uniformly converges on an open set , then the sequence of the derivatives also locally uniformly converges on Ω to the derivative of the limit. Thus, if ) and ) locally uniformly converge on an open set , then ) and ) locally uniformly converge in the space ( ) (holomorphic functions). Nevetheless, we cannot deduce from this fact that and , whenever We will ﬁnd an upper bound for || || . Let us deﬁne, for every , the sequence of positive numbers as (1 + =1 || (2 || Recall that the quasi-Banach space satisﬁes the following condition =1 =1 || || for every and ,...,v Therefore, || || + 1) =1 || || exp =1 d

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN Let us ﬁx and deﬁne ) for = 1 ··· ,n as ) = exp =1 dτ. Let us ﬁnd an upper bound for ) whenever . The sequence will be chosen to be increasing and satisfying 1, for every . If then either 1 + 2 or there is i such that . Let us study both cases. (1) Consider for j . Then ) is the product of the integrals ) = d 1+ d k d (a) Assume that 1 + 2 . Then, for every = ( ··· , we have that . Then, dt k (b) Now assume that there is i such that . Then for every = ( ··· , we have that . In the case dt i,j In the case te dt (2) Consider . Then, ) = 1+ d k d (a) Assume that 1 + 2 . Then, te dt k (b) Finally assume that there is i such that . Then d k

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6 M. JIM ENEZ-SEVILLA Therefore, for , we have || || =1 + 1) || || =1 + 1) || || It is now clear that we can proceed to select large enough such that + 1) + 1 which implies + 1) || || =1 || || whenever Fact 2.2. For every v,w (2.1) || || ≤|| (2 || || (2 || Proof. If , then ) = ρz ) = ρz , where is the canonical basis of . Then, || || || ρz || ρz It is enough to check that for every v,w || || ≤|| (2 || || (2 || For = 1, we have that ≤| ≤| . Now, the function ) = n> is convex. Then, (2 ) + (2 (2 ) + (2 ) and inequality (2.1) follows easily. From inequality (2.1), we obtain (2.2) || || ≤|| (2 || || (2 || for every Let us now consider . Since is locally ﬁnite, there is such that 6 , for n>i . Thus, n>i + 1) || || n>i =1 || || n>i =1 || (2 || || (2 || Let us denote =1 || (2 || . Then, n>i + 1) || || n>i nr || (2 || n>i (2.3)

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN Since nr and , we deduce that || || || || and || || || || for every . Therefore for every Now it is clear that, for every ε> 0, we can -approximate with a real analytic function whose set of derivatives is in Finally, let us mention that if is deﬁned in an open subset of , the proof is analogous with and deﬁned in an open subset of containing Let us consider, for every 0 1, the quasi-Banach spaces with the quasi- norm || || = ( /p , for . Recall that || =1 || =1 || || for every ··· ,v . We shall deduce the following corollaries. Corollary 2.3. Let us consider real numbers < p < q < . Then, every real- valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated by analytic functions such that the set of derivatives is in Proof. The proof of Theorem 2.1 works not only for but also for every 0 p < 1. With the notation of Theorem 2.1, if we select large enough so that + 1) || || , and follow the rest of the proof (Fact 2.2 holds for every p > 0), we obtain a 1-approximating analytic function such that . Next, for 0 < p < q < 1, we choose and deﬁne a real-analytic function 1) such that 0 for every , for instance ) = 1+ 1. Then, ( and ( . Thus is a real analytic function which 2-approximates Corollary 2.4. (1) Every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated by analytic functions such that the set of derivatives is in p> (2) Consequently, every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated by analytic functions such that the set of derivatives is in span p> }\ p> for any pre-ﬁxed \ p> Proof. (1) The proof of Theorem 2.1 works with some modiﬁcations: We ﬁnd a upper bound for || || /n /n whenever , such that (2.4) /n + 1) || || /n /n for every . In order to do that (and following the same notation) we consider (1 + )(1 +

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8 M. JIM ENEZ-SEVILLA where =1 || (2 || /n /n and . The calculations are similar except in the step corresponding to equation (2.3) where we have n>i /n + 1) || || /n /n n>i nr || (2 || /n /n n>i Since n>i we only need to prove that n>i nr || (2 || /n /n We need an additional bound for || (2 || /n /n /n . Let us denote by | and . Then /n /n || (4 || In addition, we have /n /n Therefore, || (2 || /n /n ≤|| (4 || Now, n>i nr || (2 || /n /n n>i nr || (4 || n>i nr || (4 || n>i nr and (2.4) is proved. From inequality (2.4) we obtain, for every 0 1 and a natural number such that and /n + 1) || || /n /n 1. Since /n whenever 0 1 and , it follows that + 1) || || /n /n + 1) || || /n /n Moreover, ||·|| ≤||·|| /n whenever . Therefore, + 1) || || + 1) || || /n and ) = p> Now in order to prove (2), it is clear that ( = 0 for every because 6 p> (where is the real analytic function considered in the proof of Corollary 2.3). Moreover, ( span p> }\ p>

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN In the next corollary we extend Theorem 2.1, Corollaries 2.3 and 2.4 to every separable real Banach space with property (K) and containing . Recall that, by the Sobczyk theorem , for some separable real Banach space . Let us consider a diﬀerentiable function deﬁned on an open subset of . For every = ( x,y , let us denote by ∂f ∂c ) the set of partial derivatives ∂f ∂x x,y ) : x,y Corollary 2.5. Let be a separable real Banach space with property (K) such that contains a copy of . Then, (1) every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated on by an analytic function such that the set ∂h ∂c p> (2) Consequently, every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated on by an analytic function such that the set ∂g ∂c span p> }\ p> , for any pre-ﬁxed \ p> Proof. We follow the proofs of [7, Theorem 1], Theorem 2.1 and Corollary 2.4 for the space and the uniformly analytic and separating function deﬁned below: (a) We consider in , the function x,y ) = ) + ), for and , where is uniformly analytic and separating in and ) = for . Clearly is uniformly analytic and separating in (b) We consider the complexiﬁed space of = , where is the complexiﬁed space of . The function x,y ) := ) + ), where is the analytic extension of to an open subset Ω of of the form sU is the open unit ball of ) for some s> 0. (c) Let us deﬁne, for every 0 <ρ< 1, the functions x,y ) = ρx,ρy ), for and (d) ∂D ∂x x,y ) = ) and ∂D ∂x x,y ) = ( ), for and Then, following the notation of Theorem 2 1, we have: (i) ∂x x,y p> , for every ( x,y and , (ii) ∂x x,y ) = ∂x x,y p> and ∂x x,y ) = ∂x x,y p> for every ( x,y , where := ( ,x and is the pre-ﬁxed dense sequence in , and thus, (iv) the analytic approximations and satisfy that ∂h ∂x x,y p> and ∂g ∂x x,y ) = ∂x x,y ) = ∂h ∂x x,y ) + )) span p> }\ p> for every ( x,y , if p> Let us now generalize Theorem 2.1, Corollaries 2.3, 2.4 and 2.5 for the case of uniformly continuous functions with values into a (ﬁnite or inﬁnite dimensional) real Banach space Corollary 2.6. Let be a real Banach, a separable real Banach space with property (K) so that contains a copy of and an open subset of . Then, (1) every uniformly continuous function can be uniformly approxi- mated by analytic functions such that ∂h ∂c p> for every

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10 M. JIM ENEZ-SEVILLA (2) Let us ﬁx any \ p> and \{ . Then, every uniformly continuous function can be uniformly approximated by an analytic function such that ∂g ∂c span p> }\ p> for every (3) In addition to (2), if is ﬁnite dimensional, we obtain that the analytic approximation has no critical points, i.e. is surjective for every Proof. (1) The proofs of Theorem 2.1 and Corollary 2.5 can by applied to obtain the above assertion. We only have to consider || || instead of and select so that || || /n + 1) || ∂x x,y || /n /n for every ( x,y (we follow the notation of the proof of Corollary 2.5). Then, the analytic function x,y ) = x,y x,y (which 1-approximates ) satisﬁes the following property: ∂h ∂x x,y p> for every and ( x,y (2) Let us consider || || ||·|| is the norm on ) and deﬁne x,y ) = ) for all ( x,y , where 1) is the analytic function already deﬁned in the proof of Corollary 2.3. On the one hand, x,y x,y | for every ( x,y . On the other hand, ∂g ∂x x,y ) = ∂h ∂x x,y )+ vw )) for every x,y . This implies ∂g ∂x x,y ) = ∂h ∂x x,y ) + ( )) for every x,y and . Moreover, ( )) = 0 whenever and x,y . Thus, ∂g ∂x x,y span p> }\ p> whenever and x,y (3) Let us consider a continuous function with components = 1 ,...,n . Let us select linear independent elements ,...,z p> such that (span ,...,z p> ) = . By Corollary 2.5, we can uniformly approximate by analytic functions satisfying the following condition: for every ( x,y there are \{ and p> such that ∂g ∂x x,y ) = . Therefore ∂g ∂x x,y ,..., ∂g ∂x x,y are linearly independent (thus x,y ,...,g x,y are l.i. as well) for every ( x,y and the analytic function := ( ,...,g ) uniformly approximate The existence of real analytic approximations with no critical points for uniformly continuous functions with the conditions of Corollary 2.6 and ﬁnite dimensional) can be obtained from the results of P. H´ajek [13]. Statement (3) in Corollary 2.6 provides an alternative proof to this assertion with an additional description of the range of the derivatives. Let us sketch how to deduce the assertion from [13] and for (the general case is deduced in an analogous way). If is a real analytic function approximating and we consider ,...,h the components of , then by the results given in [13] there is a set (namely a countable union of compact sets) of , such that ∪··· ⊂K We may consider that is a linear subspace of ; otherwise we consider the span {K} which is also a subset. Let us follow the notation [ ,...,a ] = span ,...,a Therefore, there is \{ such that [ ∩K . Since [ ] + is again a subspace of , there is \{ such that [ ([ ] + ) = and then , z ∩K . Proceeding in this way, we obtain linear independent elements = 1 ,...,n in , such that [ ,...,z ∩K . Now we proceed as in the proof of Corollary 2.6(3) and deﬁne := = 1 ,...,n and ε> 0 ( is

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN 11 the auxiliary real analytic function considered in the proof of Corollary 2.3). Then, ,...,g are linearly independent for every and := ( ,...,g is a real analytic function with no critical points and uniformly approximates In addition, it can be obtained that ∩M for any pre-ﬁxed subset M ,E ) = ×··· ···× (linear continuous function from into ). Ir order to do that, we denote by the linear span of the projection of over the ﬁrst . Then, is a subset of . Following the above arguments, we select elements = 1 ,...,n such that [ ,...,z ) = . In this case, ∈M and then ∈M for every Let us ﬁnish this note with a remark on the separation of closed sets and a question on the separable Hilbert space. Recall that, whenever a real Banach space has the property that every uniformly continuous, real-valued function on can be uniformly approximated by real-analytic functions with no critical points, one can deduce the following non-linear separation and approximation result. An open subset of is said to be real-analytic smooth provided its boundary ∂U is a real-analytic smooth one-codimensional submanifold of Remark 2.7. Let be a real and separable Banach space containing with property (K). (1) Let and be two disjoint and closed subsets of such that dist( A,B Then, there is a real-analytic function with no critical points such that the level set (0) is a 1-codimensional analytic submanifold of that separates and in the following sense: Deﬁne and , then and are disjoint real- analytic open subsets of with common boundary ∂U ∂U such that and (2) Every closed subset of can be approximated by real-analytic smooth open subsets of in the following sense: For every closed set and every ε > , there is a real-analytic open set so that εB where is the closed unit ball of Proof. In the ﬁrst case, let us deﬁne := dist( A,B 0 and consider the function ) = dist( x,A ), which is Lipschitz. We can -approximate by a real-analytic function with no critical points. Then := 0 on g > 0 on , and (0) is a real-analytic 1-codimensional submanifold of separating and . In the second case, we consider in (1) the sets := and := εU ), where is the open unit ball of Question. Recall that it is an open problem whether every real-valued and contin- uous function on can be uniformly approximated by analytic functions with no critical points. Let us consider in a non negative separating polynomial with (0) = 0 and a continuous function . A similar strategy could be ap- plied to following the proof of Kurzweil [15] to obtain: (1) real-analytic functions so that span ··· ,Q for every , where is the auxiliary sequence of considered in the proof, (2) if ||·|| is the Hilbertian norm and ) = || || , then span x,x ··· ,x } span x,c 00 , whenever the auxiliary sequence } 00 (this can be assumed). In that case, we could con- struct the real analytic function ) = so that span x, p>

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12 M. JIM ENEZ-SEVILLA for every and 1-approximates . Thus, it would be enough to ﬁnd a bounded real-analytic function satisfying span x, p> for every , to obtain a real-analytic function λT with no critical points, that 2-approximates for some λ > 0. Nevertheless, we do not know the existence of such a function Acknowledgements The author is indebted to the reviewer and the Editor for their constructive com- ments. The author would like to thank the Department of Mathematics at Ohio State University and very specially to David Goss for his kind hospitality. References [1] D. Azagra and M. Cepedello, Uniform approximation of continuous mappings by smooth map- pings with no critical points on Hilbert manifolds , Duke Math. J. 124 (2004) 207–226. [2] D. Azagra and R. Deville, James’ theorem fails for starlike bodies , J. Funct. Anal. 180 (2001), 328–346. [3] D. Azagra and M. Jim´enez-Sevilla, The failure of Rolle’s theorem in inﬁnite-dimensional Ba- nach spaces , J. Funct. Anal. 182 (2001) 207–226. [4] D. Azagra and M. Jim´enez-Sevilla, Approximation by smooth functions with no critical points on separable Banach spaces , J. Funct. Anal. 242 (2007), no. 1, 1–36. [5] D. Azagra, R. Deville and M. Jim´enez-Sevilla, On the range of the derivatives of a smooth mapping between Banach spaces , Proc. Cambridge Phil. Soc. 134 (2003), no. 1, 163–185. [6] J. M. Borwein, M. Fabian and P.D. Loewen, The range of the gradient of a Lipschitz smooth bump in inﬁnite dimensions , Isr. J. Math. 132 (2002), 239–251. [7] M. Cepedello and P. H´ajek Analytic approximations of uniformly continuous functions in real Banach spaces , J. Math. Anal. Appl. 256 (2001), 80–98. [8] R. Deville and P. H´ajek, On the range of the derivative of Gˆateaux-smooth functions on sepa- rable Banach spaces , Isr. J. Math. 145 (2005), 257–269. [9] M. Fabian, O. F. K. Kalenda and J. Kol´aˇr, Filling analytic sets by the derivatives of -smooth bumps , Proc. Amer. Math. Soc 133 (2005), no. 1, 295–303. [10] J. Ferrer, Rolle’s Theorem for polynomials of degree four in a Hilbert space , Journal of Mathe- matical Analysis and Applications 265 (2002), 322–331. [11] R. Fry, Analytic approximation on , J. Funct. Analysis 158 (1998), no. 2, 509-520. [12] T. Gaspari, On the range of the derivative of a real-valued function with bounded support Studia Math. 153 (2002), no. 1, 81–99. [13] P. H´ajek, Smooth functions on , Israel Journal of Mathematics 104 (1998), 17-27. [14] P. H´ajek and M. Johanis, Smooth approximations without critical points , Cent. Eur. J. Math. (2003), no. 3, 284–291. [15] J. Kurzweil, On approximations in real Banach spaces , Studia Math. 14 (1954), 214–231. [16] J. M´ujica, Complex analysis in Banach spaces , North-Holland Mathematical Studies 120 , El- sevier (1986). [17] S.A. Shkarin, On Rolle’s theorem in inﬁnite-dimensional Banach spaces , Translated from: Mat. Zametki 51 (3) (1992) 128-136. Departamentos de An alisis Matem atico, Facultad de Matem aticas, Universidad Complutense, 28040 Madrid, Spain E-mail address marjim@mat.ucm.es

JIM ENEZSEVILLA Abstract A real Banach space satis64257es property K de64257ned in 7 if there exists a realvalued function on which is uniformly real analytic and separat ing We obtain that every uniformly continuous function where is an open subse ID: 22197

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A NOTE ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS OF UNIFORMLY CONTINUOUS FUNCTIONS ON M. JIM ENEZ-SEVILLA Abstract. A real Banach space satisﬁes property (K) (deﬁned in [7]) if there exists a real-valued function on which is uniformly (real) analytic and separat- ing. We obtain that every uniformly continuous function , where is an open subset of a separable Banach space with property (K) and containing (thus for some Banach space ) can be uniformly approximated by (real) analytic functions such that ∂g ∂c p> (where ∂f ∂c is the set of partial derivatives ∂f ∂x x,y ) : ( x,y ). Similar statements are obtained for uniformly continuous functions with values in a (ﬁnite or inﬁnite dimensional) Banach space . Some consequences of these results are studied. 1. Introduction Several properties related to the size and shape of the set of derivatives of smooth functions between Banach spaces have been studied in [6], [2], [12], [8], [9] and [3] among others. Azagra and Deville [2] constructed smooth bump functions , in any Banach space with a Lipschitzian and smooth bump function, such that the set of derivatives ﬁlls the dual space, that is ) = (a bump function is a function with non-empty and bounded support). Some generalizations of this result are given in [5]. If has the Radon-Nikod´ym property it follows from Stegall’s variational principle that the cone generated by the set of derivatives of any continuous and Gˆateaux diﬀerentiable bump function is residual. In turn, P. H´ajek [13] proved that every Fr´echet diﬀerentiable function with locally uniformly continuous derivative from into a Banach space with non trivial type has locally compact derivative. Azagra and Cepedello [1] proved that every continuous function from into can be uniformly approximated by smooth functions with no critical points. P. H´ajek and M. Johanis [14] stated a related result for separable Banach spaces with a smooth bump function and containing : every real- valued continuous function deﬁned on can be uniformly approximated by smooth functions whose range of derivatives is of ﬁrst category and avoids a pre- ﬁxed subset. Recently, it was proved in [4] that every real valued continuous function deﬁned on a separable Banach space with separable dual (with a LUR and smooth equivalent norm, respectively) can be approximated by smooth ( smooth, respectively) functions with no critical points. Date : July, 2008. 2000 Mathematics Subject Classiﬁcation. Primary 46B20, 46T30. Secondary 58E05, 58C25. Key words and phrases. Approximation by analytic functions, range of the derivatives. The author has been supported by a fellowship of the Ministerio de Educacion y Ciencia, Spain.

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2 M. JIM ENEZ-SEVILLA Let us focus now on the (real) analytic case. J. Kurzweil [15] proved that every continuous function from into a Banach space can be uniformly approximated by analytic functions. R. Fry [11] proved the same assertion for uniformly continuous functions deﬁned on . Independently, M. Cepedello and P. H´ajek [7] proved the same result for and for the class of separable Banach spaces with property (K). A real separable Banach space satisﬁes property (K) if there is a function which is uniformly analytic and separating: that is, is real analytic at every point with radius of convergence uniformly bounded below by some constant M > and there is some such that the set < } and is not empty. In this note, we will prove that every uniformly continuous function , where is an open subset of a separable Banach space with property (K) and containing (thus for some Banach space ), can be uniformly approximated by (real) analytic functions such that ∂g ∂c p> (where ∂f ∂c ) is the set of partial derivatives ∂f ∂x x,y ) : ( x,y )). Our proof in based on the construction of real analytic approximations for a uniformly continuous function given by M. Cepedello and P. H´ajek [7]. We shall prove that, under additional conditions, we can obtain that p> . This result allows us to obtain real analytic approximations such that ∂h ∂c for any pre-ﬁxed 0 < p < q < 1 (or even ∂h ∂c span p> }\ p> , for any pre-ﬁxed \ p> ). A similar estimation of the set of derivatives is given for vectorial functions , where is a (ﬁnite or inﬁnite) dimensional Banach space. Let us point out that from the results of P. H´ajek [13] we can obtain an approx- imate Morse-Sard theorem, namely the existence of real analytic approximations with no critical points for uniformly continuous functions with open subset of (separable real Banach space with property (K) containing ) and ﬁnite dimensional. The results given in this note provide an alternative proof of this assertion, but with the advantage that it gives an additional description of the range of the derivatives. Finally, let us remark that, whenever has the property that every uniformly continuous function can be approximated by real analytic functions with no critical points, a real analytic separation result can be applied to any pair of closed subsets at positive distance in (via 1-codimensional real analytic submanifolds of ). This can be viewed as a real analytic approximation result for closed subsets in 2. Main results Let us ﬁrst state the following result for . Then we will state an analoguous result for (real) separable Banach spaces containing with property ( ). Theorem 2.1. Every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated by analytic functions whose set of deriva- tives is included in Proof. From now on, let us follow the notation given in [7]. Let us consider a uniformly continuous function . Let us ﬁx := 1 and take ρ > satisfying | whenever || || . We shall denote the complex Banach space by . Recall that ) := for and the complex

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN function ) = deﬁned in are real-analytic and analytic respectively, with radius of convergence one at every point [16, Example 5.5]. We will consider the real Banach space , the complex Banach space (which we shall denote by ) and the real quasi-Banach space with the quasi-norm ||·|| deﬁned as || || for all . Notice, that for every , we have that || || and thus ( . Let us consider, following the proof of [7], (1) the real-analytic function ) := ρx ) and the analytic function ) := ρz ) deﬁned in and respectively, with radius of convergence at every point . Let us denote by x,r ) = (2) a dense sequence in (3) a strictly decreasing sequence of real numbers converging to 0 with < (4) the two locally ﬁnite coverings of formed by the open sets and deﬁned as 1) 1 + 2 i 1) ) ) for n> i 1 + 2 ) ) for n> 1; (5) the -dimensional intervals deﬁned as =[ 1 + =[1 1) ··· × [1 1 + for n> 1; (6) the holomorphic functions deﬁned on as ) := + 1) exp =1 dτ, where = ( ··· , and are sequence of positive numbers and is the norming factor exp =1 d n/ n/ ··· In order to simplify the notation, let us write the following := and := M. Cepedello and P. H´ajek proved in [7] that whenever the sequence of positive numbers is taken small enough so that n>m m,n for every where m,n = sup {| ) + e is the closed unit ball of ) and the sequence is taken large enough so that (1) , whenever (2) + 1) , whenever (3) + 1) exp(

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4 M. JIM ENEZ-SEVILLA then the function + 1) ) locally uniformly converges in and the functions ) := ), ) := ) and ) := are well deﬁned and holomorphic on an open subset of containing . Therefore, the restrictions of the above functions to are real-analytic on . Moreover, they prove that 1 for every , that is 1-approximates in An additional condition we set for is that it is a decreasing sequence (oth- erwise, we change min ··· ,a by ). Once we ﬁx the sequence , we shall prove in the next that, under additional conditions on , the derivatives ) = ) = and , for every First, let us compute the derivative of ), for . Then, ) = + 1) =1 ) exp =1 d The derivative ) is a ﬁnite linear combination of ) : = 1 ,...,n . Since ) : = 1 ,...,n } , we obtain that . Moreover, , p> for every If a sequence of holomorphic functions locally uniformly converges on an open set , then the sequence of the derivatives also locally uniformly converges on Ω to the derivative of the limit. Thus, if ) and ) locally uniformly converge on an open set , then ) and ) locally uniformly converge in the space ( ) (holomorphic functions). Nevetheless, we cannot deduce from this fact that and , whenever We will ﬁnd an upper bound for || || . Let us deﬁne, for every , the sequence of positive numbers as (1 + =1 || (2 || Recall that the quasi-Banach space satisﬁes the following condition =1 =1 || || for every and ,...,v Therefore, || || + 1) =1 || || exp =1 d

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN Let us ﬁx and deﬁne ) for = 1 ··· ,n as ) = exp =1 dτ. Let us ﬁnd an upper bound for ) whenever . The sequence will be chosen to be increasing and satisfying 1, for every . If then either 1 + 2 or there is i such that . Let us study both cases. (1) Consider for j . Then ) is the product of the integrals ) = d 1+ d k d (a) Assume that 1 + 2 . Then, for every = ( ··· , we have that . Then, dt k (b) Now assume that there is i such that . Then for every = ( ··· , we have that . In the case dt i,j In the case te dt (2) Consider . Then, ) = 1+ d k d (a) Assume that 1 + 2 . Then, te dt k (b) Finally assume that there is i such that . Then d k

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6 M. JIM ENEZ-SEVILLA Therefore, for , we have || || =1 + 1) || || =1 + 1) || || It is now clear that we can proceed to select large enough such that + 1) + 1 which implies + 1) || || =1 || || whenever Fact 2.2. For every v,w (2.1) || || ≤|| (2 || || (2 || Proof. If , then ) = ρz ) = ρz , where is the canonical basis of . Then, || || || ρz || ρz It is enough to check that for every v,w || || ≤|| (2 || || (2 || For = 1, we have that ≤| ≤| . Now, the function ) = n> is convex. Then, (2 ) + (2 (2 ) + (2 ) and inequality (2.1) follows easily. From inequality (2.1), we obtain (2.2) || || ≤|| (2 || || (2 || for every Let us now consider . Since is locally ﬁnite, there is such that 6 , for n>i . Thus, n>i + 1) || || n>i =1 || || n>i =1 || (2 || || (2 || Let us denote =1 || (2 || . Then, n>i + 1) || || n>i nr || (2 || n>i (2.3)

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN Since nr and , we deduce that || || || || and || || || || for every . Therefore for every Now it is clear that, for every ε> 0, we can -approximate with a real analytic function whose set of derivatives is in Finally, let us mention that if is deﬁned in an open subset of , the proof is analogous with and deﬁned in an open subset of containing Let us consider, for every 0 1, the quasi-Banach spaces with the quasi- norm || || = ( /p , for . Recall that || =1 || =1 || || for every ··· ,v . We shall deduce the following corollaries. Corollary 2.3. Let us consider real numbers < p < q < . Then, every real- valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated by analytic functions such that the set of derivatives is in Proof. The proof of Theorem 2.1 works not only for but also for every 0 p < 1. With the notation of Theorem 2.1, if we select large enough so that + 1) || || , and follow the rest of the proof (Fact 2.2 holds for every p > 0), we obtain a 1-approximating analytic function such that . Next, for 0 < p < q < 1, we choose and deﬁne a real-analytic function 1) such that 0 for every , for instance ) = 1+ 1. Then, ( and ( . Thus is a real analytic function which 2-approximates Corollary 2.4. (1) Every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated by analytic functions such that the set of derivatives is in p> (2) Consequently, every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated by analytic functions such that the set of derivatives is in span p> }\ p> for any pre-ﬁxed \ p> Proof. (1) The proof of Theorem 2.1 works with some modiﬁcations: We ﬁnd a upper bound for || || /n /n whenever , such that (2.4) /n + 1) || || /n /n for every . In order to do that (and following the same notation) we consider (1 + )(1 +

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8 M. JIM ENEZ-SEVILLA where =1 || (2 || /n /n and . The calculations are similar except in the step corresponding to equation (2.3) where we have n>i /n + 1) || || /n /n n>i nr || (2 || /n /n n>i Since n>i we only need to prove that n>i nr || (2 || /n /n We need an additional bound for || (2 || /n /n /n . Let us denote by | and . Then /n /n || (4 || In addition, we have /n /n Therefore, || (2 || /n /n ≤|| (4 || Now, n>i nr || (2 || /n /n n>i nr || (4 || n>i nr || (4 || n>i nr and (2.4) is proved. From inequality (2.4) we obtain, for every 0 1 and a natural number such that and /n + 1) || || /n /n 1. Since /n whenever 0 1 and , it follows that + 1) || || /n /n + 1) || || /n /n Moreover, ||·|| ≤||·|| /n whenever . Therefore, + 1) || || + 1) || || /n and ) = p> Now in order to prove (2), it is clear that ( = 0 for every because 6 p> (where is the real analytic function considered in the proof of Corollary 2.3). Moreover, ( span p> }\ p>

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN In the next corollary we extend Theorem 2.1, Corollaries 2.3 and 2.4 to every separable real Banach space with property (K) and containing . Recall that, by the Sobczyk theorem , for some separable real Banach space . Let us consider a diﬀerentiable function deﬁned on an open subset of . For every = ( x,y , let us denote by ∂f ∂c ) the set of partial derivatives ∂f ∂x x,y ) : x,y Corollary 2.5. Let be a separable real Banach space with property (K) such that contains a copy of . Then, (1) every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated on by an analytic function such that the set ∂h ∂c p> (2) Consequently, every real-valued, uniformly continuous function deﬁned on an open subset of can be uniformly approximated on by an analytic function such that the set ∂g ∂c span p> }\ p> , for any pre-ﬁxed \ p> Proof. We follow the proofs of [7, Theorem 1], Theorem 2.1 and Corollary 2.4 for the space and the uniformly analytic and separating function deﬁned below: (a) We consider in , the function x,y ) = ) + ), for and , where is uniformly analytic and separating in and ) = for . Clearly is uniformly analytic and separating in (b) We consider the complexiﬁed space of = , where is the complexiﬁed space of . The function x,y ) := ) + ), where is the analytic extension of to an open subset Ω of of the form sU is the open unit ball of ) for some s> 0. (c) Let us deﬁne, for every 0 <ρ< 1, the functions x,y ) = ρx,ρy ), for and (d) ∂D ∂x x,y ) = ) and ∂D ∂x x,y ) = ( ), for and Then, following the notation of Theorem 2 1, we have: (i) ∂x x,y p> , for every ( x,y and , (ii) ∂x x,y ) = ∂x x,y p> and ∂x x,y ) = ∂x x,y p> for every ( x,y , where := ( ,x and is the pre-ﬁxed dense sequence in , and thus, (iv) the analytic approximations and satisfy that ∂h ∂x x,y p> and ∂g ∂x x,y ) = ∂x x,y ) = ∂h ∂x x,y ) + )) span p> }\ p> for every ( x,y , if p> Let us now generalize Theorem 2.1, Corollaries 2.3, 2.4 and 2.5 for the case of uniformly continuous functions with values into a (ﬁnite or inﬁnite dimensional) real Banach space Corollary 2.6. Let be a real Banach, a separable real Banach space with property (K) so that contains a copy of and an open subset of . Then, (1) every uniformly continuous function can be uniformly approxi- mated by analytic functions such that ∂h ∂c p> for every

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10 M. JIM ENEZ-SEVILLA (2) Let us ﬁx any \ p> and \{ . Then, every uniformly continuous function can be uniformly approximated by an analytic function such that ∂g ∂c span p> }\ p> for every (3) In addition to (2), if is ﬁnite dimensional, we obtain that the analytic approximation has no critical points, i.e. is surjective for every Proof. (1) The proofs of Theorem 2.1 and Corollary 2.5 can by applied to obtain the above assertion. We only have to consider || || instead of and select so that || || /n + 1) || ∂x x,y || /n /n for every ( x,y (we follow the notation of the proof of Corollary 2.5). Then, the analytic function x,y ) = x,y x,y (which 1-approximates ) satisﬁes the following property: ∂h ∂x x,y p> for every and ( x,y (2) Let us consider || || ||·|| is the norm on ) and deﬁne x,y ) = ) for all ( x,y , where 1) is the analytic function already deﬁned in the proof of Corollary 2.3. On the one hand, x,y x,y | for every ( x,y . On the other hand, ∂g ∂x x,y ) = ∂h ∂x x,y )+ vw )) for every x,y . This implies ∂g ∂x x,y ) = ∂h ∂x x,y ) + ( )) for every x,y and . Moreover, ( )) = 0 whenever and x,y . Thus, ∂g ∂x x,y span p> }\ p> whenever and x,y (3) Let us consider a continuous function with components = 1 ,...,n . Let us select linear independent elements ,...,z p> such that (span ,...,z p> ) = . By Corollary 2.5, we can uniformly approximate by analytic functions satisfying the following condition: for every ( x,y there are \{ and p> such that ∂g ∂x x,y ) = . Therefore ∂g ∂x x,y ,..., ∂g ∂x x,y are linearly independent (thus x,y ,...,g x,y are l.i. as well) for every ( x,y and the analytic function := ( ,...,g ) uniformly approximate The existence of real analytic approximations with no critical points for uniformly continuous functions with the conditions of Corollary 2.6 and ﬁnite dimensional) can be obtained from the results of P. H´ajek [13]. Statement (3) in Corollary 2.6 provides an alternative proof to this assertion with an additional description of the range of the derivatives. Let us sketch how to deduce the assertion from [13] and for (the general case is deduced in an analogous way). If is a real analytic function approximating and we consider ,...,h the components of , then by the results given in [13] there is a set (namely a countable union of compact sets) of , such that ∪··· ⊂K We may consider that is a linear subspace of ; otherwise we consider the span {K} which is also a subset. Let us follow the notation [ ,...,a ] = span ,...,a Therefore, there is \{ such that [ ∩K . Since [ ] + is again a subspace of , there is \{ such that [ ([ ] + ) = and then , z ∩K . Proceeding in this way, we obtain linear independent elements = 1 ,...,n in , such that [ ,...,z ∩K . Now we proceed as in the proof of Corollary 2.6(3) and deﬁne := = 1 ,...,n and ε> 0 ( is

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ON THE RANGE OF THE DERIVATIVES OF ANALYTIC APPROXIMATIONS IN 11 the auxiliary real analytic function considered in the proof of Corollary 2.3). Then, ,...,g are linearly independent for every and := ( ,...,g is a real analytic function with no critical points and uniformly approximates In addition, it can be obtained that ∩M for any pre-ﬁxed subset M ,E ) = ×··· ···× (linear continuous function from into ). Ir order to do that, we denote by the linear span of the projection of over the ﬁrst . Then, is a subset of . Following the above arguments, we select elements = 1 ,...,n such that [ ,...,z ) = . In this case, ∈M and then ∈M for every Let us ﬁnish this note with a remark on the separation of closed sets and a question on the separable Hilbert space. Recall that, whenever a real Banach space has the property that every uniformly continuous, real-valued function on can be uniformly approximated by real-analytic functions with no critical points, one can deduce the following non-linear separation and approximation result. An open subset of is said to be real-analytic smooth provided its boundary ∂U is a real-analytic smooth one-codimensional submanifold of Remark 2.7. Let be a real and separable Banach space containing with property (K). (1) Let and be two disjoint and closed subsets of such that dist( A,B Then, there is a real-analytic function with no critical points such that the level set (0) is a 1-codimensional analytic submanifold of that separates and in the following sense: Deﬁne and , then and are disjoint real- analytic open subsets of with common boundary ∂U ∂U such that and (2) Every closed subset of can be approximated by real-analytic smooth open subsets of in the following sense: For every closed set and every ε > , there is a real-analytic open set so that εB where is the closed unit ball of Proof. In the ﬁrst case, let us deﬁne := dist( A,B 0 and consider the function ) = dist( x,A ), which is Lipschitz. We can -approximate by a real-analytic function with no critical points. Then := 0 on g > 0 on , and (0) is a real-analytic 1-codimensional submanifold of separating and . In the second case, we consider in (1) the sets := and := εU ), where is the open unit ball of Question. Recall that it is an open problem whether every real-valued and contin- uous function on can be uniformly approximated by analytic functions with no critical points. Let us consider in a non negative separating polynomial with (0) = 0 and a continuous function . A similar strategy could be ap- plied to following the proof of Kurzweil [15] to obtain: (1) real-analytic functions so that span ··· ,Q for every , where is the auxiliary sequence of considered in the proof, (2) if ||·|| is the Hilbertian norm and ) = || || , then span x,x ··· ,x } span x,c 00 , whenever the auxiliary sequence } 00 (this can be assumed). In that case, we could con- struct the real analytic function ) = so that span x, p>

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12 M. JIM ENEZ-SEVILLA for every and 1-approximates . Thus, it would be enough to ﬁnd a bounded real-analytic function satisfying span x, p> for every , to obtain a real-analytic function λT with no critical points, that 2-approximates for some λ > 0. Nevertheless, we do not know the existence of such a function Acknowledgements The author is indebted to the reviewer and the Editor for their constructive com- ments. The author would like to thank the Department of Mathematics at Ohio State University and very specially to David Goss for his kind hospitality. References [1] D. Azagra and M. Cepedello, Uniform approximation of continuous mappings by smooth map- pings with no critical points on Hilbert manifolds , Duke Math. J. 124 (2004) 207–226. [2] D. Azagra and R. Deville, James’ theorem fails for starlike bodies , J. Funct. Anal. 180 (2001), 328–346. [3] D. Azagra and M. Jim´enez-Sevilla, The failure of Rolle’s theorem in inﬁnite-dimensional Ba- nach spaces , J. Funct. Anal. 182 (2001) 207–226. [4] D. Azagra and M. Jim´enez-Sevilla, Approximation by smooth functions with no critical points on separable Banach spaces , J. Funct. Anal. 242 (2007), no. 1, 1–36. [5] D. Azagra, R. Deville and M. Jim´enez-Sevilla, On the range of the derivatives of a smooth mapping between Banach spaces , Proc. Cambridge Phil. Soc. 134 (2003), no. 1, 163–185. [6] J. M. Borwein, M. Fabian and P.D. Loewen, The range of the gradient of a Lipschitz smooth bump in inﬁnite dimensions , Isr. J. Math. 132 (2002), 239–251. [7] M. Cepedello and P. H´ajek Analytic approximations of uniformly continuous functions in real Banach spaces , J. Math. Anal. Appl. 256 (2001), 80–98. [8] R. Deville and P. H´ajek, On the range of the derivative of Gˆateaux-smooth functions on sepa- rable Banach spaces , Isr. J. Math. 145 (2005), 257–269. [9] M. Fabian, O. F. K. Kalenda and J. Kol´aˇr, Filling analytic sets by the derivatives of -smooth bumps , Proc. Amer. Math. Soc 133 (2005), no. 1, 295–303. [10] J. Ferrer, Rolle’s Theorem for polynomials of degree four in a Hilbert space , Journal of Mathe- matical Analysis and Applications 265 (2002), 322–331. [11] R. Fry, Analytic approximation on , J. Funct. Analysis 158 (1998), no. 2, 509-520. [12] T. Gaspari, On the range of the derivative of a real-valued function with bounded support Studia Math. 153 (2002), no. 1, 81–99. [13] P. H´ajek, Smooth functions on , Israel Journal of Mathematics 104 (1998), 17-27. [14] P. H´ajek and M. Johanis, Smooth approximations without critical points , Cent. Eur. J. Math. (2003), no. 3, 284–291. [15] J. Kurzweil, On approximations in real Banach spaces , Studia Math. 14 (1954), 214–231. [16] J. M´ujica, Complex analysis in Banach spaces , North-Holland Mathematical Studies 120 , El- sevier (1986). [17] S.A. Shkarin, On Rolle’s theorem in inﬁnite-dimensional Banach spaces , Translated from: Mat. Zametki 51 (3) (1992) 128-136. Departamentos de An alisis Matem atico, Facultad de Matem aticas, Universidad Complutense, 28040 Madrid, Spain E-mail address marjim@mat.ucm.es

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