An extension of Whitney approximation theorem

An extension of Whitney approximation theorem Carlos ANDRADAS, Eva ANEIROS, and Antonio D´IAZ-CANO ´ IMI and Departamento de Algebra Facultad de Ciencias Matem´ aticas Universidad Complutense de Madrid E-28040 Madrid, Spain [email protected] [email protected] [email protected]

Dedicado al Profesor Juan Tarr´ es

ABSTRACT The classical Whitney approximation theorem states that a smooth function f : Rn → R can be approximated by analytic functions. We prove that these analytic functions approximating f can be taken as real holomorphic functions on the whole space Cn . Key words: Whitney approximation theorem, approximation by analytic functions 2010 Mathematics Subject Classification: 58C25, 26E05, 41A30, 32C05, 14P15

1. Introduction In 1934 H. Whitney proved that a smooth function f : Rn → R can be approximated by analytic functions, cf. [7]. This result is known as Whitney approximation theorem and is of fundamental importance in the study of differentiable and analytic manifolds. We refer to [1] or the survey [6] and the many references therein for applications and generalizations of it. The aim of this paper is to extend Whitney approximation theorem proving that the analytic functions approximating f can be taken as real holomorphic functions on Cn . In other words, the classical Whitney approximation theorem asserts that the ring O(Rn ) of analytic functions on Rn is a dense subset of the space C m (Rn ) of real-valued functions on Rn with respect to the so called Whitney topology. In this paper, we found a strictly smaller ring which is also a dense subset of C m (Rn ). This is the ring Authors partially supported by Spanish MTM2011-22435 and Grupo UCM 910444

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Actas Homenaje J. Tarr´ es

C. Andradas/E. Aneiros/A. D´ıaz-Cano

An extension of Whitney approximation theorem

of real holomorphic functions, denoted as OR (Cn ), whose elements are holomorphic functions on Cn which are invariant under conjugation or, alternatively, real analytic functions on Rn which can be extended analytically to Cn . Real holomorphic functions are of interest in the study of real analytic sets, that is, subsets of Rn defined as the zero set of some real analytic functions. For example, the full normalization of some real analytic sets can have complex parts, see [4], and then real analytic functions which can be extended analytically to these complex parts arise. Real holomorphic functions also appear in the solution of Hilbert 17th problem for non-coherent analytic surfaces, cf. [2].

2. Preliminaries We consider in Rn the usual topology. Given a compact subset K ⊂ Rn , an integer m n p ∈ N and a function f ∈ C m (Rn ), the seminorm k f kK p of f ∈ C (R ) is defined as: k f kK p =

X 1 supx∈K |Dα f (x)| , α!

|α|≤p

where we use the standard notations Dα f =

∂ |α| f n αn , α = (α1 , . . . , αn ) ∈ N , |α| = α1 + . . . + αn 1 ∂xα 1 . . . ∂xn

and we suppose |α| ≤ m ≤ ∞. A fundamental system of neighborhoods of f ∈ Cm (Rn ) in the weak topology is the collection: {g ∈ C m (Rn ) | k f − g kK p < ε} , where K is a compact subset of Rn , p ∈ N (of course, p ≤ m) and ε is a positive real number. Intuitively, we can say that in the weak topology two functions are close if in a compact subset K the values of both functions and of their p first derivatives are close. As Rn is not compact, the closeness of functions at infinity is not controlled by the weak topology. For that reason, we introduce the Whitney topology as a limit of the weak topology. In this topology, a fundamental system of neighborhoods of f ∈ Cm (Rn ) is given by the sets: Ki 1 {g ∈ C m (Rn ) | k f − g kK m1 < ε1 , ..., k f − g kmi < εi , ...} ,

where {mi } is a sequence of positive integers (mi ≤ m, ∀i), {εi } is an arbitrary n sequence of positive real numbers and the Ki are compact S subsetsn of R verifying n Ki ⊂ Vi for some locally finite covering {Vi } of R (that is, Vi = R and every point of Rn has a neighborhood which intersects only finitely many sets in the covering). Now, we can state the Theorem 2.1. (Classical Whitney aproximation theorem) Consider the sequences {Ki }, {mi } and {εi } as above, and let f ∈ Cm (Rn ). Then there exists an analytic function g ∈ O(Rn ) such that

—————————— Actas Homenaje J. Tarr´ es

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i k f − g kK mi < i ,

for each i ≥ 1.

The proof can be found in the paper of H. Whitney, cf. [7], or in the book by R. Narasimhan [5]. A similar theorem is proved in [3] for analytic submanifolds of Rn . An immediate corollary of this theorem is the following: Corollary 2.2. The ring O(Rn ) of analytic functions on Rn is a dense subset of the space C m (Rn ) with respect to the Whitney topology.

3. Extended Whitney approximation theorem First we are going to see how to approximate in the weak topology a smooth function f having compact support (that is, a function f ∈ C m (Rn ) such that the closure of {x ∈ Rn | f (x) 6= 0} is a compact set) by an analytic function. This can be achieved by convolution with a suitable exponential function. For the proof of this key lemma we will closely follow the proof of lemma 1.6.1 in [5]. A similar result can be also found in [1]. Lemma 3.1. Let f ∈ C m (Rn ) be a function with compact support, 0 ≤ m < ∞. For λ > 0, set: Z 2 n 2 f (y)e−λkx−yk dy gλ (x) ≡ Iλ (f )(x) := cλ Rn

where c = π

−n 2

, so that cλ

n 2

2

R Rn

gλ ∈ O(Rn )

e−λkxk dx = 1. Then n

lim k gλ − f kR m = 0.

and

λ→∞

Proof. By a change of variables, we can write gλ (x) in the form: Z 2 n 2 gλ (x) = cλ f (x − y)e−λkyk dy Rn

and so, for any α ∈ Nn such that |α| ≤ m, we have Z Z 2 n n (Dα f )(x − y)e−λkyk dy = cλ 2 Dα gλ (x) = cλ 2 Rn

2

(Dα f )(y)e−λkx−yk dy

Rn

Therefore, α

α

D gλ (x) − D f (x) = cλ

n 2

Z

2

(Dα f (y) − Dα f (x))e−λkx−yk dy.

Rn

Now, since f has compact support and |α| ≤ m < ∞ there is M > 0 such that |Dα f (y)| < M

for all y.

Also, given > 0 there exists δ > 0 such that |Dα f (y) − Dα f (x)| < /2

for k x − y k< δ.

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Thus Z 2 |D gλ (x) − D f (x)| = cλ (Dα f (y) − Dα f (x))e−λkx−yk dy kx−yk , ∀x ∈ R such that k x k≥ p + 1 , 2

n

k=1

where z = (z1 , . . . , zn ) ∈ Cn , x = (x1 , . . . , xn ) ∈ Rn , Re(ζ) denotes the real part of the complex number ζ and Int denotes the interior with respect to the usual topology in Cn . Then [ Up = Cn . p≥1

Proof. To have an idea of how the Up are, in figure 1 we can see the shape that they have in dimension 1.

Im

6 q i qp2 + 2p + 12

Re -(p+1)

-p

-

p

p+1

Figure 1 Take z ∈ C n and we will prove that z ∈ Up for some p. We write z = (z1 , . . . , zn ) = (a1 + ib1 , . . . , an + ibn ), ai , bi ∈ R. Thus we have

Re

X n

(zk −xk )

k=1

2

X X n n 2 2 2 = Re (ak +ibk −xk ) = Re (ak −xk ) −bk +2(ak −xk )bk i k=1

k=1

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=


X n n n X X (ak − xk )2 − b2k = (ak − xk )2 − b2k k=1

k=1

k=1 n

We search for p ∈ N such that for every x = (x1 , . . . , xn ) ∈ R verifying k x k≥ p + 1: n X

(ak − xk )2 −

k=1

n X

b2k >

k=1

1 . 2

If we consider points a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) of Rn , the previous inequality can be rewritten as: k a − x k2 >

1 + k b k2 . 2

(1)

Take p such that p + 1 >k a k + k b k +1 . If k x k≥ p + 1 then k a − x k≥ p + 1− k a k> 1+ k b k and inequality (1) is satisfied. Hence for this p the point z is in Up , so that the lemma is proved. Now we can prove our main theorem. Without loss of generality, in the definition of the Whitney topology we take the sequence of compact sets Ki = {x ∈ Rn | i−1 ≤k x k≤ i}. Our proof follows the classical one of [5], but our particular election of the sequence {Ki } allow us to prove that the approximating function is, in fact, a real holomorphic function. Theorem 3.4. (Extended Whitney approximation theorem) Let f ∈ C m (Rn ), 0 ≤ m ≤ ∞, and let Ki = {x ∈ Rn | i − 1 ≤k x k≤ i}. Let {mi } be an arbitrary sequence of positive integers (mi ≤ m) and {i } any sequence of strictly positive real numbers. Then there exists a real holomorphic function g ∈ OR (Cn ) verifying that i k f − g kK mi < i

for each i ≥ 1. Proof. Since for any subset S ⊂ Rn and any pair of integers k ≤ k 0 we have the inequality k f kSk ≤ k f kSk0 , for the proof of the theorem we can suppose mp+1 ≥ mp , for every p ≥ 1. The following property of seminorms will be used: if ϕ, ψ ∈ C m (Rn ) and S ⊂ Rn , then we have that: k ϕψ kSk ≤ k ϕ kSk k ψ kSk . We take a sequence of functions ϕp ∈ C ∞ (Rn ), p ≥ 1, with compact support, such that ϕp (x) = 1 if x ∈ Kp and ϕp (x) = 0 if x ∈ / {y ∈ Rn | p − 23 ≤k y k≤ p + 21 }. Corollary 1.2.6 of [5] assures that such functions ϕp exist. Let n Mp = 1+ k ϕp kR mp . Note that 1 < Mp < ∞, because ϕp has compact support. Take a sequence {δp } of positive numbers such that δp+1 ≤ δp /2 and X

δq Mq+1 ≤

q≥p


6

1 p , for p ≥ 1. 4


An extension of Whitney approximation theorem ...

For example, we can take δp = 14 2p M21 ...Mp p+1 , as we can suppose that i < 1, ∀i ≥ 1. Sp We define Bp := {x ∈ Rn | k x k≤ p}, so that Bp = i=1 Ki . By lemma 3.1, there exists λ1 > 0 such that if g1 := Iλ1 (ϕ1 f ), then 1 k g1 − ϕ1 f kB m1 < δ1 .

Also, by lemma 3.1, there exists λ2 > 0 such that if g2 := Iλ2 (ϕ2 (f − g1 )), then 2 k g2 − ϕ2 (f − g1 ) kB m2 < δ 2 .

By induction, we define λp > 0 and gp such that if gp := Iλp (ϕp (f − g1 − . . . − gp−1 )), then p k gp − ϕp (f − g1 − . . . − gp−1 ) kB (2) mp < δ p . As ϕp = 0 in a neighbourhood of Bp−2 , (2) implies that p−2 k gp kB mp < δp .

(3)

P∞ The function we are looking for is g := q=1 gq . First, we are going to prove that g ∈ C mp (Rn ) for every p. As ϕp (x) = 1 in a neighbourhood of Kp , from (2) we have that p k f − g1 − ... − gp kK mp < δ p .

(4)

Substituting p by p + 1 in (2), using (4) and that mp ≤ mp+1 , we have: p k gp+1 kK mp ≤ k ϕp+1 (f −

p X

p gq ) kK mp + k gp+1 − ϕp+1 (f −

q=1

≤k

n ϕp+1 kR mp

kf−

p X

p gq kK mp

p X

p gq ) kK mp

q=1

+ k gp+1 − ϕp+1 (f −

q=1

p X

p+1 gq ) kB mp+1 ≤ Mp+1 δp + δp+1

q=1

and Bp−1 p p + k gp+1 kK k gp+1 kB mp ≤ δp+1 + (Mp+1 δp + δp+1 ) mp ≤ k gp+1 kmp

= Mp+1 δp + 2δp+1 ≤ Mp+1 δp + δp ≤ 2δp Mp+1 . Therefore k

X

p gq kB mp ≤ 2

q>p

X

δq Mq+1
p

1 p , 2

(5)

which implies that, for every p, g=

∞ X

gq ∈ C mp (Rn ).

q=1

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Now, using (4) and (5) we see that g approximates f as desired: p k f − g kK mp ≤ k f −

p X

p gq kK mp + k

X

p gq kK mp < δp +

q>p

q=1

1 p < p . 2

Finally, we will see that, with an adequate choice of the λp , the function g is analytic over Cn , that is, g is a real holomorphic function. As seen in lemma 3.1 the function gr (x) is analytic on Rn and, moreover, it can be extended holomorphically to Cn by the same formula gr (z) =

cλn/2 r

r−1 X Pn 2 ϕr (y) f (y) − gq (y) e−λr i=1 (zi −yi ) dy.

Z supp ϕr

q=1

We take open (complex) subsets Up of Cn as in lemma 3.3. If y ∈ supp ϕr , with r > p + 2, then k y k≥ p + 1 and for z ∈ Up we have that Re

X n 1 (zj − yj )2 > , 2 j=1

so −λr /2 |gr (z)| ≤ cλn/2 r e

Z supp ϕr

r−1 X dy ≤ cλn/2 ϕr (y) f (y) − e−λr /2 Hr g (y) q r

(6)

q=1

where Hr depends only of λ1 , ..., λr−1 , because gq depends only on the λj with j ≤ q. Thus, we can choose inductively, the λq large enough to assure that X

−λq /2 λn/2 Hq < ∞. q e

q≥1

With this choice of the {λq }, we have from (6) that the series g(z) =

X

gq (z)

converges uniformly over each Up for every p. In fact, for any z ∈ Up we have |g(z)| ≤

p+2 X

sup |gq (z)| +

q=1 z∈Up

∞ X

−λq /2 λn/2 Hq . q e

q=p+3

From this it can be deduced, by the Weierstrass convergence theorem and lemma S (3.3), that g is holomorphic over Cn = Up .

Corollary 3.5. The ring OR (Cn ) is dense in Cm (Rn ), 0 ≤ m ≤ ∞, with respect to the Whitney topology.


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References [1] F. Acquistapace, F. Broglia, A. Tognoli, “Smooth and Analytic Solutions for Analytic Linear Systems”, Rev. Mat. Complut., 9 (1996), 17–54. [2] E. Aneiros, Ph. D. Dissertation, Universidad Complutense de Madrid, in preparation. [3] J. Bochnak, W. Kucharz, M. Shiota, “On equivalence of ideals of real global analytic functions and the 17th Hilbert problem”, Invent. Math., 63 (1981), 403–421. [4] F. Guaraldo, P. Macr`ı, A. Tancredi, Topics on Real Analytic Spaces, Advanced Lectures in Mathematics, Vieweg (1986). [5] R. Narasimhan, Analysis on Real and Complex Manifolds, North Holland (1968). [6] A. Tognoli, “Approximation theorems in real analytic and algebraic geometry”, Lectures in real geometry (Madrid, 1994), de Gruyter Exp- Math. 23, Berlin 1996. [7] H. Whitney, “Analytic extensions of differentiable functions defined in closed sets”, Trans. Amer. Math. Soc. 36 (1934), 63–89.

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