SOME NONCOMPACT BOREL SUBSETS OF R

MARKOV-TYPE MOMENT PROBLEMS FOR ARBITRARY COMPACT AND

SOME NONCOMPACT BOREL SUBSETS OF R

JANINA MIHAELA MIH ˘AIL ˘A, OCTAV OLTEANU and CONSTANTIN UDRIS¸TE

We consider a class of positive polynomials on the whole interval [0,∞[, related to the functions exp(−kt),t∈[0,∞[,k∈Z+. We prove that for any positive Borel regular measure ν on [0,∞[ with moments of all orders, these polynomials are dense in (L¹_ν([0,∞[))+. We solve a determinate Markov-type moment problem on [0,∞[. The Markov moment problem on arbitrary compactsK⊂Rⁿ is reduced to the Markov problem for semi-algebraic compactsKA⊂Rⁿ. The “non-compact case” is reduced to the “compact case” for some Borel subsets ofRⁿ.

AMS 2000 Subject Classification: 41A10, 26D15, 28A20, 46A22, 46A40.

Key words: approximation by polynomials; Markov-type moment problem.

1. SOME RESULTS ON APPROXIMATION

BY POLYNOMIALS ON UNBOUNDED SUBSETS OF Rⁿ Lemma1.1 (Theorem 3.7 [18]). For any s∈R we have

exp(s)−

1 + s

1! +· · ·+s^m m!

= exp(s) m!

_s

0

exp(−τ)·τ^mdτ, m∈Z+. Corollary1.2. For any α >0, k∈Z+ we have

0<exp(−αt)≤1−αt

1! +α²t²

2! − · · ·+α^2kt^2k

(2k)! , ∀t≥0 and

exp(−αt)≥1−αt

1! + α²t²

2! − · · ·+α^2kt^2k

(2k)! −α^2k+1t^2k+1

(2k+ 1)! , ∀t≥0.

Corollary 1.3. Let e_k(t) = exp(−kt), t ≥ 0, k ∈ Z+, and let x be an element of the linear hull generated by {e_k, k ∈Z+}. Then there exists a sequence of polynomial functions(p_l)_l∈Z+,p_l(t)> x(t), ∀t≥0, and lim

l p_l =x uniformely on compact subsetsK ⊂[0,∞[.

REV. ROUMAINE MATH. PURES APPL.,52(2007),6, 655–664

Lemma 1.4. Let x˜ : [0,∞[→ R+ be a continuous function such that

t→∞lim x˜(t)∈R+exists. Then there exists a sequence(x_l)_l∈Z+,x_l ∈Sp{exp(−kt), t≥ 0;k ∈ Z+}, such that x_l(t) > x˜(t) ≥ 0, ∀t ≥ 0, ∀l ∈ Z+ and lim

l x_l = ˜x uniformly on [0,∞[. By Corollary 1.3, there also exists a sequence of polyno- mials (˜p_l)_l∈Z+, p˜_l ≥x_l > x˜ ≥0, ∀l ∈ Z+ (on [0,∞[), and p˜_l → x˜ uniformly on compact subsets of[0,∞[.

Proof. Let X0 be the space of all continuous functions ˜x : [0,∞[→ R, such that lim

t→∞x˜(t) ∈ R exists. For such ˜x ∈ X0, let ˆx˜ : [0,∞] → R be the continuous extension of ˜x to the Alexandroff compactification [0,∞] of [0,∞[. The idea is to apply the Stone-Weierstrass theorem to the subalgebra Aˆ = {ˆa; a ∈ A} ⊂ C([0,∞]), where A ⊂ X0, A := Sp{exp(−kt);k ∈ Z+, t∈[0,∞[}. Then the conclusions follow quite easily.

Lemma 1.5. (a) Let ν be a determinate positive regular Borel measure on[0,∞[, such that

[0,∞[t^jdν <∞,∀j∈Z+. If(˜p_l)_l is a sequence of polyno- mials with the properties stated in Lemma1.4, then there exists a subsequence ( ˜p_lm)_m, such that p˜_lm →x˜ in L¹_ν([0,∞[). In particular, for such measures ν, the positive polynomials are dense in (L¹_ν([0,∞[))₊.

By Luzin’s theorem, for every positive simple functions involving inter- valsIj ⊂[0,∞[, j = 1, N, there exists p˜_lm ↓sin L¹_ν([0,∞[).

(b)ReplaceSp{exp(−kt);k∈Z+, t≥0}in Lemma1.4bySp{exp(−kt²);

k∈Z+, t≥0} and define q_l(t) := ˜p_l(t²)>x˜(t²)≥0, ∀t∈R, where (˜p_l)_l∈Z+ are as in Lemma 1.4. Then q_l, l ∈ Z+ are even positive polynomials on the whole real line such that q_l(t) = q²_1,l(t²) +t²q²_2,l(t²), t ∈ R, q_1,l, q_2,l ∈ R[t] (see [1]). So, every even function x ∈ (C_c(R))₊ can be uniformly approx- imated on the compact subsets of R and in L¹_ν(R) by positive even poly- nomials q_lm, m ∈ Z+. Further, in several variables, every special function y ∈ (Cc(Rⁿ))₊, y(t1, . . . , tn) := ˜xn

i=1t²_i

, where x˜ ∈ (Cc([0,∞[))₊, can be uniformly approximated on compact subsets of Rⁿ and in L¹_ν(Rⁿ) by special (positive) polynomialsr_lm(t1, . . . , t_n) =q_1,l² _mn

i=1t²_i +

_n

i=1t²_i

·q²_2,lmn

i=1t²_i , (t1, . . . , t_n) ∈ Rⁿ, q_1,lm, q_2,lm ∈ R[t], m ∈ Z+ (ν is assumed to be determi- nated).

Proof. Let ˜x∈(C_c[0,∞[)₊ and (˜p_l)_l∈Z+ as in Lemma 1.4. Define P1(˜x) := inf

l∈Z+

[0,∞[p˜_l(t)dν ≥

[0,∞[x(t)dν˜ ≥0.

There exists a subsequence ( ˜p_lm)_m such that P1(˜x) = lim

m

[0,∞[p˜_lm(t)dν.

LetP2(˜x) := inf

m∈Z+ [0,∞[p˜²_lm(t)dν_1/2

≥0. There exists a subsequence of ( ˜p_lm)_m (which will be denoted by ( ˜p_lm)_m too (by abuse of notation)), such that P2(˜x) = lim

m

[0,∞[p˜²_lm(t)dν_1/2

. It follows that ( ˜p_lm)_m is bounded in L²_ν([0,∞[), hence the set {p˜_lm}_m is relatively weakly compact. By Theo- rem 11.2, [25], we have

P1(˜x) := lim

m lim

n

_n

0

p˜_lm(t)dν = lim

m lim

n p˜_lm, χ_[0,n]=

= lim

n lim

m p˜_lm, χ_[0,n]= lim

n lim

m

_n

0

p˜_lm(t)dν=

= lim

n

_n

0

x˜(t)dν =

[0,∞[x˜(t)dν,

since ˜p_lm → x˜ uniformly on the compact [0, n] for any fixed n ∈ Z+. The conclusion is that lim

m

[0,∞[p˜lm(t)dν =

[0,∞[x˜(t)dν, and, since ˜plm >˜x,∀m∈ Z+, (a) follows, using also the fact that (C_c([0,∞[))₊is dense in (L¹_ν([0,∞[))₊ (see [24]). The assertions stated at (b) follow quite easily from (a), using the

“geometric” intuition related to the graph of an even function from (C_c(R))₊, and the well-known representation of a polynomial which is positive on the whole nonnegative semiaxis (R)₊.

2. MARKOV-TYPE MOMENT PROBLEMS ON COMPACT AND ON SOME NON-COMPACT BOREL SUBSETS OF Rⁿ

Using results of Section 1, we can solve some determinate Markov mo- ment problems on [0,∞[.

Theorem2.1.Let dν =ρ_ν(t)dt, where ρ_ν is a continuous positive func- tion on [0,∞[, such that _∞

0 t^jρ_ν(t)dt < ∞, ∀j ∈ Z+. Assume that ν is M- determinate. Let{y_j;j∈Z+} ⊂R. The following statements are equivalent.

(a)There exists a Borel function h on [0,∞[ such that (1)

_∞

0

t^jh(t)dν =y_j, ∀j∈Z+, and

(2) 0≤h(t)≤1 ν-a.e.

(b)For any finite subset J0⊂Z+ and any {λj;j∈J0} ⊂R we have

(3.1) 0≤

i,j∈J0

λiλjyi+j ≤

i,j∈J0

λiλj

_∞

0

t^i+jdν,

(3.2) 0≤

i,j∈J0

λ_iλ_jy_i+j+1≤

i,j∈J0

λ_iλ_{j ∞}

0

t^i+j+1dν.

Proof. (a)⇒ (b) is obvious.

(b)⇒(a). LetX:={x∈L¹_ν([0,∞[). There exists a polynomialp∈P+, such that |x(t)| ≤ p(t), ∀t ≥ 0}. On the subspace P ⊂ X of all polynomial functions, define a linear functionalF0 by

F0 j∈J0

λjxj

:=

j∈J0

λjyj, J0 ⊂Z+ finite subset.

From (3.1) and (3.2) we obtain

(4) 0≤F0(˜p)≤

_∞

0

p˜(t)dν, ∀p˜∈P+

(since by [1], any such ˜p can be represented as ˜p(t) = p²₁(t) +tp²₂(t), t ≥ 0, p1, p2 ∈ R[t]). Thus F0 is also positive on the majorizing subspace P of X. By [9], pp. 160, there exists a linear positive extensionF of F0 to the whole spaceX. Define σ: [0,∞[→R+ by σ(t) :=F(χ_[0,t]), t≥0.

By Lemma 1.5 (a), if ˜p_lm ↓χ_[t1,t2], 0≤t1< t2, we have σ(t2)−σ(t1) = F(χ_[t1,t2])≤infF0(˜p_lm)⁽⁴⁾≤

(4)≤ lim _∞

0

p˜_lm(t)dν= _∞

0

χ_[t1,t2](t)dν =

= _t2

t1

ρ_ν(t)dt=σ_ν(t2)−σ_ν(t1), where σ_ν(t) := _t

0ρ_ν(τ)dτ, t > 0, is the primitive of ρ_ν which vanishes at t0 = 0. From this, we infer that for any finite or countable collection of open intervals ]t_j, t_j+1[⊂ [0,∞[, j∈J ⊂Z+, we have:

(5)

j∈J

[σ(t_j+1)−σ(t_j)]≤

j∈J

[σ_ν(t_j+1)−σ_ν(t_j)] =

=

Ë

j∈J ]tj,tj+1[

ρ_ν(t)dt→0, when

j∈J

(t_j+1−t_j)→0

(since ρν ∈ L¹_dt([0,∞[: _∞

0 ρν(t)dt < ∞ by hypothesis). In particular, it follows that σ_ν is absolutely continuous and, by (5), so is σ. Using Theo- rem 7.18, [24], (a) ⇒ (b) ⇒ (c), we conclude that σ is differentiable dt-a.e.

andρ:= ^dσ_dt ∈L¹_dt([0,∞[) since _∞

0

ρ(t)dt= _∞

0

dσ⁽⁵⁾≤ _∞

0

dσ_ν = _∞

0

ρ_ν(t)dt <∞.

We also have_x

0 ρ(t)dt=σ(x)−σ(0)∀x >0.

On the other hand, the proof of Theorem 2.6.3 in [1] shows that (6)

_∞

0

t^jdσ =yj, ∀j∈Z+.

Using again (5) and the definition ofσ, too, we obtain 0≤dσ ≤dσν, i.e.,

(5) 0≤

Ddσ ≤

Ddσ_ν, ∀D⊂[0,∞[ open subset.

Since dσ = ρdt, dσν(t) = ρν(t)dt, both are regular measures, so that (5) leads to

(5) 0≤

Edσ≤

Edσ_ν, ∀E ⊂ [0,∞[ Borel subset.

In particular, dσ dσ_ν. It follows that there exists ˜h ∈ L¹_ν([0,∞[) such that

dσ = ˜hdν, 0≤˜h≤1 dt-a.e.

(this follows from (5)). Changing (if necessary) the values of ˜h on a set of Lebesgue measure zero, we obtain a Borel functionh= ˜h dt-a.e. The conclu- sion is that

yj (6)

= _∞

0

t^jdσ= _∞

0

t^j˜h(t)dν = _∞

0

t^jh(t)dν, ∀j∈Z+, 0≤h(t)≤1 dt-a.e. The proof is complete.

Next, we consider the Markov moment problem on arbitrary compact subsetsKof Rⁿ, using the fact that any such compact is contained in a semi- algebraic compact subsetK_AofRⁿ, for which one knows the “representation”

of positive polynomials onK_A.

Using results of Schm¨udgen [26] (see also [6], [16]), for semi-algebraic compacts K_A ⊂Rⁿ given by K_A:={t∈Rⁿ;r_j(t)≥0, j = 1, . . . , m},where r1, . . . , rm∈R[t1, . . . , tn], we can state

Theorem 2.2. Let Y be an order-complete vector lattice endowed with a linear topology such that the positive cone Y+ is closed and normal. Let F2 ∈L+(C(K_A), Y) be a linear positive operator. Let {x_j;j∈Zⁿ₊} ⊂C(K_A), xj(t) =x_(j1,...,jn)(t1, . . . , tn) := t^j₁¹. . . t^jnⁿ, t ∈KA, j = (j1, . . . , jn) ∈Zⁿ₊. Let {yj;j ∈Zⁿ₊} ⊂Y. The following statements are equivalent.

(a)There exists a continuous linear operatorF ∈L(C(KA), Y)such that (1) F(x_j) =y_j, ∀j∈Zⁿ₊,

and

(2) 0≤F(ϕ)≤F2(ϕ), ∀ϕ∈(C(K_A))₊.

(b)For any finite subset J0⊂Zⁿ₊, any {λ_j;j ∈J0} ⊂R and any (7) p_r(t) =r_i1(t). . . r_ir(t) =

k∈F

α_kt^k,{i1, . . . , i_r} ⊂ {1, . . . , m},

(is1 = is2 for s1 = s2, s1, s2 ∈ {1, . . . , r}), with F ⊂ Zⁿ₊ a finite subset, we have

(3.1) 0≤

i,j∈J0

λ_iλ_jy_i+j ≤

i,j∈J0

λ_iλ_jF2(x_i+j),

(3.2) 0≤

i,j∈J0

λiλj k∈F

αkyi+j+k

≤

i,j∈J0

λiλj k∈F

αkF2(xi+j+k)

. Proof. (b) ⇒ (a). By the Weierstrass theorem, for any x ∈ C(K_A), there exists a sequence (pl)_l of polynomials functions such that pl →xin the usual norm-topology on C(K_A), i.e., p_l → x uniformly in K_A. It is an easy exercise to prove that there exists another sequence (˜p_l)_l of polynomials such that ˜p_l > x, ∀l ∈Z+, ˜p_l >p˜_l+1, ∀l ∈Z+, ˜p_l ↓ x in C(K_A). In particular, it follows that the subspaceP of polynomial functions onK_Ais dense inC(K_A) and, if x ∈ (C(KA))₊, the polynomials ˜p_l, l ∈ Z+ are positive on KA (this observation is essential, since a result of Schm¨udgen [26] gives the “form” of positive polynomials onKA). On the other hand, by the hypothesis “Y+closed and normal”, any linear positive operator fromC(K_A) into Y is continuous.

DefineF0 :P →Y, F0 i∈J0

λ_ix_j

:=

j∈J0

λ_iy_j, for any finite subsetJ0 ⊂Zⁿ₊. By [26], any polynomial function ˜p which is positive in K_Acan be written as (8) p˜(t) =

k1∈J1

p²_k(t) +

lr∈J2

p²_lr(t)·p_r(t), ∀t∈K_A,

where p_r is given by (7). p_k, p_lr ∈ R[t1, . . . , t_n], J1, J2 ⊂ Z+ finite subsets.

Thus, (3.1), (3.2), (8) and the definition ofF0 yield

(9) 0≤F0(˜p)≤F2(˜p), ∀p˜∈P, p˜(t)>0, ∀t∈K_A.

Using again the fact that Y+ is closed, it is easy to extend (9) to any p˜∈P for which ˜p(t)≥0,∀t∈KA. HenceF0 defines a linear positive operator fromP intoY. SinceP is a majorizing subspace ofC(K_A), by [9], pp. 160, we can extendF0 to F ∈(L(C(KA), Y))₊ andF is continuous. By the definition

ofF0 andF, (1) is obviously verified. Letϕ∈(C(KA))₊ and (˜p_l)_la sequence of polynomials such that p_l ↓ ϕ in C(K_A). From the positivity of F and from (9) we get

0≤F(ϕ)≤F(˜p_l) =F0(˜p_l)⁽⁹⁾≤ F2(˜p_l)→F2(ϕ), l→ ∞, i.e., 0≤F(ϕ)≤F2(ϕ),∀ϕ∈(C(KA))₊.

Thus (b)⇒ (a) is proved, while (a)⇒ (b) is obvious. Note that the ex- tensionF ofF0to the whole ofC(K_A) is unique by the density of polynomials inC(K_A) and because of the continuity of F.

An important particular case of Theorem 2.2 is that of the algebraic compacts KA := {t ∈ Rⁿ;p1(t) = 0, . . . , p_k(t) = 0} = {t ∈ Rⁿ; p_l(t) ≥ 0,−p_l(t)≥0, l= 1, . . . , k}, when (3.2) is superfluous. The first such Markov- type result was proved by Ambrozie, see [2], Theorem 4.3, without using any Hahn-Banach extension type theorem.

On the other hand, as any compact K ⊂ Rⁿ can be embedded into a semi-algebraic compact K_A⊂Rⁿ (K_A may be a closed ball, a simplex, etc.), Theorem 2.2 leads (in the caseY =R) to the following consequence.

Corolary2.3. Let K ⊂Rⁿ be an arbitrary compact and KA a semi- algebraic compact such that K ⊂K_A⊂Rⁿ. Letν be a positive regular Borel measure onK. Let{y_j;j∈Zⁿ₊} ⊂R. The following statements are equivalent.

(a)There exists a Borel function h on K such that (1)

Kt^jh(t)dν =y_j, ∀j∈Zⁿ₊,

(2) 0≤h(t)≤1 ν-a.e. in K.

(b) If K_A := {t ∈Rⁿ;r_j(t) ≥ 0, j = 1, . . . , m}, with the notation from Theorem2.2 we have

(3.1) 0≤

i,j∈J0

λiλjyi+j ≤

i,j∈J0

λiλj

Kt^i+jdν, (3.2) 0≤

i,j∈J0

λ_iλ_j

k∈F

α_ky_i+j+k

≤

i,j∈J0

λ_iλ_j

k∈F

α_k

Kt^i+j+kdν

. Proof. We apply Theorem 2.2 withY =R, F2 :C(KA)→ R,F2(x) :=

Kx|_K(t)dν, where x|_K is the restriction of x to K ⊂K_A. By Theorem 2.2, (b)⇒ (a), there exists a linear (positive) form F ∈(C(K_A))^∗₊ such that (1) and (2) hold. From (2) and the definition ofF2, using the Riesz representation theorem and then the Radon-Nicodym theorem, we conclude that there exists a positive regular Borel measureµon K which represents F|_C(K), with dµ=

hdν, 0≤h(t)≤1,ν-a.e. inK. Finally, we observe that (1) follows from (1), and the proof is complete.

We next consider the “non-compact case”, that can be solved by solving

“countable many compact cases”, when we have a Borel subsetS ⊂Rⁿ such thatS =

m∈ZK_m, whereK_m ⊂Rⁿis a compact subset∀m∈Z. Assume that a positive regular Borel measureν onS is given such that ν(Km1 ∩Km2) = 0 for any m1, m2∈Z,m1=m2.

Theorem2.4. Let{y_j;j∈Zⁿ₊} ⊂R\ {0}. The following assertions are equivalent.

(a)There exists a Borel function h on S such that (1)

St^jh(t)dν =yj, ∀j∈Zⁿ₊,

(2) 0≤h(t)≤1 ν-a.e. in S.

(b)For anym∈Zthere exists a Borel function h_m on K_m andε_m ∈R such that

(1^iv)

Km

t^jh_m(t)dν_m=ε_my_j, ∀j∈Zⁿ₊, ∀m∈Z, (2) 0≤h_m(t)≤1ν_m-a.e. inK_m, ∀m∈Z,

m∈Z

ε_m = 1,

where νm is defined by νm(B) :=ν(B) for any Borel subset B⊂Km, m∈Z. Proof. (b)⇒(a) For anym∈Zdefine ˜h_m:S →[0,1] by ˜h_m(t) :=h_m(t) if t ∈ K_m\

m=mK_m, ˜h_m(t) := 0 if t ∈

m∈Z\{m}K_m. Then define h(t) :=

m∈Z

˜h_m(t),t∈S (this sum has at most one nonzero term). Thush(t)∈[0,1]

ν-a.e. inS. The implication (b)⇒ (a) is proved (see below).

Next, (a) ⇒ (b) is almost obvious. Take h_m :=h|_Km,m∈Z. Then, for any m∈Z, there exists ε_m ∈R such that (1^iv) holds (see below). Using the notation from above and the assumptionν(K_m1 ∩K_m2) = 0 form1=m2, by (a) we have

y_j =

St^jh(t)dν =

m∈Z

Km

t^jh_m(t)dν_m ⁽¹=^iv)

m∈Z

ε_m

y_j, for any fixedj∈Zⁿ₊.

Because of the hypothesisy_j = 0, the conclusion

m∈Zε_m = 1 follows (note that the same assumption allows to choose ε_m :=

Kmt^jh_m(t)dν_m

·y_j⁻¹,

such that (1^iv) is verified for any j ∈ Zⁿ₊ and any m ∈ Z). The proof is complete.

Examples 2.5. (a) Let S1 :=

(t1, . . . , tn) ∈ Rⁿ; t²_n = ⁿ⁻¹

k=1a²_kt²_k , ν1 a positive measure on S1, absolutely continuous with respect to the (n−1)- surface measure dσ on S1.

(b) S2 :=

(t1, . . . , t_n) ∈ Rⁿ; t²_n ≥ ⁿ⁻¹

k=1a²_kt²_k

, a_k ∈ R\ {0}, k = 1, . . . , n−1, ν2 a positive measure on S2, absolutely continuous with respect to the volume measure dt= dt1. . .dt_n on S2.

One can write S_i =

m∈ZK_m,i, where K_m,i := {(t1, . . . , t_n) ∈ S_i; m ≤ t_n≤m+ 1},i= 1,2, m∈Z.

Remark 2.6. The identities t^j = ¹₂

t^j+¹₂₂

−¹₂

t^j−¹₂₂

, j = 1, . . . , n,

∀t∈R, yield p∈

R −

R for any polynomialp∈R[t1, . . . , t_n], where

R is the cone of the sums of squares of polynomials with real coefficients innvariables.

Remark 2.7. The first three moment problems solved in Section 2 are determinate while the last one (solved in Theorem 2.4) seems to be indeter- minate.

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Received 8 January 2007 Ecological University of Bucharest

Department of Mathematics and Computer Science Str. Francez˘a nr. 22, Bucharest, Romania

[email protected] and

University Politehnica of Bucharest Department of Mathematics I

Splaiul Independent¸ei 313 060042 Bucharest, Romania

[email protected] [email protected]