Ukrainian Mathematical Journal, Vol. 55, No. 5, 2003
BRIEF COMMUNICATIONS
CRITERION FOR THE DENSENESS OF ALGEBRAIC POLYNOMIALS IN THE SPACES L p ( R , d µ ) , 1 ≤ p < ∞ A. G. Bakan
UDC 517.5
The criterion for the denseness of polynomials in the space L 2 ( R , d µ ) established by Hamburger in 1921 is extended to the spaces Lp ( R , d µ) , 1 ≤ p < ∞ .
1. Introduction and Main Result Let M ∗ ( R ) denote the set of all positive Borel measures on the real axis all moments of which are finite and whose supports are unbounded. For µ ∈ M ∗ ( R ), we assume that { P n }n ≥ 0 is the corresponding sequence of orthonormal polynomials of the first kind [1] and −1
∞ ρ( µ , z ) : = ∑ | Pn (z) |2 , n=0
z ∈ C.
(1)
A measure µ ∈ M ∗ ( R ) is called definite (in the Hamburger sense) if there is no other measure from M ∗ ( R ) that has the same moments as µ ; otherwise, it is called indefinite. We denote by P [ C ] and P [ R ] all algebraic polynomials with complex and real coefficients, respectively. It is known that, for every 1 ≤ p ≤ ∞, algebraic polynomials with real coefficients are dense in the space of real-valued functions Lp ( R , d µ ) if and only if
P [ C ] is dense in the corresponding space of complex-valued functions [1]. Hence, without loss of generality, we can consider only the real case. In 1921, Hamburger [2] proved the following theorem: ∗ Theorem A. If µ ∈ M ( R ) is definite, then ρ ( µ , z ) = 0 for all z ∈ C exception at most countably
many points z ∈ R where µ ({ z }) > 0. If µ ∈ M ∗( R ) is indefinite, then ρ( µ , z ) > 0 for all z ∈ C . It was also shown in [2] that ρ( µ , z ) = inf ∫ | P( x ) |2 d µ( x ) | P(z) | = 1, P ∈P [ C ] . R
(2)
Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 55, No. 5, pp. 701–705, May, 2003. Original article submitted September 17, 2002. 0041–5995/03/5505–0847 $25.00
© 2003
Plenum Publishing Corporation
847
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A. G. B AKAN
The following fundamental result was obtained by Riesz [3] in 1923: Theorem B. The set P [ R ] is dense in L2 ( R , d µ ) if and only if the measure ( 1 + | x | )– 2 d µ ( x ) is definite. Thus, all well-known criteria for the definiteness of a measure µ can be reformulated as criteria for the denseness of polynomials in the space L2 ( R , ( 1 + | x | )2 d µ ( x ) ) . For arbitrary µ ∈ M ∗ ( R ) , z ∈ C, and 1 ≤ p < ∞, we define the measures d µp ( x ) : = ( 1 + | x | )– pd µ ( x ) ,
d µ ( p) ( x ) : = | x | pd µ ( x )
(3)
and the functions
{
}
ρ p ( µ , z ) = inf || P ||L p (µ p ) | P(z) | = 1, P ∈P [ R ] ,
(4)
Mp ( µ , z ) = sup | P(z) | || P ||L (µ ) ≤ 1, P ∈P [ C ] . p p
(5)
Here and in what follows, || ⋅ ||L p (µ ) , 1 ≤ p ≤ ∞ , denotes the norm in the space Lp ( R , d µ ) . It should be noted that the functions Mp were introduced in connection with the investigation of the Bernstein problem on weight polynomial approximation on the real axis [4] and were called the Hall – Mergelyan majorants. It is easy to verify that 1 2 ≤ Mp ( µ , z ) ≤ ρ p (µ , z) ρ p (µ , z)
(6)
and
{
}
1 = inf || P ||L p (µ p ) | P(z) | = 1, P ∈P [ C ] . M p (µ , z)
(7)
Hence, in terms of polynomial denseness, Theorem A can be reformulated as follows: Theorem A*. Let µ ∈ M ∗ ( R ) and let supp µ be its support. If P [ R ] is dense in L 2 ( R , d µ ) , then ρ2 (µ, z) = 0 for any z ∈ C \ supp µ . If there exists z ∈ C \ supp µ such that ρ 2 ( µ , z ) = 0, then
P [ R ] is dense in L2 ( R , d µ ) . In 1989, Levin (Theorem 1.1 in [5]) and, in 1996, Berg [6, p. 22] extended this result to all spaces Lp ( R , d µ ) for measures µ absolutely continuous with respect to the Lebesgue measure on the straight line and for arbitrary measures µ ∈ M ∗ ( R ), respectively. In the theorems of these authors, the Hall – Mergelyan ma-
CRITERION FOR THE DENSENESS OF ALGEBRAIC POLYNOMIALS IN THE SPACES Lp ( R , d µ) , 1 ≤ p < ∞
849
jorants Mp were used, but, by virtue of (6), the Berg theorem can be formulated in the following equivalent form: Theorem C. Suppose that µ ∈ M ∗ ( R ) and 1 ≤ p < ∞. If P [ R ] is dense in L p ( R , d µ ) , then ρ p (µ, z) = 0 for any z ∈ C \ supp µ . If there exists z ∈ C \ supp µ such that ρ p ( µ , z ) = 0, then P [ R ] is dense in Lp ( R , d µ ) . In 1921, Hamburger [2] also established another criterion for the definiteness of a measure, which, by virtue of Theorem B, can be formulated in the following form: Denote by Cl L p (µ ) A the closure of A ⊆ Lp ( R , dµ) in the space Lp ( R , dµ) . Theorem D. Let µ ∈ M ∗ ( R ) . Then Cl L2 (µ ) P [R] ≠ L2 ( R , d µ ) ⇔ ρ 2 ( µ , 0 ) > 0
and
ρ2 ( µ
(2 )
, 0 ) > 0.
(8)
A simpler proof of Theorem D was obtained by Riesz [7] in 1923 (see also [8, p. 69]). Using the proof of Theorem C obtained by Berg in [6], we extend criterion (8) to all spaces L p ( R , dµ) , 1 ≤ p < ∞, in the following way: Theorem 1. Suppose that µ ∈ M ∗ ( R ) and 1 ≤ p < ∞. Algebraic polynomials are not dense in the space Lp ( R , d µ ) if and only if ρp ( µ , 0 ) > 0
and
ρp ( µ
( p)
, 0 ) > 0.
(9)
2. Preliminary Results In the statement below, we establish a simple but essential property of Hall – Mergelyan majorants. Proposition 1. For arbitrary 1 ≤ p < ∞ and a ∈ R, the function M p ( µ , a + i y ) of variable y ∈ R is even on R and nondecreasing on [ 0 , + ∞ ) . In particular, 1 ≤ Mp ( µ , x ) ≤ Mp ( µ , x + i y ) ρ p (µ , x )
∀x , y ∈ R , 1 ≤ p < ∞.
(10)
Here, 1 / 0 : = + ∞ and + ∞ ≤ + ∞ . Proof. Arbitrarily replacing zeros of a certain polynomial P ∈ P [ C ] by complex conjugate ones, we obtain a set of polynomials π ( P ) that contains only one polynomial P ∗ ∈ π ( P ) all zeros of which lie in the lower half-plane.
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A. G. B AKAN
It is obvious that | Q ( x ) | = | P ( x ) | ∀x ∈ R ∀Q ∈ π ( P ) , and, hence, for every p ≥ 1, we have
|| Q ||L p (µ p ) = || P ||L p (µ p ) ∀Q ∈ π ( P ) . Moreover, for any a ∈ R and y ≥ 0, we establish that | P ∗ ( a + i y ) | ≥ | Q ( a + i y ) | ∀Q ∈ π ( P ) and | P ∗ ( a + i y ) | is a nondecreasing function of y ≥ 0. Hence, for all p ≥ 1, a ∈ R, and y ≥ 0, we get Mp ( µ , a + i y ) = sup | P∗ (a + iy) | || P ||L (µ ) ≤ 1, P ∈P [ C ] . p p
(11)
This equality proves that Mp ( µ , a + i y ) is a nondecreasing function of y ≥ 0. Finally, the obvious equality Mp ( µ , z ) = M p (µ, z ) ∀z ∈ C and inequalities (6) complete the proof of Proposition 1. Below, in the proof of Theorem 1, we use Proposition 1 and the inequality ( p) Mp ( µ , z ) ≥ | z | Mp ( µ , z )
∀z ∈ C ,
p ≥ 1,
which easily follows from the restriction of the polynomial set in (5) to all polynomials vanishing at zero. 3. Proof of Theorem 1 For a function F : R → [ 0 , + ∞ ), we use the notation SF : = { x ∈ R | F ( x ) > 0 }. Sufficiency. Assume that relation (9) is true. Then µ ( p ) ⬅ 0 and, hence, supp µ \ { 0 } ≠ ∅ . By virtue of (4), we have
| P (0 ) | ≤
|| P ||L p (µ p ) ρ p (µ , 0)
,
| P (0 ) | ≤
|| P ||L
( p) p (µ p )
ρ p (µ ( p) , 0)
∀P ∈ P [ R ] .
(13)
By virtue of the Hahn – Banach theorem and inequalities (13), there exist functions fq ∈ Lq ( R , d µp ) \ { 0 } and
(
gq ∈ Lq R, d µ (pp)
) \ {0 },
1 / p + 1 / q = 1 ( 1 < q ≤ ∞ ) , such that
P (0 ) =
∫ P(t) fq (t)d µ p (t)
R
=
∫ P(t)gq (t)d µ p
( p)
(t )
∀P ∈ P [ R ] .
(14)
R
Applying the first equality in (14) to polynomials vanishing at point zero, we obtain t fq (t )
∫ P(t) (1 + | t |) p d µ(t)
= 0
∀P ∈ P [ R ] .
(15)
R
If
(
)
µ S| fq | \ {0} > 0, then t f q ( t ) / (1 + | t | )p ∈ Lq ( R , d µ) \ {0},
(
)
and, by virtue of (15),
Cl L p (µ ) P [R] ≠
Lp ( R , d µ) . In the case where µ S| fq | \ {0} = 0, we have µ ( { 0 } ) > 0, 0 ∈ S| fq | , and, by virtue of (14), µ ({ 0 }) = 1 / fq ( 0 ) . Then equalities (14) mean that
CRITERION FOR THE DENSENESS OF ALGEBRAIC POLYNOMIALS IN THE SPACES Lp ( R , d µ) , 1 ≤ p < ∞
0 =
∫ P(t) ϕ p (t)d µ(t)
∀P ∈ P [ R ] ,
ϕp( t ) : =
fq (t ) − | t | p gq (t )
R
It is easy to verify that ϕp ∈ Lq ( R , d µ ) and if d µ0 ( x ) : = d µ ( x ) –
(1 + | t |) p
851
.
1 δ( x ) , where δ ( x ) is the Dirac funcfq (0)
tion at point zero, then, for any ε > 0, we have
|| ϕ p ||qLq (µ) ≥
+ε
∫ | ϕ p (t)|
−ε
q
dµ 0 (t ) +
1 | ϕ p (0) |q ≥ fq ( 0 ) q – 1 > 0 fq (0)
if p > 1, and || ϕ1 ||L∞ (µ ) ≥ f∞ (0 ) > 0 if p = 1. Thus, ϕ p ∈ Lq ( R , d µ ) \ { 0 } and, hence, Cl L p (µ ) P [R] ≠ Lp ( R , d µ ) .
Necessity. Let Cl L p (µ ) P [R] ≠ Lp ( R , d µ ) . Then, by virtue of the Hahn – Banach theorem, there exists a function gq ∈ Lq ( R , d µ ) \ { 0 } , 1 / p + 1 / q = 1, 1 < q ≤ ∞ , such that
∫ P(t)gq (t)d µ(t)
∀P ∈ P [ C ] .
= 0
(16)
R
Under these conditions, the function ϕq ( z ) : =
∫
R
gq (t ) t−z
d µ(t )
is analytic on C \ R and is not identically equal to zero. Hence, there exists λq ∈ [ 1, 2 ] such that ϕq ( i λq ) ≠ 0. Furthermore, it is easy to conclude from (16) that, for any z ∈ C \ R and P ∈ P [ C ] , one has P ( z ) ϕq ( z ) : =
∫
R
P(t )gq (t ) t−z
d µ(t ) .
(18)
Setting z = i λq in (18), we obtain
|| gq ||Lq (µ) | P ( i λq ) | ≤ ϕ q (i λ q )
1/ p
| P(t ) | p d µ(t ) ∫ p R | t − i λ q |
≤
2 || gq ||Lq (µ ) ϕ q (i λ q )
|| P ||L p (µ p ) .
(19)
Thus, Mp ( µ , i λq ) < ∞ and, in view of (10), 1 / ρp ( µ , 0 ) ≤ Mp ( µ , i λq ) < ∞. This relation, by virtue of the inequalities 1 1 ≤ Mp ( µ ( p ) , i λq ) ≤ M (µ , i λ q ) < ∞ , |i λ q | p ρ p (µ ( p) , 0) which follow from (10) and (12), yields inequalities (9). Theorem 1 is proved.
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A. G. B AKAN
REFERENCES 1. N. I. Akhiezer, Classical Moment Problem [in Russian], Fizmatgiz, Moscow (1961). 2. H. Hamburger, “Über eine Erweiterung des Stieltjesschen Momentenproblems,” Math. Ann., 81, 235–319 (1920); 82, 120–164, 168–187 (1921). 3. M. Riesz, “Sur le probleme des moments et le theoreme de Parseval correspondant,” Acta Litt. Acad. Sci. Szeged., 1, 209–225 (1923). 4. P. Koosis, The Logarithmic Integral I, Cambridge Univ. Press, Cambridge (1988). 5. B. Ya. Levin, “Completeness of systems of functions, quasianalyticity, and subharmonic majorants,” Zap. LOMI, 170, 102–156 (1989). 6. C. Berg, “Moment problems and polynomial approximation,” Ann. Fac. Sci. Univ. Toulouse Stieltjes Spec. Issue, 9–32 (1996). 7. M. Riesz, “Sur le probleme des moment. Troisieme note,” Arc. Mat. Astronom. Fys., 17, No. 16 (1923). 8. J. Shohat and J. Tamarkin, The Problem of Moments, American Mathematical Society, Providence (1950).
