A Characterization of Polynomial Density on Curves via Matrix Algebra

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces $L^{2}(\mu)$, with $\mu$ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix to the measure $\mu$. In order to do it, in the more general context of Hermitian positive semidefinite matrices we introduce three indexes $\gamma(\mathbf{M})$, $\lambda(\mathbf{M})$ and $\alpha(\mathbf{M})$ associated with different optimization problems concerning these matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non empty interior using the index $\gamma$ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated to a measure in terms of the index $\gamma$ that will allow us to give an alternative proof of Thomson's theorem in \cite{Brennan} by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the frame of Hermitian Positive Definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.


Introduction
Thorough the paper we consider positive Borel measures µ which are finite and compactly supported in the complex plane.We always consider nontrivial measures, that is, measures with an infinite amount of points in their support.The problem of completeness of polynomials in the Hilbert space L 2 (µ) is the following: for a certain measure µ, are polynomials dense in the space L 2 (µ)?In other words, denote by P 2 (µ) the closure of the polynomials in the space L 2 (µ), the question is under what conditions the equality L 2 (µ) = P 2 (µ) is true.In the particular case of µ being the two-dimensional Lebesgue measure on an arbitrary domain G and L 2 (G) the associated Hilbert space, the classical results of approximation by polynomials can be seen in e.g.[9], where it is explored the question of which assumptions on G will be assumed in order to have polynomials density in L 2 (G).The related questions about the existence of approximation rational, entire, or meromorphic were solved by the great theorem of Mergelyan in 1951 which completes a long chain of theorems about approximation by polynomials.
The problem of density of polynomials is also an interest topic in the Theory of orthogonal polynomials associated with a measure.Indeed, in the particular case of orthogonal polynomials in the unit circle the well known Szegö theory ( see e.g.[11], [12]) deals with the problem of polynomial approximation using proper tools of orthogonal polynomials.
On the other hand, in [7] a necessary condition was provided to assure polynomial approximation using the behaviour of the smallest eigenvalues of the finite sections of the moment matrix associated to a measure.Along this work we follow this matrix approach in order to obtain the main results in this paper.
Thorough this paper we consider infinite positive definite hermitian matrices M = (c i,j ) ∞ i,j=0 .As in [1], [7] an Hermitian positive definite matrix (in short, an HPD matrix) defines an inner product , in the space P[z] of all polynomials with complex coefficientes in the following way: if p(z) = n k=0 v k z k y q(z) = m k=0 w k z k then, p(z), q(z) = vMw * , being v = (v 0 , . . ., v n , 0, 0, . . .), w = (w 0 , . . ., w m , 0, 0, . . . ) ∈ c 00 , where c 00 is the space of all complex sequences with only finitely many non-zero entries.The associated norm is p(z) 2 = p(z), p(z) for every p(z) ∈ P[z].An interesting class of HPD matrices are those which are moment matrices with respect to a measure µ, i.e., HPD matrices M = (c i,j ) ∞ i,j=0 such that there exists a representating measure µ with infinite support on C and finite moments for all i, j ≥ 0, Our aim here is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 (µ), with µ a compactly supported measure in the complex plane, in terms of properties of the associated matrix M. In order to do it, in the more general context of Hermitian positive semidefinite matrices we introduce three matrix indexes γ(M), λ(M) and α(M), each one related with different optimization matrix problems.Among these indexes we highlight the index γ that as we have realized will be essential to characterize the polynomial density in our context.The other index λ is related to the asymptotic behaviour of the smallest eigenvalues in our previous works (see [7]).These indexes will be introduced in the first section some properties of them will be given.We also provide an application to the index λ to some problems of perturbations of measures in the same direction as in [3].
In the following section we consider the case when the Hermitian semidefinite positive matrices are moment matrices associated with a measure µ with compact support in the complex plane.Our main result is a characterization of completeness of polynomials in the associated space L 2 (µ), in the case of Jordan curves with 0 in its interior, in terms of the index γ of the moment matrix associated with the measure µ.
In the last section we give our main result which is characterization of density of polynomials on Jordan curves with non empty interior in terms of an specific index related to the index γ.Moreover, we provide a matrix algebra point of view of the notion of bounded point evaluation of a measure.This will lead us to obtain a new proof of Thomson's theorem in [2], in the particular context of Jordan curves with non empty interior, using our techniques and our results.
Finally, we point out that our approach is based in matrix algebra tools in the frame of general HPD and in the computation of certain indexes related to some optimization problems for infinite matrices.This point of view would allow solving certain matrix optimization problems in terms of the Theory of orthogonal polynomials and on the other hand would let obtaining results of interest concerning orthogonal polynomials using the matrix optimization tools.

New indices of an HPD matrix and connections with the polynomial approximation
In this section we introduce some indices associated with general Hermitian semipositive definite matrices.Let M = (c i,j ) ∞ i,j=0 be an infinite Hermitian matrix, i.e. c i,j = c j,i .We say that an infinite Hermitian matrix M is positive definite if (in short an HPD matrix) if |M n | > 0 for all n ≥ 0, where M n is the truncated matrix of size (n + 1) × (n + 1) of M. In an analogous way, if |M n | ≥ 0 for all n ≥ 0 we say that M is an Hermitian semi-positive definite matrix (in short HSPD).In the sequel we use the same notation as in [7], we denote by (1, v) ≡ (1, v 1 , . . ., v n , 0, 0, ) for every v = (v 1 , v 2 , . . . ) ∈ c 00 and by (v, 1, 0, . . . ) ≡ (v 0 , . . ., v n−1 , 1, 0, . . . ) Definition 1.Let M be an infinite Hermitian semi-definite positive matrix.We define This index always exists and γ(M) ≥ 0. Definition 2. Let M be an infinite Hermitian semi-definite positive matrix.We define λ(M) = inf{vMv * ; vv * = 1, v ∈ c 00 }.This index always exists and λ(M) ≥ 0.
Remark 1.Note that there is an important link between eigenvalue problems and optimization is the Rayleigh quotients.Indeed, for Hermitian matrices M n it is well known that if we define We denote by λ n the smallest eigenvalue of M n as in [7]; that is if Moreover, the sequence {λ n } ∞ n=0 is an non non decreasing sequence and λ(M) = lim n→∞ λ n .
Definition 3. Let M be an infinite Hermitian semi-definite positive matrix.We define α(M) This index always exists and α(M) ≥ 0.
Next we relate these three indexes.
Proposition 1.Let M be an infinite Hermitian semi-definite positive matrix.Then, Proof.We first show i).Let v = 0 and consider the normalized vector v (vv * ) 1/2 .By the definition of λ(M) we have that By taking in mind that for any t > 0 it holds (tv)M(tv) * = t 2 vMv * , then it easily follows that vMv * ≥ λ(M)vv * . Consequently, By taking the infimum we obtain λ(M) ≤ γ(M).In the same way, we have that for every v ∈ c 00 , and consequently λ(M) ≤ α(M).
Remark 2. The equality is not true in general, even for Toeplitz matrices.For instance, consider the matrix .

By induction we obtain that |T
, where κ is the leading coefficient of orthonormal polynomial sequence.We get the associated weight w(t) of the Toeplitz matrix T, It follows by [10] Therefore by [7] it can be obtained Remark 3. Using the results in [7] it can be easily obtained that in the case of Toeplitz matrices T the indexes γ(T), α(T) coincide, that is: Let T be an infinite Toeplitz SHPD matrix.Then, In the set of Hermitian semi-definite positive infinite matrices we may define an order in the following way: we say that We have the following results: We give some applications of the above result to some perturbation results in the same lines as [3].Let σ be a non trivial positive measure with support in T, in [3] it is obtained that if the measure σ verifies Szegö condition and σ is the perturbed measure of σ by the normalized Lebesgue measure in the unit circle, that is d σ = dσ + r dθ 2π for r > 0, then σ also verifies Szegö condition.Using our techniques we generalize this result pointing out that there is no need to require that σ verifies Szegö condition since the conclusion is true always.Indeed we have: Corollary 1.Let σ 1 , σ 2 positive positive measures with support on T. Assume that one of them verifies Szegö condition, then the measure σ := σ 1 + σ 2 verifies Szegö condition.In particular, if d σ = dσ + r dθ 2π with r > 0 for some positive measure with support in T then σ verifies Szegö condition.
Remark 4. Note that in Corollary 1 it is not required that both measures are non trivial; indeed, we may consider a perturbation by a finite amount of atomic points.
In the particular case of d σ = dσ +r dθ 2π , obviously the normalized Lebesgue measure in the unit circle verifies Szego condition and consequently σ also verifies Szegö condition independently of σ.
From now on we consider now an infinite HPD matrix M.This matrix induces an inner product in the vector space c 00 .In this way the space c 00 endowed with such a norm is a prehilbert space with the prehilbertian norm v 2 M = vMv * .We consider the completion of this space with such norm that we denote by P 2 (M); we may apply Gram-Schmidt orthogonalization procedure to the canonical algebraic basis {e n } ∞ n=0 in c 00 and we obtain the unique orthonomal basis (2) α(M) = inf{ dis 2 (e n , [e 0 , . . ., e n−1 ]), n ∈ N}.
From the results in [7] we have the following infinite dimensional version of the result in the case of Hermitian positive definite matrices.
Proposition 4. Let M be an HSPD matrix and let {v 0 , v 1 , v 2 , . . .} be the orthonormal basis in P 2 (M) with respect the inner product induced by M with v i = (v 0,i , . . ., v i,i , 0, . . . ) for i ∈ N 0 and v i,i > 0.Then, 3. When the Hermitian semi-definite positive matrices are moment matrices.
In this section we consider the most important example of Hermitian definite positive matrices which are the moment matrices with respect a Borel non trivial compactly supported measure µ in the complex plane M(µ).In this case, the space c 00 is replaced by the space of polynomials P[z] via the identification v = (v 0 , . . ., v n , 0, 0, . . .
The associated norm in P[z] with respect to M := M(µ) is the usual norm of the polynomials in the space L 2 (µ); that is for every p(z As usual the completion of the space of polynomials in the space L 2 (µ) is denoted by P 2 (µ), {ϕ n (z)} ∞ n=0 is the sequence of orthonormal polynomials and {Φ n (z)} ∞ n=0 is the associated sequence of monic orthogonal polynomials.We denote by P 2 0 (µ) the completion of polynomials vanishing at zero.The well known extremal properties of the monic polynomials and the n-kernels are just obtained by reformulating in this context Proposition 4 above, which, as we have pointed out, are results obtained by algebraical proofs in the more general context of the general Hermitian definite matrices.Indeed, reformulating Lemma 1 we obtain: Lemma 2. Let µ be a measure compactly supported measure with infinite support in the complex plane and let M := M(µ) be the associated moment matrix.Then, γ(M) = dis 1, P 2 0 (µ) .We need the following lemma: Lemma 3. Let µ be a non trivial positive compactly supported measure in C with 0 / ∈ supp(µ).The following are equivalent, (1) γ(M) = 0. ( Proof.First of all there exist R > 0 and α > 0 such that α ≤ |z| ≤ R for every z ∈ supp(µ).Consequently, for every k ∈ Z and for every v 0 , v 1 , . . ., v n ∈ C, n ∈ N it follows As a consequence, for compactly supported measures with 0 / ∈ supp(µ) the condition γ(M) characterizes completeness of polynomials in the closed subspace of Laurent : Corollary 2. Let µ be a non trivial positive compactly supported measure in C with 0 / ∈ supp(µ).The following are equivalent, (1) γ(M) = 0. ( In particular, for non trivial positive measures σ supported in the unit circle it is well known that Laurent polynomials are dense in L 2 (σ) and therefore the condition γ(T) = 0 characterizes completeness of polynomials in L 2 (σ).More generally this result will be true whenever Laurent for measures µ such that polynomials are dense in L 2 (µ).Moreover, we have: Let Γ be a Jordan curve such that 0 ∈ int Γ and let µ be a measure with support in Γ and associated moment matrix M. The following are equivalent: (1) γ(M) = 0.
Proof.In [9] the following consequence of Mergelyan's theorem is given: if Γ is a Jordan curve, 0 ∈ int Γ, f is continuous on Γ, then for every ǫ > 0 there exists a P (z) = N n=−N a n z n such that |f (z) − P (z)| < ǫ for every z ∈ Γ.This means that C[z, z −1 ] is dense in the space of continuous functions on Γ with the uniform norm, that is, for every f continuous in Γ and ǫ > 0 there exists g and consequently C[z, z −1 ] is dense in the space of continuous functions in L 2 (µ).Since for compactly supported measures continuous functions are dense in L 2 (µ) we obtain that As a consequence of the above results we have the well known consequence of Szegö theorem for measures supported in the unit circle: ( The result is a consequence of Theorem 1 and Proposition 4 since 4. Bounded point evaluations from the matrix algebra point of view.Thomson's theorem revisited. We first recall the definitions of bounded point evaluation.Let µ be a non trivial positive measure with support on C. Recall ( see e.g.[5] ) that a point z 0 ∈ C is a bounded point evaluation (in short, bpe) for P 2 (µ) if there exists a constant C > 0 such that for every polynomial p(z) Moreover, the point z 0 ∈ C is an analytic bounded point evaluation (in short an abpe) if there exists a constant C > 0 and ǫ > 0 such that for every w ∈ C with |w − z 0 | < ǫ and for every polynomial p(z) it holds Remark 5. Of course, an analytic bounded point evaluation is a bounded point evaluation.The converse is not true; indeed, any atomic isolated point is a bounded point evaluation but it is not an analytic bounded point evaluation.
It is well known that if a point z 0 ∈ supp(µ) is an atomic point of µ, that is µ({z 0 }) > 0, then it is a bounded point evaluation for P 2 (µ).We prove it for the sake of completeness Lemma 4. Let z 0 be an atomic point of a measure µ with µ({z 0 }) = α > 0. Then z 0 is a bounded point evaluation for P 2 (µ) with constant C = α −1/2 .
Proof.Let p(z) be a polynomial.Then, Therefore, Remark 6.It is important to point out the following remark in order to avoid confusions.In several references (see e.g.[13], [2]) as a consequence of Thomson's theorem a dichotomy is established in the following way: let µ be a compactly supported measure in C: then either P 2 (µ) = L 2 (µ) or there exist bounded point evaluations.This is not a dichotomy in the strict sense of the word, that is, in the sense that if one of them is true then the other must not happen.Indeed, there are examples of compactly supported measures µ such that P 2 (µ) = L 2 (µ) and nevertheless there exist bounded point evaluations.Consider the example in [7] of any measure with 0 ∈ supp(µ) being a point mass and such that supp(µ) is a compact set with empty interior and with K c a connected set.In particular, consider the sequence of infinite points z n = e 2iπ n in the unit circle, for example with weights p n = 1/2 n , n ≥ 1.In this case by Mergelyan theorem (see e.g.[9]) P 2 (µ) = L 2 (µ) and nevertheless there exists a bounded point evaluation.Indeed, every z n is a bounded point evaluation for P 2 (µ) since it has been proved in Lemma 1.We give the formulation of Thomson's theorem as appears in (see e.g.[13], [2]): Theorem 2. Let µ be a compactly supported measure in C. If P 2 (µ) = L 2 (µ), then there exist a bounded point evaluation for P 2 (µ).
Our aim in this section is to prove this theorem for measures supported in Jordan curves but with the novelty of using techniques from the matrix algebra and using infinite HPD matrices.In order to do it, we first give a new approach of bounded point evaluations for a measure, and more generally, for infinite HPD matrices.Definition 4. Let M be an HPD matrix and let P 2 (M) the closure of the polynomials with the inner product induced by M. Let z 0 ∈ C, we say that z 0 is a bounded point evaluation for P 2 (M) if there exists a constant C > 0 such that for every polynomial p(z) it holds Remark 7. Obviously, in the case of M being a moment matrix associated with a measure µ the notion of bounded point evaluation for P 2 (M) coincides with the usual of bounded point evaluation for P 2 (µ).
We need to introduce a new index for a given z 0 ∈ C: Definition 5. Let M an HPD matrix and z 0 ∈ C and k z 0 = {z k 0 } ∞ k=0 , we define Remark 8.Note that γ z 0 (M) ≥ 0 for every z 0 ∈ C and in the particular case that z 0 = 0 then γ z 0 (M) = γ(M).
Next we prove: Lemma 5. Let M be an HPD matrix.Then the following statements are equivalent: (1) z 0 is a bounded point evaluation for P 2 (M).
Proof.Assume first that z 0 is a bounded point evaluation of P 2 (M) with constant C, then for every (v 0 , v 1 , . . ., v n , 0, 0, . . . ) and p(z In particular, if On the other hand, if γ z 0 (M) > 0 and p(z) = n k=0 v k z k with (v 0 , . . ., v n , 0, . . . ) ∈ c 00 , either p(z 0 ) = 0 and obviously p(z 0 ) ≤ p(z) 2  P 2 (M) , or p(z 0 ) = 0 and the vector w = (w i ) i=0 ∈ c 00 defined by for each i ≥ 0 verifies wk z 0 = 1 and consequently Therefore, and z 0 is a bounded point evaluation for P 2 (M) with constant 1 (γ z 0 (M)) 1/2 .Remark 9. Note that proof of the above Lemma gives us information about the constant of the bounded point evaluation; indeed, z 0 is bounded point evaluation for P 2 (M) with constant γ z 0 (M).
We may generalize the notion of kernels in the context of infinite HPD matrices.More precisely, for an infinite HPD matrix M we may define the associated kernels: for every z, w such that the series converges.In this context, the extremal property for polynomials can be reformulated as: Lemma 6.Let M be an HPD matrix and let {ϕ n (z)} ∞ n=0 the sequence of orthonormal polynomials associated with M.Then, the following are equivalent: Using the notation for polynomials we may rewrite: First consider a polynomial q(z) = n k=0 v k z k and we express it in terms of the orthonormal basis, that is, q(z) = n k=0 w k ϕ k (z), then by using the Cauchy-Schwartz inequality Then, And by taking the infimum all over the polynomials of degree n, On the other hand, if we consider the polynomial q We summarize all the equivalent notions of bounded point evaluations for an HPD matrix in the following proposition: Corollary 4. Let M be an HPD matrix and let {ϕ n (z)} ∞ n=0 the sequence of orthonormal polynomials associated with M.Then, the following are equivalent: (1) z 0 is a bounded point evaluation of P 2 (M). ( Proof.We only need to prove (2) ⇐⇒ (4 the last identity is the Schur complement, that says As a consequence of this result we obtain our main result which is the following proof of Thomson's theorem for measures supported in Jordan curves via an algebraical way.This let us to provide an algebraical characterization of density of polynomials in terms of an index of the moment matrix associated with the measure: Theorem 3. Let Γ be a Jordan curve such that z 0 ∈ int Γ and let µ be a measure with infinite support in Γ with associated moment matrix M.Then, the following statements are equivalent (1) γ z 0 (M) > 0.
(3) z 0 is a bounded point evaluation of P 2 (M).
Proof.Let M be the moment matrix associated with image measure μ obtained after a similarity map, ϕ(z) = αz + β onto C, is applied to the measure µ and Γ the image Jordan curve.We first prove that γ( M) = γ z 0 (M).
It is know ( see e.g.[6]) that the expression that relates the matrices M n and M n , it is given by M n = A n (α, β)M n A * n (α, β), where A n (α, β) is defined as in [6], .
Note that if we choose α = 1 and a translation β = −z 0 , then 0 ∈ int( Γ), we obtain: ).Since M is HPD, all its sections are invertible and we can write It is clear that We finish with some applications to our results: Corollary 5. Let Γ be an analytic Jordan curve with non empty interior and let µ be a measure with support in Γ with weight function w(z) defined on Γ positive and continous and.Then L 2 (µ) = P 2 (µ).
And also from Theorem 3 the following result is obvious: Corollary 6.Let Γ be a Jordan curve and µ be a measure with support in Γ. Assume that L 2 (µ) = P 2 (µ), then every z 0 ∈ int Γ is a bounded point evaluation of µ.
every n ∈ N 0 .We denote by [e 1 , e 2 , . . .] M the closed vector subspace generated by the set of vectors e ′ n s with the norm induced by the matrix M. Proposition 3. Let M be an Hermitian definite positive matrix.Let {e n } ∞ n=0 be the canonical basic sequence in c 00 , then (1) γ(M) = dis 2 (e 0 , [e 1 , e 2 , . . .] M ).

Corollary 3 .
Let σ be a non trivial positive measure with support in C and {ϕ n (z)} ∞ n=0 the associated sequence of orthonormal polynomials associated.Then (1) Polynomials are dense in L 2 (T).