In this paper, we consider positive Borel measures
, which are finite and compactly supported in the complex plane. We always consider non-trivial measures, that is, measures with an infinite amount of points in their support. The problem of completeness of polynomials in the Hilbert space
is the following: For a certain measure
, are polynomials dense in the space
? In other words, denote by
the closure of the polynomials in the space
; the question is under what conditions the equality
is true. In the particular case of
being the two-dimensional Lebesgue measure on an arbitrary domain G
the associated Hilbert space, the classical results of approximation by polynomials can be seen in, e.g., the work of Gaier [1
], who explored the question of which assumptions on G
will be assumed to have polynomials density in
. 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., [2
]) deals with the problem of polynomial approximation using proper tools of orthogonal polynomials.
On the other hand, in [4
], a necessary condition was provided to assure polynomial approximation using the behavior of the smallest eigenvalues of the finite sections of the moment matrix associated with a measure. Along this work, we follow this matrix approach in order to obtain the main results in this paper.
Throughout this paper, we consider infinite Hermitian positive definite matrices
. As in [4
], a Hermitian positive definite matrix (HPD matrix) defines an inner product
in the space
of all polynomials with complex coefficients in the following way: if
is the space of all complex sequences with only finitely many non-zero entries. The associated norm is
An interesting class of HPD matrices are those which are moment matrices with respect to a measure
, i.e., HPD matrices
such that there exists a measure
with infinite support on ℂ and finite moments for all
Our aim here is to obtain conditions to assure polynomial approximation in Hilbert spaces
a compactly supported measure in the complex plane, in terms of properties of the associated matrix
. To do it, in the more general context of Hermitian positive semidefinite
matrices, we introduce two matrix indexes
, each related with different optimization matrix problems. Among these indexes, we highlight the index
that is essential to characterize the polynomial density in our context. The other index
is related to the asymptotic behavior of the smallest eigenvalues in our previous works (see [4
]). These indexes are introduced in Section 2
and some of their properties are given. We also provide an application to the index
to some problems of perturbations of measures in the same direction as in [6
In Section 3
, we consider the case when the Hermitian positive semidefinite matrices are moment matrices associated with a measure
with compact support in the complex plane. Our main result is a characterization of completeness of polynomials in the associated space
, in the case of Jordan curves with 0 in its interior, in terms of the index
of the moment matrix associated with the measure
In Section 4
, we give our main result, which is the characterization of density of polynomials on Jordan curves with non-empty interior in terms of another index related to the index
. Moreover, we provide a matrix algebra point of view of the notion of bounded point evaluation of a measure. This leads us to obtain a new proof of Thomson’s theorem in [7
], 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.
2. New Indices of an HPD Matrix and Connections with the Polynomial Approximation
In this section, we introduce some indices associated with general Hermitian positive semidefinite matrices. Let
be an infinite Hermitian matrix, i.e.,
. We recall that an infinite HPD
matrix verifies that
is the truncated matrix of size
. In an analogous way, if
, we say that
is a Hermitian positive semidefinite matrix (HPSD). In the sequel, we use the same notation as in [4
]; we denote by
Definition 1. Letbe an infinite Hermitian positive semidefinite matrix. We define
This index always exists and.
Definition 2. Letbe an infinite Hermitian positive semidefinite matrix. We define
This index always exists and.
Remark 1. Note that an important link between eigenvalue problems and optimization is the Rayleigh quotients. Indeed, for Hermitian matrices, it is well-known that, if we definefor, thenandgives the extreme eigenvalues of. We denote bythe smallest eigenvalue ofas in ; that is, ifis the euclidean norm in,
moreover, the sequenceis an non-increasing sequence and
Next, we relate these indexes.
Proposition 1. Letbe an infinite Hermitian semi-definite positive matrix. Then, Proof.
and consider the normalized sequence
. By the definition of
we have that
By taking in mind that, for any
holds, then it easily follows that
By taking the infimum, we obtain □
Remark 2. The equalityis not true in general, even for Toeplitz matrices. For instance, consider the matrix
By induction, we obtain that; consequently,, whereis the leading coefficient of orthonormal polynomial sequence and. We obtain, which is the absolutely continuous part of the associated measure of the Toeplitz matrix,
It follows from  that
Therefore, according to , it can be obtained that
Note that this example shows how to use our techniques to solve certain matrix optimization problem.
In the set of infinite Hermitian positive semidefinite matrices, we may define an order in the following way: we say that if for every . The following results are directly obtained from the definition.
Letbe infinite Hermitian positive semidefinite matrices with; then:
We give some applications of the above result to some perturbation results in the same lines as [6
be a non-trivial positive measure with support in
, which verifies Szegö condition, that is
is the absolutely continuous part of
in the Lebesgue decomposition (see [10
]). In [6
], it is obtained that, if the measure
verifies Szegö condition and
is the perturbed measure of
by the normalized Lebesgue measure in the unit circle, that is
also verifies Szegö condition. Using our techniques, we generalize this result pointing out that there is no need to require that
verifies Szegö condition since the conclusion is true always. Indeed we have the next result.
Letpositive measures with support onAssume that one of them verifies Szegö condition, then the measureverifies Szegö condition. In particular, ifwithfor some positive measure with support in, thenverifies Szegö condition.
Let be the Toeplitz positive semidefinite moment matrices associated with . Assume that verifies Szegö condition, then . By Lemma 1, it follows that and consequently verifies Szegö condition. □
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 , obviously the normalized Lebesgue measure in the unit circle verifies Szegö condition and consequently also verifies Szegö condition independently of .
From now on, we consider an infinite HPD matrix
. This matrix induces an inner product in the vector space
. In this way, the space
endowed with such a norm is a vector space with an inner product that is not necessarily complete. The norm induced by the inner product
We consider the completion of this space with such norm that we denote by ; we may apply Gram–Schmidt orthogonalization procedure to the canonical algebraic basis in and we obtain the unique orthonormal basis with for and . Consider the orthogonal monic vector. It is clear that for every . We denote by the closed vector subspace generated by the set of vectors with the norm induced by the matrix . We denote the distance of a vector v to a subspace as .
Proposition 2. Letbe a Hermitian definite positive matrix. Letbe the canonical basic sequence in; then,
From the results in [4
], we have the following infinite dimensional version of the result in the case of Hermitian positive definite matrices.
Proposition 3. Letbe an HPSD matrix and letbe the orthonormal basis inwith respect the inner product induced bywithforand. Then,
where the left side is zero if.
3. HPSD Matrices Which Are Moment Matrices
In this section, we consider the most important example of HPD matrices which are the moment matrices with respect to a Borel non-trivial compactly supported measure
in the complex plane
. In this case, the space
is replaced by the space of polynomials
via the identification
The associated norm in
with respect to
is the usual norm of the polynomials in the space
, that is for every
As usual, the completion of the space of polynomials in the space is denoted by , is the sequence of orthonormal polynomials, and is the associated sequence of monic orthogonal polynomials. We denote by 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 3 above, which, as pointed out, are results obtained by algebraic proofs in the more general context of the general Hermitian definite matrices. Indeed, rewriting the definition of in terms of polynomials, we obtain:
Lemma 2. Let μ be a measure compactly supported measure with infinite support in the complex plane and letbe the associated moment matrix. Then,
We need the following lemma:
Let μ be a non-trivial positive compactly supported measure in ℂ with. The following are equivalent,
First, there exist
. Consequently, for every
and for every
, it follows that
if and only if
, for all
Consequently, for compactly supported measures with , the condition characterizes completeness of polynomials in the closed subspace of Laurent polynomials in denoted by :
Let μ be a non-trivial positive compactly supported measure inwith. The following are equivalent,
In particular, for non-trivial positive measures supported in the unit circle, it is well-known that Laurent polynomials are dense in and therefore the condition characterizes completeness of polynomials in . More generally, Corollary 2 will be true whenever Laurent polynomials are dense in . Moreover, we have:
Let Γ be a Jordan curve such that and let μ be a measure with support in Γ and associated moment matrix . The following are equivalent:
], the following consequence of Mergelyan’s theorem is given: if
is a Jordan curve,
, and f
is continuous on
, then for every
there exists a
. This means that
is dense in the space of continuous functions on
with the uniform norm. Therefore,
is dense in the space of continuous functions in
. Since, for compactly supported measures, continuous functions are dense in
, we obtain that
if and only if
as we required. □
As a consequence of the above results, we have the well-known consequence of Szegö theorem for measures supported in the unit circle:
Let σ be a non-trivial positive measure with support inandthe associated orthonormal polynomial sequence. Then, the following conditions are equivalent
Polynomials are dense in.
The result is a consequence of Theorem 1 and Proposition 3 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
be a non-trivial positive measure with support on ℂ. Recall (see, e.g., [11
]) that a point
is a bounded point evaluation
if there exists a constant
such that for every polynomial
Moreover, the point
is an analytic bounded point evaluation
) if there exists a constant
such that for every
and for every polynomial
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 is an atomic point of , that is , then it is a bounded point evaluation for . We prove it for the sake of completeness
Letbe an atomic point of a measure μ with. Then,is a bounded point evaluation forwith constant.
be a polynomial. Then,
Our aim in this section is to prove a theorem that is close in spirit to Thomson’s theorem [7
], but for measures supported in Jordan curves, with the novelty of using techniques from the matrix algebra and using infinite HPD matrices. To do it, we first give a new approach of bounded point evaluations for a measure, and, more generally, for infinite HPD matrices.
Definition 3. Letbe an HPD matrix and letbe the closure of the polynomials with the inner product induced by. Let; we say thatis a bounded point evaluation forif there exists a constantsuch that for every polynomialit holds that
Obviously, in the case ofbeing a moment matrix associated with a measure μ, the notion of bounded point evaluation forcoincides with the usual of bounded point evaluation for.
We need to introduce a new index for a given :
Definition 4. Letbe an HPD matrix,, and. We define
Note that for every and in the particular case that , then .
Next we prove:
Let be an HPD matrix. Then, the following statements are equivalent:
is a bounded point evaluation for .
Assume first that
is a bounded point evaluation of
with constant C
; then, for every
In particular, if
, it holds that
. On the other hand, if
and the vector
is a bounded point evaluation for
Note that proof of the above Lemma gives us information about the constant of the bounded point evaluation; indeed, is bounded point evaluation for with constant .
We may generalize the notion of kernels in the context of infinite HPD matrices. More precisely, for an infinite HPD matrix
, we may define the associated kernels:
such that the series converges. In this context, the extremal property for polynomials can be reformulated as:
Lemma 6. Let be an HPD matrix and let the sequence of orthonormal polynomials associated with . For every ,
Using the notation for polynomials, we may rewrite:
First, consider a polynomial
and we express it in terms of the orthonormal basis, that is,
. Then, by using the Cauchy–Schwartz inequality,
By taking the infimum all over the polynomials of degree n
On the other hand, if we consider the polynomial
, the above infimum is reached at this polynomial since
Then, for every
We summarize all the equivalent notions of bounded point evaluations for an HPD matrix in the following proposition:
Let be an HPD matrix and let be the orthonormal polynomial sequence associated with . Then, the following are equivalent:
is a bounded point evaluation of .
We only need to prove
. In [2
], the expression of the n
-kernel by determinants is given as follows
the last identity is the Schur complement, which 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 algebraic way. This allows us to provide an algebraic characterization of density of polynomials in terms of an index of the moment matrix associated with the measure:
Let Γ be a Jordan curve such that and let μ be a measure with infinite support in Γ with associated moment matrix . Then, the following statements are equivalent
is a bounded point evaluation of .
Let be the moment matrix associated with image measure obtained after a similarity map, onto ℂ, is applied to the measure and the image Jordan curve.
We first prove that
An expression that relates the matrices
is known, which is given by (see, e.g., [12
is defined as in [12
Note that, if we choose
and a translation
, and we obtain:
is HPD, all its sections are invertible, and we can write
It is clear that
It is immediate that
We have also that
Now, the result is consequence of Theorem 1. □
We finish with some applications to our results:
Let Γ be an analytic Jordan curve with non-empty interior and let μ be a measure with support in Γ. Let , where is positive and continuous function on Γ. Then, .
be an arbitrary interior point of
; by using the results in [2
], it follows that
, and, by using Theorem 1, we may conclude
In addition, from Theorem 3, the following result is obvious:
Let Γ be a Jordan curve and μ be a measure with support in Γ. Assume that , then every is a bounded point evaluation of μ.