On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants

: In this article, we analyze a rate of attraction of poles of an approximated function to poles of incomplete multipoint Padé approximants and use it to derive a sharp bound on the geometric rate of convergence of multipoint Hermite-Padé approximants to a vector of approximated functions in the Montessus de Ballore theorem when a table of interpolation points is Newtonian.


Introduction
The study of convergence of sequences of rational functions with a fixed number of free poles is a classical problem in approximation theory.Montessus de Ballore's classical theorem is one of the main objects in this problem.To clarify the goal of our work, let us remind the definition of classical Padé approximants: Let n be the space of all polynomials of degree at most n.Consider a power series For given ∈ ∪ n m , 0, { } we can find ∈ P n and ∈ The rational function  We denote by Q m f the monic polynomial whose zeros are the poles of f in R f 0, m ( ( )) counting their multi- plicities and by f m ( ) the set of all distinct zeros of Q .The pioneering result in the study of row sequences of classical Padé approximants is the Montessus de Ballore theorem [1] stated that if > R f 0 0 ( ) and f has exactly m poles in R f 0, , m ( ( )) then for each compact and where ⋅ K ‖ ‖ denotes a sup-norm on K and ⋅ ‖ ‖ denotes the coefficient norm in the space m .The reason behind the convergence in equation (1.3) is that as → ∞ n , all poles of the approximants R n m , converge to all poles of f in R f 0, m ( ( )) as shown in equation (1.4).This allows R n m , to imitate f in R f 0, m ( ( )) with the geometric rate of convergence in (1.3).For a precise history, we would like to point out that Montessus de Ballore [1] verified the ≤ inequalities in equations (1.3) and (1.4) and the ≥ inequalities in equations (1.3) and (1.4) were later obtained by Gonchar [2,3].
Let E be a bounded continuum subset of the complex plane such that ⧹E is simply connected.By E , ( ) we denote the space of all functions holomorphic in some neighborhood of E. Set The vector of rational functions is called a multipoint Hermite-Padé (MHP) f with respect to m and α.
For given n m , ( ), the approximant in equation (1.5) always exists but is not uniquely determined.Without loss of generality, we can assume that Q n m , is a monic polynomial that has no common zero simultaneously with all P n k m , , .In all what follows, m remains fixed and When E is a disk centered at 0 and all interpolation points α n k , are zero, this MHP approximation becomes classical type II Hermite-Padé approximation [9].Most studies of classical Hermite-Padé approximants concentrated on diagonal or near-diagonal sequences and their applications.There are few studies devoted to row sequences [7,8,[10][11][12].In the direction of row sequences, Graves-Morris and Saff [10] proved an analogue of the Montessus de Ballore theorem for classical Hermite-Padé approximants under the concept of polewise independence.Later, such study received a renewed interest by Cacoq et al. [7,8].In [7], they refined the estimates of convergences of

{
} in [10].In [8], they proved a reciprocal of the Montessus de Ballore theorem for classical Hermite-Padé approximants.See also further investigations [11,12] involving the relationship between asymptotic properties of zeros of Q n m , and singularities of f.Recently, convergence problems of various generalizations of classical Hermite-Padé approximants on row sequences were considered in [13][14][15][16].The results in [16] generalize the ones in [8] to MHP approximants.In particular, Bosuwan et al. [16] computed the exact rate of convergence of and estimated the rate of convergence of . The aim of this study is to show that the rate of convergence of [16] (see (1.11) below) is sharp.
Let Φ be the conformal bijection from ⧹E to ⧹ 0, 1 . It is commonly known that there exist tables of points α satisfying the condition: uniformly on compact subsets of ⧹E , where c denotes some positive constant and G z ( ) is some holomorphic and nonvanishing function on ⧹E , see Chapters 8-9 in [17].Moreover, it is easy to check that equation (1.6) implies that uniformly on compact subsets of ⧹E.
The shape of domain of convergence of

{
} can be explained using the map Φ.For each > ρ 1, we define as the level curve of index ρ and the canonical domain of index ρ, respectively.Given ∈ f E , ( ) denote by ρ f 0 ( ) the index of the largest canonical domain D ρ to which f can be holomorphically extended and by ρ f m ( ) the index of the largest canonical domain D ρ to which f can be meromorphically extended with at most m poles counting multiplicities.Additionally, denote by D f m ( ) the canonical domain of the index ρ f .m ( ) In the study of row sequences of MHP approximants of f , we consider "a system pole" of f instead of a pole of f.The concept of system pole was originated by Cacoq et al. [8].In [16], Bosuwan et al. modified their definition to fit the study of MHP approximants.
) if τ is the largest positive integer such that for each = s τ 1,…, , there exists at least one polynomial combination of the form: for a pole at = z λ of exact order s.
Note that a concept of system pole depends on both a vector of functions f and a multi-index m.However, in this study, we consider a row sequence of MHP of f with respect to m, so sometimes, we will omit the multi- index m.When = d 1, the concepts of pole and system pole coincide.However, when > d 1, the situation is not quite the same.Poles of the individual functions f k may not be system poles of f and system poles of f may not be poles of any of the functions f k as given in examples in [8].More importantly, when studying the con- vergence of

{
} instead of considering poles of each individual function f , k we will consider poles of the polynomial combinations of all components f in equation (1.8).
Let τ be the order of λ as a system pole of f .For each = s τ 1,…, , let ρ f m , λ s , ( )denote the largest of all the numbers ρ g , s ( ) where g is a polynomial combination of type (1.8) that is holomorphic on a neighborhood of ) except for a pole at = z λ of order s.Then, we define and To explain the domain of convergence of ) be the largest canonical domain in which all the poles of f k are system poles of f with respect to m, their order as poles of f k does not exceed their order as system poles, and f k has no other singularity.By ρ f m , k ( ), we denote the index of this canonical domain.
).For each = j N 1,…, , let τ ˆj be the order of λ j as pole of f k and τ j be its order as a system pole.By assumption, ≤ τ τ and let D f m * , k ( ) be the canonical domain with this index.By Q , m f we denote the monic polynomial whose zeros are the system poles of f with respect to m taking account of their order.The set of distinct zeros of Q m f is denoted by m f .The Montessus de Ballore theorem for MHP is the main result in [16].
where ⋅ ‖ ‖ denotes the coefficient norm in the space | | m .
Moreover, if either (a) or (b) takes place, then and for any compact subset where ⋅ K ‖ ‖ denotes the sup-norm on K, and if Let us give one concrete example explaining the definition of system pole and Theorem 1.1.
⊂ α E be a table of interpolation points satisfying (1.6), and where the branch of the square root is chosen so that [ ] It is not difficult to see that 2 and 6 are system poles of order 1 of f with respect to m, which further implies that f has exactly m | | system poles with respect to m.Notice that 6 is not in the domains of meromorphy of f 1 and f 2 (extended from the set E) because of the essential singularity at 5. However, by (1.9) and (1.10), the zeros of Q n m , can detect both 2 and 6 with the following rate of convergence: ) contain only one pole of order 1 at 2. Consequently, The purpose of this article is to show that the inequality in equation (1.11) is equality for a certain class of (as defined below) and a Newtonian table α.Let B be a subset of the complex plane .Denote by B ( ) the class of all coverings of B by at most a countable collection of disks.We define the 1-dimensional Hausdorff content of the set B as follows: where U j | | is the radius of the disk U j .
Definition 3. We say that a compact set ⊂ K is h-regular if for each ∈ z K 0 and for each > δ 0, it holds Our main result is stated in the following: An outline of this article is as follows.Section 2 contains a study of incomplete multipoint Padé approximants.We will use the study in Section 2 to prove Theorem 1.2 in Section 3.

Incomplete multipoint Padé approximants
An incomplete multipoint Padé approximation defined below plays an important role in proving Theorem 1.2 (see also [7] and [8], where the authors used a concept of incomplete Padé approximation to study a similar Multipoint Hermite-Padé approximants  5 problem for classical Hermite-Padé approximation).Our proofs in this section are strongly influenced by the methods employed in [7] We normalize Q n m , in the aforementioned definition to be monic.Note that for ) corre- sponding to f .k We will make use of this fact in the proof of Theorem 1.2 below.The concept of convergence in Hausdorff content is defined as follows.
be a sequence of functions defined on a domain ⊂ D and g be a function defined on D. We say that converges in 1-dimensional Hausdorff content to the function g inside D if for every compact subset K of D and for each > ε 0, we have ) and q n m , is normalized in terms of its zeros λ n j , so that Now, we discuss some upper and lower estimates on the normalized polynomials q n m , in equation (2.1).Let , n ℓ (they are not necessarily distinct and ≤ m n ℓ ).We cover each λ j with the open disk centered at λ j of radius ∕ ε m 6 ( ) and denote by J f ε 0, ( ) the union of these disks.For each ≥ n m, we cover each λ n j , with the open disk centered at λ n j ) and denote by J f n ε , ( ) the union of these disks.Set Applying the monotonicity and subadditivity of h, we have The normalization of q n m , produces useful upper and lower bounds of q n m , , which are given in the following α E be a table of points, m and m* be nonnegative integers with ≥ ≥ m m* 1, ⊂ K be a compact set, and > ε 0 be fixed.Then, there exist constants > c 0 1 and > c 0 2 independent of n such that for all sufficiently large n, where ⋅ K ‖ ‖ is the sup-norm on K and the second inequality is meaningful when K ε ( ) is a nonempty set.
Proof.The proof is very standard so we leave this to the reader.□ From now on, we will denote some constants that do not depend on n by c c c , , ,…. ) for f.Then, Proof.Let Q m* be the monic polynomial whose zeros are poles of f in D f m* ( ) taking account of their order.Clearly, where the second integral in equation (2.5) is zero because has a zero at ∞ of order at least two.
Let K be a compact subset of D f m* ( ), > ε 0, and ∈ ρ ρ f 1, m* ( ( ))be the index such that D ρ contains K and all poles of f in D f .m* ( ) From (2.6), we have for all ∈ z K ε , ( ) Therefore, by the uniform convergence of (1.7) and Lemma 1, and letting → ρ ρ f , m* ( ) where A n m , is some constant and ) is a polynomial of degree at most − m m* normalized as in equa- tion (2.1).

Proof. By (ii) in Definition 4, we know
.
By equation (2.8), 2 For any compact set ⊂ B , we put Note that using the computation similar to equation (2.2), we have and it is easy to check that the normalization of − b n m m , * implies that for any compact set K of and for any > ε 0, there exist constants ) for f.Then,

Moreover, the sequence
3), (2.9), and (2.10), it follows that and the series diverges.Therefore, the sequence 3), (2.9), and (2.10), we have From this, the series in equation (2.11) converges uniformly on K ε ( ) ) for some function φ.By Gonchar's lemma (see Lemma 1 in [3]), φ is (except on a set of h-content zero) a meromorphic function with at most m poles in D f * .m ( ) By the uniqueness of the limit In [2], Gonchar defined two indicators used to quantify a rate of attraction of a point to poles of classical Padé approximants.In what follows, we use the same indicators to quantify such rate of attraction of a pole of ℓ denote the collection of zeros of q n m , (repeated according to their multiplicity).Define ℓ , the product is taken to be 1).To define the second indicator, we suppose that for each n, the points in are enumerated in nondecreasing distance to the point λ.We set The second indicator, a nonnegative integer μ λ , ( ) is defined as follows.
Proof.Let > ε 0 and λ be a pole of f in D f * m ( ) of order τ.Let > r 0 be sufficiently small so that λ r , ( ) contains no other pole of f and , has more than τ zeros in λ r , , ( ) so we name these zeros λ λ λ , ,…, , n indexed in nondecreasing distance from λ, i.e., For any ρ with it follows from (2.7), (2.11), and (2.12) that , n n It is known that the norm of the holomorphic component of a meromorphic function may be bounded in terms of the norm of the function and the number of poles (see Theorem 1 in [18]).Thus, using this and (2.14), we obtain (2.15) for some constant c 2 and for sufficiently large n.Therefore, from (2.15) and the maximal modulus principle, we have for some constant c 3 and for sufficiently large n.Since ≠ p λ 0, ( ) replacing z by λ in equation (2.16) and taking the limit supremum of the n-root from both sides as → ∞ n , we obtain Because of the arbitrariness of r ε , , and ρ, we obtain To do that, we will show that Using (2.16) and Cauchy's integral formula, we obtain that its kth derivative satisfies an inequality like (2.16).If we put = z λ in the corresponding inequality, we have λ λ liminf 0.
In particular, Proof.We assume that Using the aforementioned equation, Inequality (2.17), and the mathematical induction on k, we obtain for some constant c 2 and for sufficiently large n.Inequality (2.19) and the expression imply that there exists ∈ Combining this and the fact that ( ) ( ) ( ) we obtain equation (2.18).Finally, let us show the last statement.Assume that We will make use of this in the proof of Theorem 1.2.
for any compact subset  ) cannot contain "a singularity of f k , which is not a system pole of f " or "a system pole of f with its order as a pole of f k greater than its order as a system pole."Consequently, ) This additionally implies that Because such λ is also a system pole of order at least τ ˆ, from Corollary 2.2 in [16], we have So, applying (2.3), (2.9), and (2.10), we obtain , are uniquely determined.For a function f as in equation (1.1), we denote by R f 0 ( ) the radius of the largest open disk at the origin to which f can be extended analytically and by R f m ( ) the radius of the largest open disk at the origin to which f can be extended meromorphically with at most m poles counting multiplicities.Set

mf
In all what follows, m remains fixed and ≥ mth row sequence of classical Padé approximants of f.

Theorem 1 . 1 .
Suppose (1.6) takes place.Let ∈ E f d ( ) and fix a multi-index ∈ m d .Then, the following two assertions are equivalent: (a) f has exactly m | | system poles with respect to m counting multiplicities.(b) For all sufficiently large n, the denominators Q n m , of MHP approximants of f are uniquely determined and there exists a polynomial Q m of degree m | | such that In order to prove our main lemmas (Lemmas 5 and 6) in this section, we will apply the spherical normalization to Q n m , in Definition 4. For fixed nonnegative integers ≥ ≥ m m* 1 and for each integer ≥ n Padé approximant, where p n m , and q n m , are the polynomials such that =

− b n m m ,*ε 0 .
is a polynomial such that it is normalized as in equation (2Using a setting similar to J f n ε , ( ), for each ≥ n m, let ′ J f ( ) denote the union of all disks centered at all zeros of − implies that all singularities of f k inside | | D f * k m ( ) are zeroes of Q m f which are system poles f by Theorem 1.1.Therefore, the boundary of D f m * , k (

) where we recall that λ λ ,…, N 1 are
all poles of f k in D f m , k () and τ ˆj is the order of λ j as a pole of f k .Since ⊂ by Lemma 2) and Q n m , is a denominator of an incomplete multipoint Padé approximant of type n from equations (1.9) and (2.20), for each pole λ of order τ ˆof f k inside D f m , , poles λ of f k in D f m , . is a set of h-content zero and the compact set K is h-regular, where we denote the denominator of R n k m , , by q n k m , , normalized as in equation (2.1).We may write − ≤ − + − .
and ⊂ α E be a Newtonian table.Fix two positive integers m and m* such that ≥ ≥ m m* 1.Consider a corresponding sequence of incomplete multipoint Padé approximants.For each ≥ n m, we have .14)for some constant c 1 and for sufficiently large n.
Let us continue using the notation in the proof of Lemma 5 in the following lemma and its proof.
) .□ is the largest index of canonical domain inside of which such h-convergence is valid,