On the Lebesgue Constant of Weighted Leja Points for Lagrange Interpolation on Unbounded Domains

This work focuses on weighted Lagrange interpolation on an unbounded domain, and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subset of the real line, and can be extended to unbounded domains with the introduction of a weight function $w:\mathbb{R}\rightarrow [0,1]$. Due to a simple recursive formulation in one dimension, such abscissas provide a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares, and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes.

1. Introduction. The Lebesgue constant for a countable set of nodes provides a measure of how well the interpolant of a function at the given points compares to best polynomial approximation of the function. We are especially interested in how this constant grows with the number of interpolation nodes, i.e., the corresponding degree of the interpolating polynomial, in an unbounded domain. As such, in this work we analyze the Lebesgue constant for a sequence of weighted Leja points on the real axis. Leveraging results from weighted potential theory [4], and orthogonal polynomials with exponential weights [2], we show that the Lebesgue constant for the weighted Leja points grows subexponentially with the number of interpolation nodes.
The standard Leja sequence on [−1, 1] ⊂ R is defined recursively: given a point x 0 , for n = 1, 2, . . . , define the next Leja point as |x − x j | . (1.1) There is still some ambiguity in this definition, since the maximum may be attained at several points. For the purposes of this work, we may choose any maximizer x n without affecting the analysis. In addition, by introducing a weight function w : R → [0, 1], we may also define the Leja sequence for weighted interpolation on the real line. Given a point x 0 , for n ≥ 1 we recursively define: As above, any maximizer is suitable, so we are not worried about the ambiguity in this definition. The works [1,3] show that a contracted version of the weighted Leja sequence (1.2) is asymptotically Fekete. Specifically, this means that we first multiply the weighted Leja sequence by a contraction factor, i.e., x n,j := n −1/α x j , j = 0, . . . , n, (1.3) for some appropriate real number α = α(w) > 1, depending on the weight w. The discrete pointmass measures µ n giving weight 1/(n + 1) to each of the first n + 1 contracted Leja points, i.e.,

Lagrange Interpolation and Leja
Points. In this section we introduce the problem of weighted Lagrange interpolation of a function on the real line. We also discuss the Lebesgue constant for a set of interpolation points, and show how it relates to the best approximation error. Finally, in §2.1 we describe our main contribution, which involves a theoretical estimate of the growth of the Lebesgue constant of the weighted Leja sequence versus of the number of interpolation points. More specifically, in Theorem 2.1 we prove that the Lebesgue constant of the weighted Leja points grows subexponentially.
To make the setting precise, assume we are given a continuous function f on R that we would like to interpolate. In other words, we have a set of n + 1 points, {x k } n k=0 ⊂ R, and the values {f (x k )} n k=0 at each of those points. Lagrange interpolation constructs a polynomial I n [f ], of degree n, that matches f at every interpolation point, i.e., The fundamental Lagrange basis functions for {x k } n k=0 are defined as: These functions satisfy l n,k (x j ) = δ j,k for all j, k = 0, . . . , n. The unique Lagrange interpolant of degree n for f is then given by Given an appropriate weight function w : R → [0, 1], to estimate the w-weighted approximation error for this interpolation scheme, we define P n = span{x j } n j=0 to be the space of polynomials of degree at most n over R, and let p n be an arbitrary element of P n . Then the error in the norm of L ∞ (R), with · ∞ := · L ∞ (R) , is given by where the quantity is called the Lebesgue constant. In contrast to the case of unweighted Lagrange interpolation on a bounded domain, here the Lebesgue constant explicitly involves the weight function w.
In the inequality (2.3), we may take the infimum over all p n ∈ P n , to see that the Lebesgue constant relates the error in interpolation to the best approximation error by a polynomial in P n : (2.5) Thus, we see that the problem of constructing a stable and accurate Lagrange interpolant consists in the construction of a set of interpolation points for which L n does not grow too quickly.
2.1. Our contribution. In this work we prove the following result: Theorem 2.1. Let α > 1 and assume w : R → [0, 1] is a weight function of the following form Then the Lebesgue constant for the weighted Leja sequence (1.2), defined on R, grows subexponentially with respect to the number of interpolation points n , i.e., The rest of this paper is devoted to the proof of Theorem 2.1. Similar to the case of unweighted Leja points [5,6], in §3, we explore the connection between polynomials and weighted potentials, and show how classical weighted potential theory can be used to understand the asymptotic behavior (with respect to n) of an n th degree polynomial with roots at the contracted Leja points. While these techniques give us most of the result, the final part of the proof requires an explicit estimate on the spacing of the weighted Leja nodes, which is developed in §4. Finally, in §5, we combine the spacing result and weighted potential theory to complete the proof of Theorem 2.1.

Weighted Potential Theory.
In this section, we state some necesary definitions and results from weighted potential theory, which will be the main tools we use to prove Theorem 2.1. For more details, we refer the interested reader to [4]. The class of weights used in this paper, defined in (2.6), are a subset of the well-studied Freud weights [2]. From (2.6), note first that we may extend Q to be a function on C, and that w has the following properties: 1. The extended weight function w : C → [0, 1] is continuous in C.
Furthermore, we also define the Mhaskar-Rhamanov-Saff number a n = a n (w), as the unique solution to the equation (see [4, Corollary IV.1.13]): This number a n has a few special properties which we use in the following analysis. First, the weighted sup-norm of an n th degree polynomial on R is realized on the compact set [−a n , a n ], i.e., for all p n ∈ P n , and |p n (x)|w(x) < p n w ∞ for |x| > a n [4]. Second, from [2, p. 27], a n → ∞ at approximately the rate n 1/α , i.e., Here, and in what follows, for two sequences a n , b n , we write a n ∼ b n if and only if there exist constants C 1 , C 2 > 0, independent of n, such that C 1 ≤ a n /b n ≤ C 2 . Let M(R) be the collection of all positive unit Borel measures µ with Supp(µ) ⊆ R. For µ ∈ M(R) and x, t ∈ R, define the weighted energy integral We also define the logarithmic potential by The goal of weighted potential theory is to find and analyze the measure µ ∈ M(R) that minimizes the weighted energy integral I w (µ). The following theorem may be found in general form in [4, Theorem I.1.3], and is presented here for the specific case (2.6) of a continuous, admissible weight w on R. Theorem 3.1. Let w be a continuous, admissible weight function on R ⊂ C, and define Then we have the following properties: • The quantity V w is finite.
• There exists a unique measure µ w ∈ M(R) such that and the equilibrium measure µ w has finite logarithmic energy, i.e., • Let F w be the modified Robin constant for w, given by The logarithmic potential U µw is continuous for z ∈ C and, moreover, for every x ∈ Supp(µ w ) ⊂ R, Proof. The first two statements are quoted directly from, and proved in, [4, Theorem I.1.3]. To prove the third statement, we note that C \ R has exactly two connected components, namely {Im(z) > 0} and {Im(z) < 0}, and that of course every point in Supp(µ w ) ⊂ {Im(z) = 0} is a boundary point for both of these sets. Thus, by [4, Theorem I.5.1], U µw is continuous on Supp(µ w ). Hence, from [4,Theorem I.4.4], U µw is continuous on all of C, and (3.7) holds for every x ∈ Supp(µ w ) ⊂ R.

Weighted Fekete Points.
In this section we describe the connection between Leja points and the weighted equilibrium measure µ w . For n ≥ 0, let T n denote a general set of points in R with cardinality |T n | = n + 1, and let w be an admissible weight on R. We say a set of n + 1 points is (weighted-)Fekete if it maximizes the quantity: . (3.8) It is known that the Lebesgue constant for a set of Fekete points F n satisfies Furthermore, we also know that for a sequence of Fekete point sets, where V w , as defined in (3.5), is the weighted logarithmic capacity for R with respect to w. For interpolation schemes, we are also interested in arrays of points with similar asymptotic properties to Fekete points in the limit as n → ∞, since this is a necessary condition for a sequence of points to have a well-behaved Lebesgue constant. Thus, we make the following definition: Note that a sequence of interpolation points may be asymptotically Fekete but not Fekete, i.e., without satisfying (3.8) for any n ∈ N. The following lemma, first proved in [1] in a more general setting than the one considered here, and later in [3], indicates that the contracted Leja sequence distributes asymptotically like the Fekete points.

Potentials and Polynomials.
Taken together, the previous two lemmas tell us that the discrete point-mass measures associated with the contracted Leja sequence converge weak * to the weighted equilibrium measure for R corresponding to the weight w given in (2.6). This fact enables us to make a key connection between potential theory and Leja points, and provides the basis for the proof of Theorem 2.1.
With {x n,j } n j=0 as in (1.3), define P n,k to be the polynomial with roots at each of the n contracted Leja points x n,j , j = 0, . . . , k − 1, k + 1, . . . , n, i.e., and let µ n,k be the measure which assigns mass 1 n to each of the roots of P n,k , i.e., Then, taking the logarithm of |P 1/n n,k w|, we convert the polynomial into a discrete logarithmic potential with respect to the measure µ n,k , i.e., By Lemma 3.3, the weighted Leja sequence is asymptotically Fekete, and therefore we have µ n,k → µ w in the weak * sense. This connections allows us to exploit potential theory to understand the asymptotic behavior of weighted polynomials. In particular, by considering polynomials with roots at the contracted Leja points (1.3), we explicitly explore this asymptotic behavior in the following two lemmas, which will be an essential part of the proof of Theorem 2.1. Lemma 3.5. Given ε > 0, there exists an N ∈ N such that, for n > N and 0 ≤ k ≤ n, Proof. See Appendix B.1. Lemma 3.6. For all ε > 0, there exist δ > 0 and N ∈ N, such that for n > N , and 0 ≤ k ≤ n, w(x n,k ) n |x n,k −xn,j|≥δ Proof. See Appendix B.2.
4. Spacing of the weighted Leja points. The goal of this section is to state and prove a result regarding the spacing of the contracted Leja sequence. This will be crucial to the final step in the proof of Theorem 2.1.
Theorem 4.1. Let w and α > 1 be as in (2.6), and let n ∈ N, with 0 ≤ i, j ≤ n. Then, for some constant C > 0, independent of n, the contracted Leja sequence (1.3) satisfies the spacing property C|x n,i − x n,j | ≥ n −1 . (4.1) To prove Theorem 4.1, the main spacing result for the contracted Leja sequence, we use a weighted version of the classical Markov-Bernstein inequalities, which relate norms of polynomials 6 to norms of their derivatives. First, for a n and Q as defined in (3.1) and (2.6), respectively, define the function ϕ n (t) = |t − a 2n ||t + a 2n | n (|t + a n | − a n ζ n )(|t − a n | + a n ζ n ) , (4.2) where ζ n = (αn) −2/3 . theorem 1. The function ϕ n plays the same role as the function  .2). The main fact we need for this proof is a Bernstein-type inequality for weighted polynomials, which can be found, for instance, in [2, Theorem 10.1]: for any polynomial p n of degree n ≥ 1, there exists some C, independent of p n and n, such that Hence, for any polynomial p n of degree n, and t ∈ R, |(p n (t)w(t)) ′ | ≤ C n a n p n w ∞ . Given 0 ≤ j < n, by the mean value theorem, there exists a point t between x j and x n such that Notice that for 0 ≤ j < n, P n (x j ) = 0 by definition. Then from (4.4),

which implies
C|x n − x j | ≥ a n n .
Now let i, j ≤ n, and assume without loss of generality that i < j. The above calculation shows that which, up to constants independent of n, is the desired result.
5. Proof of Theorem 2.1. In this section, we prove our main theorem concerning the growth of the Lebesgue constant of the weighted Leja sequence. Similar to the proof in the unweighted case given in [5,6], we separate the proof of the theorem into several smaller components.
To begin, we first show that the Lebesgue constant of the weighted Leja sequence on the real line is equal to a weighted Lebesgue constant of the contracted Leja sequence (1.3) on a fixed compact set. To do this, we first use the fact from (3.2) that supremum a w-weighted, n th degree polynomial is realized in the compact set [−a n , a n ]. Then, we exploit the specific form (2.6) of our weight function to show that which in turn implies that Finally, let 0 < c < ∞ be the smallest constant such that sup n (n −1/α a n ) ≤ c. Furthermore, for any n = 1, 2, . . . , let q n ∈ P n . Define q n (x) ∈ P n to be the unique polynomial such that q n (x) = n −n/α q n (n 1/α x) Then we calculate w(n −1/α y) n q n (n −1/α y) From this string of inequalities we have that for any n ≥ 1, and q n ∈ P n , sup x∈R w(x) n |q n (x)| = sup x∈K w(x) n |q n (x)| .
Thus, to show that this Lebesgue constant grows at a subexponential rate, the above calculation indicates that we only need to show that lim n→∞ n max k=0,...,n sup y∈Kw |w(y) n n j=0, j =k (y − x n,j )| w(x n,k ) n n j=0, j =k |x n,k − x n,j | .
We have reduced the proof to essentially a problem in weighted potential theory. The convergence of the limits (5.7) and (5.9) follow directly from Lemmas 3.5 and 3.6, respectively, which are proven in the appendix. Thus, we have left to show statement (5.10), which requires a more direct approach. We explicitly use the spacing of the contracted Leja sequence from Theorem 4.1, and find that the remainder of the estimate involving A 2 (n, k, δ) follows from this spacing lemma.
By assuming δ < 1, it is clear that the product A 2 (n, k, δ) is always less than one. Therefore, the following theorem will complete the proof of Theorem 2.1.
At least one of these sets may be empty, and in that case we simply set the corresponding product equal to one. Now, let m 1 , m 2 be the cardinality of the sets X 1 (k, δ) and X 2 (k, δ), resp., and label these points in the following way x n,k − δ ≤ x n,im 1 ≤ . . . ≤ x n,i1 < x n,k < x n,j1 < . . . < x n,jm 2 < x n,k + δ.
(5.12) Now, using (5.11) and Sterling's approximation, we see that As τ → 0 + , the function ( τ C ) 2τ → 1. Thus, we let τ < min{C, 1 e } be small enough so that Let m be the number of Leja points within the interval {t ∈ R : |x n,k − t| < δ}. According to [5,Theorem 2.4.5], for our chosen τ > 0, we can choose N ∈ N, and δ 0 > 0 such that if n > N , and We know f (x) = x x is a decreasing function on (0, 1 e ), and hence from (5.16) and (5.17), this implies that xn,j∈X1(k,δ) which is the desired result for X 1 (k, δ) and X 2 (k, δ). This completes the proof.
6. Conclusion. In this work, we considered the properties of Leja points for weighted Lagrange interpolation on an unbounded domain. Due to their nested structure, simple recursive formulation, and generally stable behavior, Leja points show promise for high-dimensional interpolation methods. Our contribution to this area was to prove that the Lebesgue constant for the weighted Leja sequence grows subexponentially with respect to the number of interpolation nodes. Furthermore, we proved a theorem regarding the separation of the weighted Leja points.
Of course, a subexponential rate encompasses a wide range of growth, potentially much bigger than the optimal Lebesgue constant O(log n). On the other hand, our experience with Leja points indicates that the Lebesgue constant grows linearly, i.e., O(n), with respect to the number of nodes. Our proof in this paper relies on potential theory, which gives only asymptotic estimates of growth. We expect that a more explicit estimate of the Lebesgue constant would require different techniques, and this is the subject of future work.
B.2. Proof of Lemma 3.6. Proof. Let ε > 0 be given, and K w = Supp(µ w ) as above. To prove the lemma, it will be enough to show that log w(x n,k ) n |x n,k −xn,j|≥δ |x n,k − x n,j | 1/n − (−F w ) < ε.
First, notice that log |x n,k −xn,j|≥δ |x n,k − x n,j | 1/n = |t−x n,k |≥δ log |t − x n,k | dµ n,k (t), and of course log (w(x n,k ) n ) 1/n = −Q(x n,k ). Furthermore, we have already seen from (3.7) that U µw (x) + Q(x) = F w , ∀x ∈ K w ⊂ R. (B.7) Thus, we estimate log w(x n,k ) n |x n,k −xn,j|≥δ |x n,k − x n,j | The last term is equal to zero, so it is left to show that there exists a δ > 0 and N ∈ N independent of n and k such that A < ε/2 and B < ε/2. The proof for the quantity A is shown in the proof of Theorem 3.5, and the proof for B follows essentially from the proof of [5, Theorem 2.4.6], so we forgo the details here.