Boundedness of Lebesgue constants and interpolating Faber bases

We investigate some conditions under which the Lebesgue constants or Lebesgue functions are bounded for the classical Lagrange polynomial interpolation on a compact subset of $\mathbb R$. In particular, relationships of such boundedness with uniform and pointwise convergence of Lagrange polynomials and with the existence of interpolating Faber bases are discussed.


Introduction
be an infinite triangular matrix whose elements (nodes) are real numbers satisfying the condition x k 1 ,n = x k 2 ,n for all distinct k 1 , k 2 ∈ {1, ..., n} and every n ∈ N. Then define the fundamental polynomials l k 0 ,n = l k 0 ,n (M, ·) as l k 0 ,n (x) = l k 0 ,n (M, x) := 1≤k≤n, k =k 0 (x − x k,n ) (x k 0 ,n − x k,n ) , x ∈ R. (1.1) The polynomials l 1,n , ..., l n,n form a basis at the linear space H n−1 of all real algebraic polynomials of degree at most n − 1. In particular, we have l 1,1 ≡ 1. Let X be an infinite compact subset of R. Denote by C X the Banach space of continuous functions f : X → R with the supremum norm f X := sup{|f (x)| : x ∈ X} and write M ⊆ X if M = {x k,n } and x k,n ∈ X for all n ∈ N and k ≤ n.
For f ∈ C X , M ⊆ X and n ∈ N, the Lagrange interpolating polynomial L n (f, M, ·) is the unique polynomial from H n which coincedes with f at the nodes x k,n+1 , k = 1, ..., n + 1. Using the fundamental polynomials we can represent L n (f, M, ·) in the form L n (f, M, ·) = n+1 k=1 f (x k,n+1 )l k,n+1 (M, ·). (1.2) For given X, M ⊆ X, and n ∈ N, the Lebesgue function λ n (M, ·) and the Lebesgue constant Λ n,X (M) can be defined as λ n (M, x) := sup{|L n (f, M, x)| : f X ≤ 1}, x ∈ R, (1.3) and, respectively, as In what follows we will denote by BLC (bounded Lebesgue constants) the set of compact nonvoid sets X ⊆ [−1, 1], for each of which there is a matrix M ⊆ [−1, 1], such that the corresponding sequence (Λ n,X (M)) n∈N is bounded, i.e., Λ n,X (M) < c (1.8) holds for some c > 0 and every n ∈ N.
In the second section of the paper we will describe some details of the well-known interplay between the boundedness of Lebesgue constants Λ n,X (M) and the uniform convergence of Lagrange polynomials L n (f, M, ·). The corresponding relationships of pointwise boundedness of Lebesgue functions λ n (M, ·) with pointwise convergence of these polynomials are also described. Moreover, the second section contains a discussion of the known results describing the smallness of sets belonging to BLC .
In the third section we obtain some new relations between the boundedness of Λ n,X (M) for special interpolating matrices M and the existence of interpolating Faber bases in the space C X .
2 Boundedness and convergence in Lagrange interpolation J. Szabados and P. Vértesi, [16], write: "... in the convergence behavior of the Lagrange interpolatory polynomials ... the Lebesgue functions ... and the Lebesgue constants ... are of fundamental importance...". Proposition 2.1. Let X be an infinite compact subset of R and let M ⊆ X. The following statements are equivalent.
is valid for every f ∈ C X .
(iii) The inequality holds for every f ∈ C X .
Proof. The linear operator L n,M is a projection of C X onto H n . Hence, by Lebesgue's lemma, see [4,Ch. 2,Pr. 4.1], we have the inequality where E n (f ) is the error of the best approximation of f by H n in C X . By the Stone-Weierstrass theorem, the continuous function f is uniformly approximable by polynomials on X, i.e., lim n→∞ E n (f ) = 0. Now (i) ⇒ (ii) follows.
The implication (ii)⇒ (iii) is trivial. Suppose that (iii) holds.To prove (iii) ⇒ (i) note that equality (2.2) implies the boundedness of sequences for every f ∈ C X . Since all L n,M : C X → C X are continuous linear operators and C X is a Banach space, the Banach-Steinhaus theorem gives us the inequality sup n∈N L n,M < ∞.
There is a pointwise analog of Proposition 2.1 Proposition 2.2. Let X be an infinite compact subset of R and let x ∈ X. The following statements are equivalent for every M ⊆ X. holds.
(ii) The limit relation is valid for every f ∈ C X .  At 1931, S. N. Bernstein [2] found that for every This equality together with Proposition 2.2 gives the existence of a point In 1980 P. Erdös and P. Vértesi [5] proved the following This theorem implies the following corollary. If there is M ⊆ [a, b] such that inequality (2.4) holds for every x ∈ X, then X is nowhere dense and Proof. Since the fundamental polynomials are invariant under the affine trasformations of R, we may suppose that a = −1 and b = +1. Now, (2.10) follows from Theorem 2.4. Equality (2.10) implies that the interior of X is empty, IntX = ∅. Since X is compact, we have X = X, where X is the closure of X. Consequently, the equality IntX = ∅ holds, it means that X is nowhere dense.
Corollary 2.6. If X belongs to BLC, then X is nowhere dense in R and its one-dimensional Lebesgue measure is zero.
and the matrix M is defined such that x k,n = x k for all n ∈ N and k ∈ {1, ..., n}, then we evidently have the equalities for every x ∈ X. Consequently, the compactness of X cannot be dropped in Corollary 2.5.
It was proved by A. A. Privalov in [13], that there are a countable set X ⊆ [0, 1] and a positive constant c 1 = c 1 (X), such that 0 is the unique accumulation point of X and the inequality Λ n,X (M) ≥ c 1 ln(n + 1) holds for every n ∈ N and every M ⊆ [−1, 1].
An example of perfect set X ∈ BLC was obtained by S. N. Mergelyan [9]. P. P. Korovkin [8] found a perfect X ⊆ [−1, 1] and a matrix M such that, for every f ∈ C X , the sequence (L n 2 (f, M, ·)) n∈N uniformly tends to f , sup n∈N Λ n 2 ,X (M) < ∞.
At the same paper [8], he wrote that there is a modification of X with bounded sequence of Lebesgue constants. Corollary 2.6 indicates that every X ∈ BLC must be small in a very strong sense. Moreover, the examples of A. A. Privalov, P. P. Korovkin and S. N. Mergelyan show that the properties "be countable" and "belong to the class BLC " not linked too closely.
In the rest of the present section we discuss the desirable smallness of sets in terms of porosity.
Let us recall the definition of the right lower porosity at a point.
Definition 2.9. Let X be a subset of R and let x 0 ∈ X. The right lower porosity of X at x 0 is the number where λ(X, x 0 , r) is the length of the largest open subinterval of the set Replacing (x 0 , x 0 +r) in the above definition by the interval (x 0 −r, x 0 ), we encounter the notion of the the left lower porosity p − (X, x 0 ). The lower porosity of X at x 0 is the number The set X is strongly lower porous if p(X, x 0 ) = 1 holds for every x 0 ∈ X.
Let us consider now a modification of the lower porosity. Write holds for every x 0 ∈ X, then X ∈ BLC.
Proof. It is known that for all n ≥ n 0 and x ∈ X. The boundedness of (Λ n,X (M)) n∈N follows. Thus, X belongs to BLC . Theorem 2.11. There is an infinite strongly lower porous compact set X ⊆ [−1, 1] such that X ∈ BLC.
Proof. Let X be the compact set, constructed by A. A. Privalov in [13]. Then X ⊆ [0, 1] and 0 is the unique accumulation point of X. Note that p − (X, x 0 ) = 1 holds if and only if x 0 is an isolated point of the set (−∞, x 0 ] ∩ X. Hence, for every x 0 ∈ X we evidently have p − (X, x 0 ) = 1. Thus X is strongly lower porous by the definition.

Faber bases and Lagrange polynomials
In what follows we study the boundedness of the Lebesgue constants Λ n,X (M) for the matrices M having the form The obtained results are inspired by some ideas of J. Obermaier and R. Szwarc [10], [11].
Let X be an infinite compact subset of R.  If n ∈ N is given, then the partial sum operator S n,p : C X → C X is a linear operator with the range H n−1 and the domain C X . Similarly, for an interpolation matrix M ⊆ X, the operator, defined by (1.5), has the same range and domain. Moreover, the linear operators S n,p and L n,M are projections on H n−1 , i.e., we have S n,p (p) = L n,M (p) = p for every p ∈ H n−1 . In what follows we study some conditions under which the operators S n,p and L n,M are the same for every n ∈ N.
holds for all f ∈ C X and k ∈ N.
Ifp and (x k ) k∈N satisfy the above condition, then we say thatp is interpolating with the nodes (x k ) k∈N . The following lemma is similar to Proposition 1.3.2 from [14]. for every k ∈ N and j < k.
Proof. Suppose thatp is interpolating with the nodes (x k ) k∈N . We must show that (3.3) holds for all k ∈ N and j < k. Since, for each f ∈ C X , the representation is unique, we have p k = 0 Sincep is interpolating with the nodes (x k ) k∈N , (3.6) implies If p k (x k ) = 0, then p k has k distinct zeros that contradicts the equality degp k = k − 1. Condition (3.3) follows. Let (3.3) hold for all k ∈ N and j < k. Then from (3.4) we obtain for every n ∈ N. Thus,p = (p k ) k∈N is interpolating with the nodes (x k ) k∈N .
Corollary 3.6. Let X be an infinite compact subset of R and letp = (p k ) k∈N be an interpolating Faber basis in C X . Then there is a unique sequence (x k ) k∈N of distinct points of X such thatp is interpolating with nodes (x k ) k∈N .
Proof. Letp be interpolating with nodes (x k ) k∈N . By Lemma 3.5 the point x 1 is the unique zero of the polynomial p 2 , the point x 2 can be characterized as the unique point of X for which p 3 (x 2 ) = 0 and p 2 (x 2 ) = 0 an so on.
Lemma 3.5 implies also the following Proposition 3.7. Let X be an infinite compact subset of R. Ifp = (p k ) k∈N be an interpolating Faber basis in C X with nodes (x k ) k∈N , then for every sequenceλ = (λ k ) k∈N of nonzero real numbers the sequencẽ is also an interpolating Faber basis with the same nodes (x k ) k∈N . Conversely, ifq = (q k ) k∈N andp = (p k ) k∈N are interpolating Faber bases with the same nodes, then there is a unique sequenceμ = (µ k ) k∈N of nonzero real numbers such thatq For given nodes (x k ) k∈N , the interpolating Faber basisp = (p k ) k∈N , if such a basis exists, can be uniquely determined by the natural normalization p k (x k ) = 1 for every k ∈ N. for all k ∈ N and j < k.
The following example gives us another condition of uniqueness of interpolating Faber basis corresponding to given nodes. Recall that a polynomial is monic if its leading coefficient is equal to 1.
Theorem 3.10. Let X be an infinite compact subset of R and let (x k ) k∈N be a sequence of distinct points of X. The following two statements are equivalent.
(i) There is an interpolating Faber basis with the nodes (x k ) k∈N .
(ii) For every f ∈ C X we have where, for each k ∈ N, π k is the Newton polynomials defined by (3.8) and f [x 1 , ..., x k ] is the divided difference of the function f, Proof. (i)⇒(ii). If (i) holds, then by Lemma 3.5π = (π k ) k∈N is an interpolating Faber basis in C X with nodes (x k ) k∈N . Consequently, for every f ∈ C X there is a unique sequence (y k ) k∈N such that Since the basisπ is interpolating, we have (3.12) The polynomial coinsides with the function f at the points x 1 , ..., x k . (See Theorem 1.1.1 and formula (1.19) in [12] for details). Since linear system (3.12) has a unique solution, we have Equality (3.10) follows.
(ii)⇒(i). Let (ii) hold. Then, the sequenceπ = (π k ) k∈N is an interpolating Faber basis in C X if and only if (3.11) implies (3.13) for every f ∈ C X and every k ∈ N, that follows from the uniqueness of solutions of (3.12).
Theorem 3.11. Let X be an infinite compact subset of R and let M = {x k,n } be an interpolation matrix with the nodes in X. The following conditions are equivalent. (i) The space C X admits a Faber basisp = (p k ) k∈N such that the equality S n,p = L n,M (3.14) holds for every n ∈ N.
(ii) The sequence (Λ n,X (M)) n∈N is bounded and there is a sequence (x k ) k∈N of distinct points of X such that for any n ≥ 2 the tuple (x 1,n , ..., x n,n ) is a permutation of the set {x 1 , ..., x n }.
Proof. (i)⇒(ii). Letp = (p k ) k∈N be a Faber basis in C X and let (3.14) hold for every n ∈ N. The partial sum operators are bounded for every Faber basis. (See, for example, [14,Proposition 1.1.4]). Hence, we have sup n S n,p < ∞.
The last inequality and (3.14) imply the boundedness of the sequence (Λ n,X (M)) n∈N . Now to prove (ii) it suffices to show that for every n ≥ 2 and every k 1 ≤ n there is k 2 ≤ n + 1 such that holds. Suppose that, on the contrary, there is n ≥ 2 and k 1 ∈ {1, .., n} such that for all integer numbers k 2 ∈ {1, ..., n + 1}. We can find a function f ∈ C X satisfying the equalities These equalities imply that holds for every f ∈ C X . (See Proposition 2.1). Since the Lagrange interpolation polynomial L n (f, M, ·) is invariant with respect to arbitrary permutation of the nodes x 1,n+1 , ..., x n+1,n+1 , we may suppose that for every n ∈ N. Using the Newton polynomials π k (see (3.8)) we may write the polynomial L n (f, M, ·) in the form Hence, we have the representation f = ∞ k=1 f [x 1 , ..., x k ]π k . Now, (i) follows from Theorem 3.10.
Corollary 3.12. Let X be an infinite compact subset of R and let M ⊆ X be an interpolation matrix with bounded (Λ n,X (M)) n∈N . Then the following conditions are equivalent. (i) There is a Faber basis of C X such that (3.14) holds for every n ∈ N. holds for every n ∈ N and every f ∈ C X .
Proof. The implications (i)⇒(ii) and (i)⇒(iii) follow directly from Definition 3.1. The proofs of (ii)⇒(i) and (iii)⇒(i) are similar to the proof (i)⇒(ii) in Theorem 3.11 Remark 3.13. Statements (ii) and (iii) of Corollary 3.12 can be considered as some special cases of Lemma 4.7 in [7] and Theorem 20.1 in [15] respectively. Lemma 3.14. Let X be an infinite compact subset of R. The following statements are equivalent for arbitrary Faber basesp = (p k ) k∈N andq = (q k ) k∈N in C X . (i)There is a sequenceλ = (λ k ) k∈N of nonzero numbers such that holds for every k ∈ N.
(ii) The equality S n,p = S n,q (3.18) holds for every n ∈ N.
The following theorem is a dual form of Theorem 3.11 and it can be considered as the main result of the third section of the paper.
Theorem 3.15. Let X be an infinite compact subset of R and letp = (p k ) k∈N be a Faber basis in C X . The following conditions are equivalent. (i) There exists an interpolation matrix M ⊆ X such that equality (3.14) holds for every n ∈ N. holds for every n ∈ X. Using Theorem 3.11 we can suppose that there is a sequence (x k ) k∈N of distinct points of X such that x k,n = x k for all n ≥ 1 and k ∈ {1, ..., n}. To prove (ii) it suffices to show thatp is interpolating with nodes (x k ) k∈N . As in the proof of implication (ii)⇒(i) from Theorem 3.11 we obtain that the basisπ = (π k ) k∈N consisting of the corresponding Newton polynomials is an interpolating Faber basis with the nodes (x k ) k∈N for which the equality L n,M = S n,π (3.20) holds for every n ∈ N. (See equality (3.16)). By Lemma 3.14, it follows from (3.19) and (3.20) that there is a sequence (λ k ) k∈N of nonzero real numbers such that p k = λ k π k holds for every k ∈ N. Sinceπ is an interpolating Faber basis with nodes (x k ) k∈N , Proposition 3.7 implies thatp is also interpolating with the same nodes.
The case of an arbitrary interpolating Faber basisp = (p k ) k∈N can be reduced to the casep =π with the help of Lemma 3.14 and Proposition 3.7.