ON CONSTRAINED UNIFORM APPROXIMATION

The problem of uniform approximants subject to Hermite interpolatory constraints is considered with an alternate approach. The uniqueness and the convergence aspects of this problem are also discussed. Our approach is based on the work of P. Kirchberger (1903) and a generalization of Weierstrass approximation theorem.


Introduction.
Let π m denote the set of all polynomials of degree less than or equal to m. Let  [a,b] h(x) . (1.1) Here we discuss uniform approximation of a prescribed f ∈ C[a, b] by the polynomials that are also Hermite interpolants to a set of given data at a finite number of preassigned points in the interval [a, b]. More precisely, we consider the following problem due to Loeb et al. [6]. satisfying a ≤ u 1 < u 2 < ··· < u k ≤ b. Then the problem is to find a best uniform approximant to a given f ∈ C [a, b] from the class Φ m,β = φ ∈ π m : φ (j) where is a subset of the real numbers with β i0 = f (u i ).
This problem originates from the work of Paszkowski [8,9] who studied it by imposing the interpolatory conditions only on the values of interpolating polynomials at k distinct points of [a, b] in the sense of Lagrange interpolation. To do this, he followed the classical Tchebycheff approach of approximating a continuous function by elements of an n-dimensional Haar subspace. In [2] Deutsch discussed Paszkowski's results [8, Theorems 2 and 5] with a different method. Deutsch's work is based on a characterization theorem of best approximation that involves extreme points of the closed unit ball in the dual of the underlying space [3, Corollary 2.6]. Later, Loeb et al. [6] extended the work of Deutsch by constraining the uniform approximants with Hermite interpolatory conditions. In fact, they discussed Problem 1.1 through the notions of the n-dimensional extended Haar subspace of order ν of C[a, b] (see [4]) and the generalized weight functions (see [7]). They also established a convergence result by giving a generalization of a theorem of de La Vallée Poussin [1, page 77].
In the present paper we continue the study of Problem 1.1. Our approach for its solution is based on the work of Kirchberger [5] that deals with extreme values of the error function. Uniqueness and convergence problems are also addressed in our work taking into account an extension of Weierstrass approximation theorem [10]. (2.1) The notation H s−1 (x, S β ), where S β is given in (1.3), stands for the polynomial of degree less than or equal to s − 1 that satisfies the following conditions: The error function corresponding to f ,g ∈ C[a, b] and the set of its extreme points in [a, b] will be denoted, respectively, as follows: , Let π * m denote the (m − s + 1)-dimensional subspace of π m generated by the polynomials x j W (x), j = 0, 1, 2,...,m − s, where W is given by (2.1). The following remark gives an explicit representation of the elements of Φ m,β (see (1.2)).
where q * ∈ π * m . This shows that φ * ∈ Φ m,β is a best approximant to f from the class Φ m,β if and only if q * ∈ π * m is a best approximant to f H,β from the class π * m . In particular, if m = s − 1 then H s−1 (x, S β ) will be the best approximation to f . For this obvious reason we assume that m ≥ s in the rest of the paper.
In view of the above remark, Problem 1.1 can be reformulated as follows.
For a given function f ∈ C[a, b], find a best approximation to f H,β (see (2.3)) in the uniform norm from the class π * m .

Characterization of best approximation.
This section deals with a necessary and sufficient condition for a solution of Problem 2.2. We note that every q ∈ π * m can be expressed as An alternate form of the characterization theorem [6, Theorem 3.1] that solves Problem 1.1 may be stated as follows.
Then q * is a best uniform approximant to f H,β from the class π * if and only if there exist N points α i ∈ crit(e f H,β ,q * ) satisfying the following conditions: Our method of proof is based on the following lemma which may be found in the standard texts of approximation theory, for example, [9, Lemma 7.1].

Lemma 3.2. Let Y be a linear subspace of C[a, b] and let
for all x ∈ crit(e f ,g * ) (see (2.3)).

Remark 3.3.
If we set Y = π * m and h = f H,β in the above lemma, then the necessary and sufficient condition for q * ∈ π * m to be a best approximation to f H,β is that there does not exist any p ∈ π * m such that for all x ∈ crit(e f H,β ,q * ) where R p (x) and E f H,β ,q * are, respectively, given in (3.1) and (3.2). To justify this, it is enough to note that  1(b) and (c) but N ≤ m−s +1. For each i = 1, 2, 3,...,N −1, fix a The choice of w i , as required above, directly follows from Remark 3.4. Now we set This can be seen by restricting α to each set (α i ,α i+1 ]∩crit(e f H,β ,q * ) for i = 0, 1,...,N − 1, where α 0 = a, and then using Theorem 3.1(c) along with (4.2). Hence by Remark 3.3, we note that q * cannot be a best uniform approximant to f H,β from the class π * m . This completes the proof.

Uniqueness.
We retain the setting of the previous sections in order to establish the uniqueness of the solution of Problem 2.2. More precisely, we prove the following theorem.
Theorem 5.1. There is exactly one best uniform approximant p * to f H,β from π * m .

Convergence.
In this section, we discuss the convergence of the sequence of best uniform approximants {q * k } ∞ k=s−1 to f H,β with the conditions that f is sufficiently differentiable and the set S β (see Problem 1.1) is replaced by (6.1) In this case, we write f H,f and Φ m,f instead of f H,β and Φ m,β (see (2.3)).
Theorem 6.1. Assume that f ∈ C n * [a, b] with n * = (max i∈I k n i ) − 1 and that the set S β (see (1.3)) is replaced by S f . If q * m ∈ π * m is the best approximant to f H,f in the sense of Theorem 3.1, then lim Consequently, the sequence {q * m + H s−1 (·,S f )} ∞ m=s−1 will converge uniformly to f .
The crux of the proof of this theorem is in an extension of a result based on the Weierstrass approximation theorem [10, page 160]. We state it in the next lemma without proof as it is a routine exercise. Lemma 6.2. For any f ∈ C r [a, b], and for a given ε > 0, there exists a polynomial p such that for all j = 0, 1, 2,...,r .
Proof of Theorem 6.1. In the notations of (2.3) and (6.1), we can write For a given ε > 0, we can fix a polynomial p of degree r > s such that (see Lemma 6.2) We set q(x) := p(x) − H s−1 (x, S p ). Then q ∈ π r and q (j) (u i ) = 0 for j ∈ N i and i ∈ I k .
Hence, W as defined in (2.1) is a factor of the polynomial q. This shows that q ∈ π * r . Using (6.4), (6.5), and (6.8) it can be seen that f H,f − q ∞ < ε. (6.9) Now consider the best uniform approximant q * r to f H from π * r (see Theorem 3.1) and note that for all m ≥ r . The last inequality follows from the relation π * r ⊆ π * m . This proves the desired result.