On the Convergence Rate of Hermite-Fejér Interpolation

By the Peano kernel theorem, this paper establishes the convergence rates for Hermite-Fejér interpolation with the observed function values and its first derivatives at Gauss-Jacobi pointsystems. These error bounds share that for a function analytic in the Bernstein ellipse 
$$\mathcal {E}_{\rho }$$

 
 
 ℰ
 
 
 ρ
 
 
 , the error decays exponentially; while for a functions of finite regularity, the error decays depending on the regularity.

To get fast convergence, the following Hermite-Fejér interpolation of f (x) at nodes (1) is considered [6,7]: where h (n) Fejér [5] and Grünwald [7] also showed that the convergence of the Hermite-Fejér interpolation of f (x) also depends on the choice of the nodes. The pointsystem while the pointsystem (1) is called strongly normal if for all n v (n) for some positive constant c.
In this paper, the following convergence rates of Hermite-Fejér interpolation H * 2n−1 (f, x) at Gauss-Jacobi pointsystems are considered.
1 In fact, Grünwald in [7] considered more general cases with any vector Comparing these results with which is sharp and attainable (see Fig. 2), we see that H * 2n−1 (f, x) converges much faster than H 2n−1 (f, x) for analytic functions or functions of higher regularities (see Fig. 1). Particularly, H 2n−1 (f, x) diverges at Gauss-Jacobi pointsystems with γ ≥ 0, whereas, H * 2n−1 (f, x) converges for functions analytic in the Bernstein ellipse or of finite limited regularity. For simplicity, in the following we abbreviate x . A ∼ B denotes there exist two positive constants c 1 and c 2 such that c 1 ≤ |A|/|B| ≤ c 2 .

Main Results
Suppose f (x) satisfies a Dini-Lipschitz condition on [−1, 1], then it has the following absolutely and uniformly convergent Chebyshev series expansion where the prime denotes summation whose first term is halved, T j (x) = cos(j cos −1 x) denotes the Chebyshev polynomial of degree j .
where σ n = +1 for even n and σ n = −1 for odd n.

Lemma 5 (Szegö [12, Theorem 8.1.2])
Let α, β be real but not necessarily greater than −1 and x k = cos θ k . Then for each fixed k, it follows where j k is the kth positive zero of Bessel function J α .
By ω n (x) = P (α,β) n (x) K n , we get In addition, by Lemma 5 with x j = cos θ j , we see that θ 1 ∼ 1 n . Similarly, by P (α,β) n n . These together yield and then by (20) it deduces the desired result.

Proof
In the following, we will focus on estimates of |E(T j , x)| for j ≥ 2n.
In the case γ > 0: From by Lemmas 3 and 6 we obtain These together with √ w i i (x)| = jτ n and then |E(T j , x)| = O j 2+2γ for j ≥ 2n, similar to the above proof in the case of γ ≤ 0, implies the desired result.
From the definition of τ n , we see that when α = β = − 1 2 the convergence order on n is the lowest. In addition, if f is of limited regularity, we have where w(f ; t) = w(t) is the modulus of continuity of f (x), andγ = max α, β, − 1 2 . and a rth derivative f (r) of bounded variation V r < ∞, then the Hermite-Fejér interpolation (5) at {x j } n j =1 has the convergence rate (11). Proof Consider the special functional L(g) = E n (g, x), where E n (g, x) is defined for ∀g ∈ C 1 ([−1, 1]) by By the Peano kernel theorem for n ≥ r (see Peano [9] or Kowalewski [8]), E n (f, x) can be represented as Moreover, noting that we get the following identity where K 2 (t) is defined by In addition, it can be easily verified that K s (−1) = K s (1) = 0 for s = 2, 3, . . .. Since f (r) is of bounded variation, directly applying the similar skills of Theorem 2 and Lemma 4 in [16], we get and |K s (t)|, for s = 2, 3, · · · , respectively. Then from (27) and (28), we can obtain that In addition, by Lemma 7, we have while by Lemmas 2-3, we get