NOTE

Inverse and direct theorems for best uniform approximation by polynomials

Shayne Waldron^{∗} (waldron@math.auckland.ac.nz)

                             Technical Report
                               January 1997

ABSTRACT

It is shown that the standard method of obtaining direct (Jackson) theorems for the order of best uniform approximation by algebraic polynomials from those for trigonometric polynomials also provides inverse (Bernstein) theorems.

Key Words: Bernstein theorem, inverse theorem, Jackson theorem, direct theorem, Ditzian–Totik moduli of smoothness

AMS (MOS) Subject Classifications: primary G41A17, 41A25, 41A27, secondary 30E10, 42A10

∗ This work was supported by the Israel Council for Higher Education.

1. Introduction and main results

The purpose of this note is to show that the standard method of obtaining direct (Jackson) theorems for the order of best uniform approximation by algebraic polynomials from those for trigonometric polynomials also provides inverse (Bernstein) theorems. The simple proof offered for the inverse theorems for approximation order O(1/n^{α}), 0 < α ≤ 1 by algebraic polynomials (which were obtained only in the early 1980’s) naturally leads to the Ditzian–Totik modulus of continuity (see Ditzian and Totik [DT87] for details). It is hoped that this elementary approach will give insight into the celebrated observation of Nikol’skii [N46] that “algebraic polynomial approximation to smooth functions is better towards the endpoints of the interval”.

Let C(TT) denote the space of continuous 2π–periodic functions (continuous functions on the circle group TT) with the uniform norm ∥·∥_{∞}. We identify g ∈ C(TT) with a function from C[−π, π] in the usual way, so, e.g., g is even means g(θ) = g(−θ), −π ≤ θ ≤ π. The error in best uniform approximation to g ∈ C(TT) from T_{n} (the trigonometric polynomials of degree ≤ n) will be denoted by E_{n}^{∗}(g), and the error in best uniform approximation to f ∈ C[−1, 1] from Π_{n} (the algebraic polynomials of degree ≤ n) by E_{n}(f).

Direct (Jackson) theorems state that if E_{n}^{∗}(g), E_{n}(f) converge to 0 at a certain rate then this implies g, f belong to certain smoothness classes, while inverse (Bernstein) theorems give results in the opposite direction. The standard method for obtaining direct results for E_{n}(f) is to make the substitution f ↦ g := f ∘cos, and use known direct results for the (simpler) case of E_{n}^{∗}(g). We now prove that if this is done carefully enough, then inverse results for E_{n}^{∗}(g) also imply inverse results for E_{n}(f).

Theorem 1.1. Suppose that r_{n} → 0, and S ⊂ C(TT). Then, the direct–inverse theorems (a) For g ∈ C(TT), E_{n}^{∗}(g) = O(r_{n}) ⟺ g ∈ S

(b) For even g ∈ C(TT), E_{n}^{∗}(g) = O(r_{n}) ⟺ g ∈ S

(c) For f ∈ C[−1, 1], E_{n}(f) = O(r_{n}) ⟺ f ∘ cos ∈ S satisfy the implications (a) ⟹ (b) ⟺ (c). These implications remain valid if the O(r_{n}) is replaced by o(r_{n}) throughout, or only the inverse (direct) part of (a),(b),(c) is considered.

Proof: Since the 1–1 onto linear map

C[−1, 1] → {g ∈ C(TT) : g is even} : f ↦ g := f ∘ cos  (1.2)

is an isometry which maps Π_{n} onto the even trigonometric polynomials of degree ≤ n, and the best approximation to an even g ∈ C(T) from T_{n} is even, it follows that E_{n}^{∗}(g) = E_{n}(f). This establishes (b) ⟺ (c). The implication (a) ⟹ (b) is immediate.

Theorem 1.1 says that direct–inverse theorems for algebraic polynomial approximation are in 1–1 correspondence with direct–inverse theorems for trigonometric polynomial approximation to even functions. The author is unaware whether, under conditions on S necessary for (a) to hold (S is translation and reflection invariant), these also correspond to direct–inverse theorems for trigonometric polynomial approximation to all functions in C(TT), i.e., the implication (a) ⟸ (b) holds. However, this is not a crucial question, since in light of the history of Jackson–Bernstein type results, it is unlikely that such a result would ever be needed to deduce direct–inverse theorems for trigonometric polynomial approximation from those for algebraic polynomial approximation.

The smoothness condition f ∘ cos ∈ S (S translation invariant) imposes less smoothness on f at the endpoints of the interval [−1, 1] (cf Lemma 1.3). Thus, Theorem 1.1 can be interpreted as Nikol’skii’s observation in the form: “the functions which can be approximated by algebraic polynomials to a given order are less smooth towards the endpoints of the interval”.

We illustrate Theorem 1.1 with some examples. These are related to the ‘Ditzian– Totik’ modulus of continuity (see [DT87])

ω_{φ}(f, t) := ω_{φ}(f, t, [a, b]) := sup_{0<h≤t} |Δ_{hφ(x)}f(x)|, f ∈ C[a, b],

where φ is a nonnegative function, with φ := 1 giving the usual modulus ω(f, t). For simplicity of exposition, we take Δ_{hφ(x)} to be the forward difference operator (instead of the symmetric difference operator), which makes no essential difference since the φ we consider is continuous. Two moduli ω_{1}, ω_{2} are said to be of the same order, written ω_{1}(t) ∼ ω_{2}(t), if there exist positive constants for which C_{1}ω_{1}(t) ≤ ω_{2}(t) ≤ C_{2}ω_{1}(t), for small t > 0. First we need a lemma.

Lemma 1.3. For f ∈ C[−1, 1],

ω_{φ}(f, t, [−1, 1]) ∼ ω(f ∘ cos, t, [−π, π]),  (1.4)

where φ(x) := \sqrt{1 − x^{2}}.

Proof: In

ω(f ∘ cos, t) := sup_{{0<h≤t \atop θ,θ+h∈[−π,π]}} |f ∘ cos(θ + h) − f ∘ cos(θ)|

we make the substitution x = min{cos θ, cos(θ + h)}, x + δ = max{cos θ, cos(θ + h)}. Since cosine is continuously differentiable (with derivative − sin),

δ = \sqrt{1 − x^{2}}(h + ε_{x}(h)) = φ(x)(h + ε_{x}(h)), |ε_{x}(h)| ≤ |o(h)|,

where |o(h)| denotes some nonnegative function of h which is independent of x and is o(h). Thus, for small t, we obtain

sup_{0<h≤t−|o(t)| ; {x,x+δ∈[−1,1] \atop δ=φ(x)h}} |f(x+δ)−f(x)| ≤ sup_{0<h≤t ; {x,x+δ∈[−1,1] \atop δ=φ(x)(h+εx(h))}} |f(x+δ)−f(x)| ≤ sup_{0<h≤t+|o(t)| ; {x,x+δ∈[−1,1] \atop δ=φ(x)h}} |f(x+δ)−f(x)|,

i.e.,

ω_{φ}(f, t − |o(t)|) ≤ ω(f ∘ cos, t) ≤ ω_{φ}(f, t + |o(t)|).  (1.5).

From (1.5), and the subadditivity property of ω, we obtain

\frac{ω_{φ}(f, t)}{ω(f ∘ cos, t)} ≤ \frac{ω(f ∘ cos, t + |o(t)|)}{ω(f ∘ cos, t)} ≤ (1 + \frac{t + |o(t)|}{t}) \frac{ω(f ∘ cos, t)}{ω(f ∘ cos, t)} → 2, t → 0,

and similarly,

\frac{ω(f ∘ cos, t)}{ω_{φ}(f, t)} ≤ \frac{ω(f ∘ cos, t)}{ω(f ∘ cos, t − |o(t)|)} ≤ (1 + \frac{t}{t − |o(t)|}) → 2, t → 0,

which establishes (1.4).

There are numerous variations of the above argument, e.g., if g : [c, d] → [a, b] is a differentiable bijection, then ω(f ∘ g, t, [c, d]) ∼ ω_{φ}(f, t, [a, b]), φ := |Dg ∘ g^{−1}|.

We now translate the classical result of Bernstein (inverse part) and Jackson (direct part) that: for g ∈ C(TT), 0 < α < 1,

E_{n}^{∗}(g) = O(1/n^{α}) ⟺ g ∈ Lip α,  (1.6)

into a direct–inverse result for algebraic polynomial approximation.

Corollary 1.7. For f ∈ C[−1, 1] and 0 < α < 1, the following are equivalent

(a) E_{n}(f) = O(1/n^{α})

(b) f ∘ cos ∈ Lip α.

(c) ω_{φ}(f, t) = O(t^{α}), where φ(x) := \sqrt{1 − x^{2}}

(d) ω(f ∘ cos, t) = O(t^{α})

(e) There exists a constant C and a sequence of polynomials p_{n} of degree n satisfying

|f(x) − p_{n}(x)| ≤ C/n^{α}, −1 ≤ x ≤ 1.

Proof: The equivalence of (a),(b),(d),(e) follows immediately from Theorem 1.1 and the result (1.6) for trigonometric approximation. The implications (c) ⟺ (d) follow from Lemma 1.3.

In the same way, the classical results for trigonometric polynomial approximation order O(1/n^{α}), α ≥ 1, in terms of the generalised Lipschitz space Lip^{∗}(α, ∞) (see DeVore and Lorentz [DL93:p51]) can be translated into results for algebraic polynomial approximation. We consider only the case α = 1, that: for g ∈ C(TT)

E_{n}^{∗}(g) = O(1/n) ⟺ g ∈ Z,  (1.8)

where Z is the Zygmund space, which consists of those functions satisfying

|g(θ) − 2g(θ + h) + g(θ + 2h)| ≤ M h,  (1.9)

for some constant M, i.e., ω_{2}(g, t) = O(t), with ω_{2} the usual second order modulus of smoothness.

Corollary 1.10. For f ∈ C[−1, 1], the following are equivalent

(a) E_{n}(f) = O(1/n)

(b) f ∘ cos ∈ Z

(c) There exists a constant M such that

|f ∘ cos(θ) − 2f ∘ cos(θ + h) + f ∘ cos(θ + 2h)| ≤ M h

(d) ω_{2}(f ∘ cos, t) = O(t)

(e) There exists a constant C and a sequence of polynomials p_{n} of degree n satisfying

|f(x) − p_{n}(x)| ≤ C/n, −1 ≤ x ≤ 1.

Proof: Follows immediately from Theorem 1.1 and (1.8), (1.9).

The isometry (1.2) can also be used to transfer sharp direct theorems to the algebraic case, e.g., Korneichuk’s sharp form of Jackson’s theorem (see [K62]),

E_{n}^{∗}(g) ≤ ω(g, \frac{π}{n + 1}),  g ∈ C(TT),  (1.11)

gives the following.

Corollary 1.12 (Sharp Jackson Theorem). For f ∈ C[−1, 1],

E_{n}(f) ≤ ω(f ∘ cos, \frac{π}{n + 1}) ≤ M ω_{φ}(f, \frac{1}{n}) ≤ M ω(f, \frac{1}{n}),  (1.13)

where the constant 1 multiplying ω(f ∘ cos, π/(n + 1)) is the best possible one independent of f and n, and M is some constant which is independent of f and n.

Proof: Since (1.11) is sharp as an inequality for even g ∈ C(T), using (1.2) we obtain the sharp inequality

E_{n}(f) = E_{n}^{∗}(f ∘ cos) ≤ ω(f ∘ cos, \frac{π}{n + 1}).

By the subadditivity property of ω, and Lemma 1.3,

ω(f ∘ cos, \frac{π}{n + 1}) ≤ (1 + \frac{nπ}{n + 1}) ω(f ∘ cos, \frac{1}{n}) ≤ M ω_{φ}(f, \frac{1}{n}) ≤ M ω(f, \frac{1}{n}),

with the last inequality following immediately from the definition of ω_{φ} (since φ ≤ 1).

The L_{p}–analogue (1 ≤ p < ∞) of Theorem 1.1 relates direct–inverse theorems for weighted L_{p}[−1, 1]–approximation by algebraic polynomials (the weight being 1/φ) to those for L_{p}(TT)–approximation of even functions by trigonometric polynomials. The corresponding version of the equivalence (a)⟺(c) of Corollary 1.7 is given in Felton [F97:Th.5.1].

References

[DL93] R. A. DeVore and G. G. Lorentz, “Constructive approximation”, Springer-Verlag, Berlin, 1993.

[DT87] Z. Ditzian and V. Totik, “Moduli of smoothness”, Springer–Verlag, New York, 1987.

[F97] M. Felton, Characterisation of best algebraic approximation by an algebraic modulus of smoothness, J. Approx. Theory 89 (1997), 1–25.

[K62] N. P. Korneichuk, The exact constant in D. Jackson’s theorem on best uniform approximation of continuous periodic functions, Soviet Math. Dokl. 3 (1962), 1040–1041.

[N46] S. M. Nikol’skii, On the best approximation of functions satisfying a Lipschitz condition by polynomials, Izv. Akad. Nauk SSSR Ser. Math 10 (1946), 295–322.