Positive Results and Counterexamples in Comonotone Approximation

Abstract

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [−1,1], that change monotonicity s≥1 times in the interval, when the degree of unconstrained polynomial approximation E _n (f)≤n ^−α, n≥1. We ask whether the degree of comonotone approximation is necessarily ≤c(α,s)n ^−α, n≥1, and if not, what can be said. It turns out that for each s≥1, there is an exceptional set A _s of α’s for which the above estimate cannot be achieved.

Appendix

For the sake of completeness, we include the proof of the fact that the inequalities (5.1) imply (2.6).

Let $\varphi(x):=\sqrt{1-x^{2}}$, and write $C^{0}_{\varphi}:=C[-1,1]$; for r≥1, we say that $f\in C^{r}_{\varphi}$ if f∈C ^(r)(−1,1) and lim _x→±1 φ ^r(x)f ^(r)(x)=0.

Finally, for $f\in C^{r}_{\varphi}$, we write

$$\omega^\varphi_{k,r}\bigl(f^{(r)},t\bigr):=\sup_{0\le h\le t}\sup_{x:|x|+\frac{kh}{2}\varphi(x)<1} K^r\biggl(x,\frac{kh}{2}\biggr)\bigl|\varDelta _{h\varphi(x)}^k\bigl(f^{(r)},x\bigr)\bigr|,$$

where K(x,μ):=φ(|x|+μφ(x)) and the symmetric difference $\varDelta _{u}^{k}$ is defined in (3.1).

Note that for r=0,

$$\omega^\varphi_{k,0}(f,t)\equiv\omega_k^\varphi(f,t),$$

the kth Ditzian–Totik modulus of smoothness.

Recall the Chebyshev knots x _j :=cos(jπ/n),0≤j≤n, and denote I _j :=[x _j ,x _j−1], 1≤j≤n. Note that J _j =I _j ∪I _j+1, 1≤j≤n−1. Let Σ _k,n be the collection of all continuous piecewise polynomials of degree <k, on the Chebyshev partition $\{ x_{j}\}_{j=0}^{n}$. Also, denote by Σ _k,n (Y _s )⊂Σ _k,n , the set of all piecewise polynomials S with the following property. Let j(i), 1≤i≤s, be such that y _i ∈I _j(i); then S∈Σ _k,n (Y _s ) if for every 1≤i≤s, the restriction of S to (x _j(i)+1,x _j(i)−2)=:O _i is a polynomial, where x _n+l :=−1, x _−l :=1. Finally, write $O:=\bigcup_{i=1}^{s}O_{i}$.

Proposition 3

([10])

For every k≥1 and s≥0, there are constants c=c(k,s) and c _∗=c _∗(k,s) such that if $Y_{s}\in \mathbb{Y}_{s}$ and S∈Σ _k,n (Y _s )∩Δ ¹(Y _s ), n≥1, then

$$E^{(1)}_{c_*n}(S,Y_s)\leq c\omega_k^\varphi(S,1/n).$$

Theorem 5

Given s∈N, let α≥1. Suppose that $f\in \varDelta ^{1}_{s}$ is such that (2.4) holds for n≥r+1 and the inequalities (5.1) are satisfied. Then

$$ n^{\alpha}E_{n}^{1,s}(f)\le c_1(\alpha,s),\quad n\ge c_2(\alpha,s)N_0.$$

(A.1)

Proof

Let f∈Δ ⁽¹⁾(Y _s ). Evidently, it suffices to prove (A.1) for $E_{n}^{(1)}(f,Y_{s})$ instead of $E_{n}^{1,s}(f)$ (recall that $E^{(1)}_{n}(f,Y_{s})$ and $E_{n}^{1,s}(f)$ are defined in (2.2) and (2.3)). First, let

$$\sigma_{r+1,n}^{(1)}(f):=\inf\bigl\{\|f-S\|: S\in \varSigma _{r+1,n}(Y_s)\cap \varDelta ^1(Y_s)\bigr\},$$

and assume that (the analog of (A.1)),

$$n^{\alpha}\sigma_{r+1,n}^{(1)}(f)\leq c_3(\alpha,s),\quad n\ge N_1.$$

Take S∈Σ _r+1,n (Y _s )∩Δ ₁(Y _s ), so that $\|f-S\|\leq2\sigma_{r+1,n}^{(1)}(f,Y_{s})$. Then by Proposition 3, we conclude that

$$E_n^{(1)}(S,Y_s)\le c\omega_{r+1}^\varphi \biggl(S,\frac{1}{n}\biggr),\quad n\geq c_*,$$

which, in turn, implies

$$E_n^{(1)}(f,Y_s)\le E_n^{(1)}(S,Y_s)+2\sigma_{r+1,n}^{(1)}(f,Y_s)\le c\omega_{r+1}^\varphi \biggl(f,\frac{1}{n}\biggr)+c\sigma_{r+1,n}^{(1)}(f,Y_s)$$

for all n≥c _∗. Since (2.4) implies that $\omega_{r+1}^{\varphi }(f,1/n)\leq cn^{-\alpha}$, we only need to verify (A.1) for N ₁≥max{c _∗,c ₃(α,s)N ₀}, c ₃(α,s) to be prescribed.

Let n≥c ₃(α,s)N ₀, and let $O_{n}=\bigcup_{i=1}^{s'}O_{i}'$ be a partition of O _n into connected intervals. First we prove that there is an m≥n/c ₂ such that:

• (i) for each 1≤i≤s′, $O_{i}'\subset J_{j(i),m}$, and

• (ii) for i ₁≠i ₂, we have that either j(i ₁)=j(i ₂) or {j(i ₁),j(i ₁)+1}∩{j(i ₂),j(i ₂)+1}=∅.

To this end, we proceed by induction on s′≤s. If s′=1, then there is just one block $O_{1}'$ of length ≤2s+1, so that we may take $m=\lceil\frac{n}{2s+1}\rceil$. If s′>1, let $n_{1}=\lceil\frac{n}{2s+1}\rceil$, then property (i) holds for m=n ₁. If property (ii) is also valid, then we are done. Otherwise,

$$O_n=\bigcup_{i=1}^{s'}O_i'\subset\bigcup_{i=1}^{s'}J_{j(i),m}=:\bigcup_{i=1}^{s''}O_i''$$

is a new partition into connected intervals, where s″<s′. We apply the induction step to s″ to get the desired value of m. It is readily seen that we may take c ₃(α,s)≥(2s+1)^s.

By virtue of (5.1), for the appropriate m, for every 1≤i≤s′, there is a polynomial p _i , of degree r, comonotone with f on J _j(i),m , satisfying p _i (x _j(i)+1,m )=f(x _j(i)+1,m ) and such that

$$\|f-p_i\|_{J_{j(i),m}}\le\frac{c(\alpha,s)}{m^{\alpha}}\le \frac{c_3(\alpha,s)}{n^{\alpha}}.$$

For j≠j(i),j(i)+1, 1≤i≤s′, f is monotone on I _j,m , so that, by (5.1), we have a polynomial $\tilde{q}_{j}$, of degree r, comonotone with it there, such that

$$\|f-\tilde{q}_j\|_{I_{j,m}}\le\frac{c_4(\alpha,s)}{n^{\alpha}}.$$

Adding the linear

we obtain q _j that is comonotone with f on I _j,m , interpolates f at endpoints of I _j,m , and satisfies

$$\|f-q_j\|_{I_{j,m}}\le\frac{3c_4(\alpha,s)}{n^{\alpha}}.$$

Now we take piecewise polynomial s _n to be defined p _i on J _j(i),m and q _j on I _j,m for the other intervals. Then s _n is comonotone with f and satisfies

$$\|f-s_n\|\leq\frac{c(\alpha,s)}{n^{\alpha}}.$$

It may have discontinuities at the points x _j(i)−1,m , but there are no more than s such points, so if we make it continuous by moving from left to right and making s _n (x _j(i)−1,m +):=s _n (x _j(i)−1,m −), then our estimate is still valid with c ₁(α,s):=sc(α,s). This completes the proof. □