Piecewise monotone pointwise approximation (Q1265168)

The authors consider constrained polynomial approximation of differentiable functions on \([-1,1]\), which change their monotonicity finitely many times, say \(s\)-times, inside the interval. The polynomials are required to change monotonicity exactly where the function does, what we call comonotone approximation. The question the paper deals with is to obtain pointwise estimates on the rate of approximation. Let \(Y:=\{y_i\}_{i=1}^s\), \(y_i\in(-1,1)\), be the points of monotonicity change of the function \(f\), and put \(\rho_n(x):=\sqrt {1-x^2}/n+1/n^2\), then it is proved that Theorem 1. There are constants \(B_{s,k}\) and \(N_{Y,k}\) depending exactly on the respective indices, so that for any \(f\in C^2[-1,1]\), which changes monotonicity at \(Y\), every \(x\in [-1,1]\) and all \(n\geq N_{Y,k}\), a polynomial \(P_n\) of degree not exceeding \(n\), such that \[ | f(x)-P_(x)| \leq B_{s,k}\rho_n^2(x)\omega_k(f'',\rho_n(x)). \] Furthermore, in general, one cannot make \(N\) dependent only on the number of monotonicity changes \(s\), rather than on their location \(Y\), while keeping the constant \(B\) dependent only on \(s\) and \(k\). If one merely assumes existence of a continuous first derivative of \(f\), then one may obtain estimates involving the third modulus of smoothness. Namely, Theorem 2. There are constants \(B_s\) and \(N_Y\) depending exactly on the respective indices, so that for any \(f\in C^1[-1,1]\), which changes monotonicity at \(Y\), every \(x\in [-1,1]\) and all \(n\geq N_Y\), a polynomial \(P_n\) of degree not exceeding \(n\), such that \[ | f(x)-P(x)| \leq B_s\rho_n(x)\omega_3(f',\rho_n(x)). \] Again, in general, one cannot make \(N\) dependent only on the number of monotonicity changes \(s\), rather than on their location \(Y\), while keeping the constant \(B\) dependent only on \(s\) and \(k\).