On the degree of piecewise shape-preserving approximation by polynomials
Let \(\mathbb{P}_{n}\) be the space of all algebraic polynomials of degree \(< n,\) and \(C[-1,1]\) the space of continuous functions on \([-1,1]\) equipped with the norm \(\| f \| = \max_{x \in [-1,1]} | f(x) |.\) Given \(s \geq 1,\) denote by \(\mathbb{Y}_{s}\) the set of all collections \(Y_{s} = \{ y_{i} \}_{i=1}^{s}\) of points \(y_{i}\) such that \(-1 = y_0 < y_1 < \dots < y_s < y_{s+1} = 1.\) We define the set of pointwise \(q\)-convex functions as \(\Delta^{(q)}(Y_s) = \{ f \in C[-1,1] \cap C^{q-2}(-1,1) \mid (-1)^{i-s} f^{(q-2)} \text{ is non-strictly convex on }[y_i,y_{i+1}], 0 \leq i \leq s \}. \) Further, we say that two functions \(f,\) \(g\) are \(q\)-coconvex if \(f, g \in \Delta^{(q)}(Y_s)\) for some \(q \geq 2,\) \( Y_s \in \mathbb{Y}_{s}.\) For \(f \in \Delta^{(q)}(Y_s)\) we denote by \(E_{n}^{(q)}(f;Y_s) = \inf _{P_n \in \mathbb{P}_{n} \cap \Delta^{(q)}(Y_s)} \| f -P_n \|\) the degree of \(q\)-coconvex approximation by algebraic polynomials of \(f\) relative to \(Y_s.\) The author proves that the degree of piecewise \(q\)-coconvex approximation \(E_{n}^{(q)}(f;Y_s)\) of a \(q\)-convex function by algebraic polynomials of degree \(< n\) asymptotically depends only on \(\alpha,\) \(Y_s\) and \(q,\) \[ n^{\alpha} E_{n}^{(q)}(f;Y_s) \leq c(\alpha, s, q), \;n \geq N^{*}(\alpha, Y_s, q), \] assuming this inequality holds for an \(N^{*}\) dependent on \(f\) and that the degree of unconstrained approximation has the same decrease rate: \(n^{\alpha} \inf_{P_n \in \mathbb{P}_{n}} \| f - P_n \| \leq 1,\) \(n \geq N\) and \(N \leq s+q+1.\)
scientific article; zbMATH DE number 6374934