New estimates of Whitney constants for \(k\leq 1000\)
For a function \(f\in C([0,1])\) let us denote NEWLINE\[NEWLINE\Delta_{h}^{k}(f,x)= \sum\limits_{j=0}^{k}(-1)^{k-1}{k\choose j}f(x+jh)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\omega_{k} (1/k;f)=\sup\limits_{x,x+kh\in[0,1]}|\Delta_{h}^{k}(f,x)|.NEWLINE\]NEWLINE The constant NEWLINE\[NEWLINEW_{k}=\sup\limits_{f\in C([0,1])}\inf\limits_{p\in P_{k-1}} {\|f-p\|_{C([0,1])}/\omega_{k}(1/k;f)},\;k\in\mathbb{N},NEWLINE\]NEWLINE is called Whitney constant, where \(P_{k-1}\) is a space of algebraic polynomials of degree less or equal to \(k-1\). The author proves that \(W_5<1.28, W_6<1.32, W_{k}<1.4\) for \(k\leq 10, W_{k}<1.5\) for \(k\leq 40, W_{k}<1.6\) for \(k\leq 200, W_{k}<1.7\) for \(k\leq 1000\).
scientific article; zbMATH DE number 1669746