On the constants in the equivalence between Ditzian-Totik moduli of smoothness and \(K\)-functionals (Q2751680)

Given a real interval \(A\) and a function \(f\in L_\infty(A)\), the Ditzian-Totik modulus of smoothness of order \(r\) with weight function \(\phi\) is defined by NEWLINE\[NEWLINE \omega_r^\phi(f,t)=\sup_{0<h\leq t} \|\Delta_{h\phi(\cdot)}^rf(\cdot)\|_\infty NEWLINE\]NEWLINE and NEWLINE\[NEWLINE K_r^\phi(f,t)=\inf_{g\in W_{r,\infty}^\phi(A)}\{\|f-g\|_\infty+t^r\|\phi^r g^{(r)}\|_\infty\} NEWLINE\]NEWLINE is the related \(K\)-functional. Here \(\Delta_{h\phi(x)}^rf(x)\) is a symmetric difference on \(A\) and \(W_{r,\infty}^\phi(A)\) is the corresponding weighted Sobolev space with weight function \(\phi^r\). The concepts of weighted modulus of smoothness and \(K\)-functional play an important role in the theory of approximation by algebraic polynomials since it puts in relief the properties of the convergence at certain points; specially at the end points of the interval. NEWLINENEWLINENEWLINEIn this paper the author deals with the most important case of weight \(\phi(x)=\sqrt{1-x^2}\) in the interval \(A=[-1,1]\). In this case, it is well known the equivalence between \(\omega_r^\phi\) and \(K_r^\phi\) in the sense that there exist the constants \(c_1,c_2>0\), which depend only on \(r\), such that NEWLINE\[NEWLINE c_1\omega_r^\phi(f,t)\leq K_r^\phi(f,t)\leq c_2\omega_r^\phi(f,t),\quad 0<t\leq {1\over 2r}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe aim of the paper is to find quantitative estimates of the constants \(c_1\) and \(c_2\). The author obtains such quantitative expressions only for the case \(r=1\) and \(r=2\). In the case \(r=1\), for all natural numbers \(n\) is used a partition with \(n+1\) knots to prove that for \(f\in C[-1,1]\) and \(t\leq {1\over 2}\) we have NEWLINE\[NEWLINE {1\over 8}\omega_1^\phi(f,t)\leq K_1^\phi(f,t)\leq c_2(n)\omega_1^\phi(f,t), NEWLINE\]NEWLINE where \(c_2(n)=1+{2c_4\over c_3}{n\over n-1}\) and \(c_3\), \(c_4\) are constants that depend on the chosen partition. Several instances of valid partitions are given with the corresponding explicit values of the constants \(c_3\) and \(c_4\). For \(r=2\), the author announces a result that will appear in a forthcoming joint paper with H. H. Gonska. Such a result when \(t\leq 1\) yields the inequality NEWLINE\[NEWLINE {1\over 16}\leq \omega_2^\varphi(f,t)\leq 404\omega_2^\varphi(f,t), NEWLINE\]NEWLINE where the weight function is now \(\varphi(x)=\sqrt{x(1-x)}\) on the interval \(A=[0,1]\). NEWLINENEWLINENEWLINEAs the author says the explicit constants obtained in the paper are not the best possible and he formulates several open problems where the computation of such best constants is proposed. Anyway, some of the constants obtained for \(r=2\) has been improved by \textit{I. Gavrea} [Rend. Circ. Mat. Palermo, Ser. II, Suppl. 68, 439-454 (2002)] recently.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00041].