On the best \(L_1\)-approximation by polynomials of functions which are fractional integrals of summable functions (Q2785845)

For any \(r \in (0, 1)\) and \(p \in [1, \infty ]\) the authors consider the class \(W_p^r\) of functions \(f\) represented by NEWLINE\[NEWLINEf(x) = \frac1{\Gamma(r)} \int_{-1}^1 (x - t)_+^{r-1}\phi(t) dt, x \in [-1, 1], NEWLINE\]NEWLINEwhere \(\Gamma\) is the Euler gamma function, \(\phi \in L_p\) and \(|\phi|_p \leq 1\). Let \(E_n(f)_1\) be the best approximation of \(f\) by algebraic polynomials of degree \(\leq n\) in the \(L_1\) norm and \(E_n(W_p^r)_1 = \sup_{ f \in W_p^r} E_n(f)_1\). The main estimates of the paper: NEWLINE\[NEWLINEE_n\left(\frac1{\Gamma(r)}(x-a)_+^{r-1}\right)_1 = \frac {K_r(1 - a^2)^{r/2}}{n^r}+O\left(\min\left(\frac1{n^{2r}}, \frac{(1-a^2)^{(r-1)/2}}{n^{r+1}}\right)\right)NEWLINE\]NEWLINEwhere \(K_r\) depends only of \(r\), and NEWLINE\[NEWLINEE_n(W_1^r)_1 = \frac{K_r}{n^r} + O\left(\frac1{n^{r+1}}\right).NEWLINE\]