Approximation by polynomials in Sobolev spaces with Jacobi weight

Polynomial approximation is studied in the Sobolev space $W_p^r(w_{\alpha ,\beta })$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{\alpha ,\beta }$. This requires simultaneous approximation of a function and its consecutive derivatives up to $s$-th order with $s\le r$. We provide sharp error estimates given in terms of $E_n(f^{(r)}{)}_{L^p(w_{\alpha ,\beta })}$, the error of best approximation to $f^{(r)}$ by polynomials in $L^p(w_{\alpha ,\beta })$, and an explicit construction of the polynomials that approximate simultaneously with the sharp error estimates.