Order of the best spline approximations of some classes of functions

The rate of decrease of the upper bounds of the best spline approximations $E_{m,n}(f)_p$ with undetermined $n$ nodes in the metric of the space $L_p(0,1)$ $(1\le p\le\infty)$ is studied in a class of functions $f(x)$ for which $\|f^{(m+1)}(x)\|_{L_q(0,1)}\le1$ $(1\le q\le\infty)$ or $\mathrm{var}\{f^{(m)}(x);0,1\}\le1$ ($m=1,2,\dots$, the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as $n\to\infty$, and asymptotic formulas are obtained for $p=\infty$ and $1\le q\le\infty$ in the case of an approximation by broken lines $(m=1)$. The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.