# Optimal subspaces for mean square approximation of classes of differentiable functions with boundary conditions

## TL;DR

This paper identifies optimal subspaces for $L_2$ approximation of certain Sobolev space functions with boundary conditions, using equidistant shifts of a single function, including spline spaces of various degrees.

## Contribution

It introduces a method to determine optimal subspaces generated by shifts of a single function for approximating Sobolev space functions with boundary conditions.

## Key findings

- Optimal subspaces are generated by equidistant shifts of a single function.
- Spline spaces of all degrees $d \,\geqslant\, r-1$ are optimal.
- The results apply to functions with boundary conditions in Sobolev spaces.

## Abstract

In this paper, we specify a set of optimal subspaces for $L_2$ approximation of three classes of functions in the Sobolev space $W^{(r)}_2$, defined on a segment and subject to certain boundary conditions. All of these subspaces are generated by equidistant shifts of a single function. In particular, we indicate optimal spline spaces of all degrees $d\geqslant r-1$ with uniform knots.