Orthogonal projectors onto spaces of periodic splines

TL;DR

This paper proves that orthogonal projections onto spaces of periodic splines converge almost everywhere to the original function as the mesh size decreases, and these projections are uniformly bounded on $L^ fty$.

Contribution

It establishes the almost everywhere convergence of orthogonal spline projections for arbitrary knots and degrees, extending previous results.

Findings

Convergence of projections to the original function as mesh size tends to zero.

Uniform boundedness of projection operators on $L^ fty$.

Applicability to arbitrary knot sequences and polynomial degrees.

Abstract

The main result of this paper is a proof that for any integrable function $f$ on the torus, any sequence of its orthogonal projections $(\widetilde{P}_n f)$ onto periodic spline spaces with arbitrary knots $\widetilde{\Delta}_n$ and arbitrary polynomial degree converges to $f$ almost everywhere with respect to the Lebesgue measure, provided the mesh diameter $|\widetilde{\Delta}_n|$ tends to zero. We also give a proof of the fact that the operators $\widetilde{P}_n$ are bounded on $L^\infty$ independently of the knots $\widetilde{\Delta}_n$.