(Joint work with S.-T. Ali, Montreal.)
One of the applications of this smoothing will be discussed in this session by A. Prymak in his talk "Three-monotone spline approximation".
Next we shall explain recent developments on greedy algorithms for the sparse solution. Mainly, we shall provide a convergence analysis of an iteratively least squares orthogonal greedy algorithm. We use this algorithm for image denoising.
Finally, we present some numerical evidence of image and movie denoising.
First we prove this estimate for $s$ with the knots that are allowed to depend on $f$ but cannot be too close to each other (``controlled'' knots). Then we use very recent results on constrained spline smoothing to achieve maximum smoothness and to move the knots to the right place.
Moreover, we also prove a similar estimate in terms of the Ditzian-Totik $4$-th modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of $3$-monotone spline approximation in the $L_p$ norm with $p<\infty$. At the same time, positive results in the $L_p$-case with $p<\infty$ are still valid for splines with ``controlled'' knots.
These results confirm that $3$-monotone approximation is the transition case between monotone and convex approximation (where most of the results are ``positive'') and $k$-monotone approximation with $k\geq 4$ (where just about everything is ``negative'').