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The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a linear program, and the lower bound, produced by a semidefinite program exploiting the method of moments, are often close enough to deduce the projection constant with reasonable accuracy. The implementation of these programs makes it possible to find the projection constant of several three-dimensional spaces with five digits of accuracy, as well as the projection constants of the spaces of cubic, quartic, and quintic polynomials with four digits of accuracy. Beliefs about uniqueness and shape-preservation of minimal projections are contested along the way.
The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain asso
We show that, on convex polytopes and two or three dimensions, the finite element Stokes projection is stable on weighted spaces $mathbf{W}^{1,p}_0(omega,Omega) times L^p(omega,Omega)$, where the weight belongs to a certain Muckenhoupt class and the
Most commonly used emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified ab
A new class of univariate stationary interpolatory subdivision schemes of dual type is presented. As opposed to classical primal interpolatory schemes, these new schemes have masks with an even number of elements and are not step-wise interpolants. A
We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n sim A alpha^{d^n}$, but li