Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDetermining Projection Constants of Univariate Polynomial Spaces
115
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 associated polynomial equations. The proposed approach also makes possible to identify conditions for the existence of the sought schemes.
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 integrability index can be different from two. We show how this estimate can be applied to obtain error estimates for approximations of the solution to the Stokes problem with singular sources.
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 absolute error tolerance for a cone, $mathcal{C}$, of non-spiky input functions in the Sobolev space $W^{2,infty}[a,b]$. Our algorithms automatically determine where to sample the function---sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function $f$ on a bounded interval with $L^{infty}$-error no greater than $varepsilon$ is $mathcal{O}Bigl(sqrt{{left|fright|}_{frac12}/varepsilon}Bigr)$ as $varepsilon to 0$. This is the same order as that of the best function approximation algorithm for functions in $mathcal{C}$. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if $f$ significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.
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 complete algebraic characterization, which covers every arity, is given in terms of identities of trigonometric polynomials associated to the schemes. This characterization is based on a necessary condition for refinable functions to have prescribed values at the nodes of a uniform lattice, as a consequence of the Poisson summation formula. A strategy for the construction is then showed, alongside meaningful examples for applications that have comparable or even superior properties, in terms of regularity, length of the support and/or polynomial reproduction, with respect to the primal counterparts.
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 little can be said about the constant $alpha$. In this paper, we show that $alpha$ is always irrational or an integer. In fact, we prove a stronger statement: if a sequence $G_n$ satisfies an asymptotic formula of the form $G_n = A alpha^n + B + O(alpha^{-epsilon n})$, where $A,B$ are algebraic and $alpha > 1$, and the sequence contains infinitely many integers, then $alpha$ is irrational or an integer.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
