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Suppose $f : [0,1]^2 rightarrow mathbb{R}$ is a $(c,alpha)$-mixed Holder function that we sample at $l$ points $X_1,ldots,X_l$ chosen uniformly at random from the unit square. Let the location of these points and the function values $f(X_1),ldots,f(X_l)$ be given. If $l ge c_1 n log^2 n$, then we can compute an approximation $tilde{f}$ such that $$ |f - tilde{f} |_{L^2} = mathcal{O}(n^{-alpha} log^{3/2} n), $$ with probability at least $1 - n^{2 -c_1}$, where the implicit constant only depends on the constants $c > 0$ and $c_1 > 0$.
The purpose of this paper is to extend the result of arXiv:1810.00823 to mixed Holder functions on $[0,1]^d$ for all $d ge 1$. In particular, we prove that by sampling an $alpha$-mixed Holder function $f : [0,1]^d rightarrow mathbb{R}$ at $sim frac{1
Let $mathsf M$ and $mathsf M _{mathsf S}$ respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $mathbb{R}^n$, and let $w$ be a nonnegative locally integrable function on $mathbb{R}^n$. We
For a wide family of even kernels ${varphi_u, uin I}$, we describe discrete sets $Lambda$ such that every bandlimited signal $f$ can be reconstructed from the space-time samples ${(fastvarphi_u)(lambda), lambdainLambda, uin I}$.
A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply this result to a problem arising from probability theory.
The $K$-hull of a compact set $Asubsetmathbb{R}^d$, where $Ksubset mathbb{R}^d$ is a fixed compact convex body, is the intersection of all translates of $K$ that contain $A$. A set is called $K$-strongly convex if it coincides with its $K$-hull. We p