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Register a new userApproximating mixed Holder functions using random samples
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Authors
Nicholas F. Marshall
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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 $c_1 > 0$.
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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 $sim frac{1}{varepsilon} left(log frac{1}{varepsilon} right)^d$ independent uniformly random points from $[0,1]^d$, we can construct an approximation $tilde{f}$ such that $$ |f - tilde{f}|_{L^2} lesssim varepsilon^alpha left(log textstyle{frac{1}{varepsilon}} right)^{d-1/2}, $$ with high probability.
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 define the associated Tauberian functions $mathsf{C}_{mathsf{HL},w}(alpha)$ and $mathsf{C}_{mathsf{S},w}(alpha)$ on $(0,1)$ by [ mathsf{C}_{mathsf{HL},w}(alpha) :=sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M chi_E(x) > alpha}) ] and [ mathsf{C}_{mathsf{S},w}(alpha) := sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M _{mathsf S}chi_E(x) > alpha}). ] Utilizing weighted Solyanik estimates for $mathsf M$ and $mathsf M_{mathsf S}$, we show that the function $mathsf{C}_{mathsf{HL},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}})^{-1}}(0,1)$ and $mathsf{C}_{mathsf{S},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}^ast})^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.
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 propose a general approach to the analysis of facial structure of $K$-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of $k$-dimensional faces, for all $k=0,dots,d-1$. We then apply our theory in the case when $A=Xi_n$ is a sample of $n$ points picked uniformly at random from $K$. We show that in this case the set of $xinmathbb{R}^d$ such that $x+K$ contains the sample $Xi_n$, upon multiplying by $n$, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding $f$-vector of the $K$-hull of $Xi_n$ to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the $f$-vector.
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