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Register a new userAn improvement of the Kolmogorov-Riesz compactness theorem
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The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result by Kolmogorov-Riesz on compact subsets of Lebesgue spaces.
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We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilberts 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions $varphi_1,ldots,varphi_5 : [,0, 1,]to [,0,1,]$ such that for any continuous function $f: [,0,1,]^2tomathbb R$ there is a continuous function $h: [,0,3,]tomathbb R$ such that for any $x,yin [,0, 1,]$ we have $$f(x,y)=sumlimits_{k=1}^5 hleft(varphi_k(x)+sqrt{2},varphi_k(y)right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.
We show that Liebs concavity theorem holds more generally for any unitary invariant matrix function $phi:mathbf{H}_+^nrightarrow mathbb{R}_+^n$ that is concave and satisfies Holders inequality. Concretely, we prove the joint concavity of the function $(A,B) mapstophibig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big] $ on $mathbf{H}_+^ntimesmathbf{H}_+^m$, for any $Kin mathbb{C}^{ntimes m}$ and any $s,p,qin(0,1], p+qleq 1$. This result improves a recent work by Huang for a more specific class of $phi$.
We consider the family of integral operators $(K_{alpha}f)(x)$ from $L^p[0,1]$ to $L^q[0,1]$ given by $$(K_{alpha}f)(x)=int_0^1(1-xy)^{alpha -1},f(y),operatorname{d}!y, qquad 0<alpha<1.$$ The main objective is to find upper bounds for the Kolmogorov widths, where the $n$th Kolmogorov width is the infimum of the deviation of $(K_{alpha}f)$ from an $n$-dimensional subspaces of $L^p[0,1]$ (with the infimum taken over all $n$-dimensional subspaces), and is therefore a measure of how well $K_{alpha}$ can be approximated. We find upper bounds for the Kolmogorov widths in question that decrease faster than $exp(-kappa sqrt{n})$ for some positive constant $kappa$.
A classical theorem of Herglotz states that a function $nmapsto r(n)$ from $mathbb Z$ into $mathbb C^{stimes s}$ is positive definite if and only there exists a $mathbb C^{stimes s}$-valued positive measure $dmu$ on $[0,2pi]$ such that $r(n)=int_0^{2pi}e^{int}dmu(t)$for $nin mathbb Z$. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We study in great detail the case of positive definite functions.
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