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Register a new userImproving estimates for discrete polynomial averages
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For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=frac{1}{N}sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $ell^{p}$-improving inequality begin{equation*} |A_N f|_{ell^q(mathbb{Z})} lesssim_{P,p,q} N^{-d(frac{1}{p}-frac{1}{q})} |f|_{ell^p(mathbb{Z})}, qquad N inmathbb{N}, end{equation*} where $1leq p leq q leq infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth.
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Let $ lambda ^2 in mathbb N $, and in dimensions $ dgeq 5$, let $ A_{lambda } f (x)$ denote the average of $ f ;:; mathbb Z ^{d} to mathbb R $ over the lattice points on the sphere of radius $lambda$ centered at $x$. We prove $ ell ^{p}$ improving properties of $ A_{lambda }$. begin{equation*} lVert A_{lambda }rVert_{ell ^{p} to ell ^{p}} leq C_{d,p, omega (lambda ^2 )} lambda ^{d ( 1-frac{2}p)}, qquad tfrac{d-1}{d+1} < p leq frac{d} {d-2}. end{equation*} It holds in dimension $ d =4$ for odd $ lambda ^2 $. The dependence is in terms of $ omega (lambda ^2 )$, the number of distinct prime factors of $ lambda ^2 $. These inequalities are discre
We exhibit a range of $ell ^{p}(mathbb{Z}^d)$-improving properties for the discrete spherical maximal average in every dimension $dgeq 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds, which extend the discrete maximal theorem of Magyar, Stein, and Wainger to weighted spaces. In particular, the sparse bounds imply that the discrete spherical maximal average is a bounded map from $ell^2(w)$ into $ell^2(w)$ provided $w^{frac{d}{d-4}+delta}$ belongs to the Muckenhoupt class $A_2$ for some $delta>0.$
Let $ Lambda $ denote von Mangoldts function, and consider the averages begin{align*} A_N f (x) &=frac{1}{N}sum_{1leq n leq N}f(x-n)Lambda(n) . end{align*} We prove sharp $ ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, Gsubset [0,N]$ there holds begin{equation*} N ^{-1} langle A_N mathbf 1_{F} , mathbf 1_{G} rangle ll frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigl( operatorname {Log} frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigr) ^{t}, end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ sup_N A_N mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.
This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call such estimates interpolatory. One important corollary of our main theorem is the following result on approximation of $fin Delta^{(2)}$, the set of convex functions, from $W^r$, the space of functions on $[-1,1]$ for which $f^{(r-1)}$ is absolutely continuous and $|f^{(r)}|_{infty} := ess,sup_{xin[-1,1]} |f^{(r)}(x)| < infty$: For any $fin W^r capDelta^{(2)}$, $rin {mathbb N}$, there exists a number ${mathcal N}={mathcal N}(f,r)$, such that for every $nge {mathcal N}$, there is an algebraic polynomial of degree $le n$ which is in $Delta^{(2)}$ and such that [ left| frac{f-P_n}{varphi^r} right|_{infty} leq frac{c(r)}{n^r} left| f^{(r)}right|_{infty} , ] where $varphi(x):= sqrt{1-x^2}$. For $r=1$ and $r=2$, the above result holds with ${mathcal N}=1$ and is well known. For $rge 3$, it is not true, in general, with ${mathcal N}$ independent of $f$.
Given a nondecreasing function $f$ on $[-1,1]$, we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at $pm 1$. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at $pm 1$). We call such estimates interpolatory estimates. In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness $omega_2(f,cdot)$ of $f$ evaluated at $sqrt{1-x^2}/n$ and were valid for all $nge1$. The current paper is devoted to proving that if $fin C^r[-1,1]$, $rge1$, then the interpolatory estimates are valid for the second modulus of smoothness of $f^{(r)}$, however, only for $nge N$ with $N= N(f,r)$, since it is known that such estimates are in general invalid with $N$ independent of $f$. Given a number $alpha>0$, we write $alpha=r+beta$ where $r$ is a nonnegative integer and $0<betale1$, and denote by $Lip^*alpha$ the class of all functions $f$ on $[-1,1]$ such that $omega_2(f^{(r)}, t) = O(t^beta)$. Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for $alphageq 2$ since 1985: If $alpha>0$, then a function $f$ is nondecreasing and in $Lip^*alpha$, if and only if, there exists a constant $C$ such that, for all sufficiently large $n$, there are nondecreasing polynomials $P_n$, of degree $n$, such that [ |f(x)-P_n(x)| leq C left(frac{sqrt{1-x^2}}{n}right)^alpha, quad xin [-1,1]. ]
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