Related papers: Improving estimates for discrete polynomial averag…

Let $f\in \ell^2(\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \ell ^{p}$-improving estimate, for $ 3/2 < p \leq 2$:…

We study averages along the integers using the divisor function $d(n)$, and defined as $$K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) \,f(x+n) , $$ where $D(N) = \sum _{n=1} ^N d(n) $. We shall show that these averages satisfy a…

In this paper we give a short proof of the $\ell^p$-improving property of the average operator along the square integers and more general quadratic polynomials. Moreover we obtain a similar result for some higher degree polynomials. We also…

Let $ \Lambda $ denote von Mangoldt's function, and consider the averages \begin{align*} A_N f (x) &=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)\Lambda(n) . \end{align*} We prove sharp $ \ell ^{p}$-improving for these averages, and sparse bounds…

We apply Christ's method of refinements to the $\ell^p$-improving problem for discrete averages $\mathcal{A}_N$ along polynomial curves in $\mathbb{Z}^d$. Combined with certain elementary estimates for the number of solutions to certain…

Let $ \lambda ^2 \in \mathbb N $, and in dimensions $ d\geq 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 $…

We prove $q$-variation estimates, $q>2$, on $\ell^{p}$ spaces for averages along primes (with $1<p<\infty$) and polynomials (with $\big| \frac1p - \frac12 \big| < \frac{1}{2(d+1)}$, where $d$ is the degree of the polynomial). This improves…

Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise…

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our…

We decompose the discrete bilinear spherical averaging operator into simpler operators in several ways. This leads to a wide array of extensions, such as to the simplex averaging operator, and applications, such as to operator bounds.

We initiate the theory of $\ell^p$-improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove $\ell^p$-improving estimates for the discrete spherical averages and some of their…

Let d>2 and let p be a prime coprime to d. Let Z_pbar be the ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. Let P be a prime ideal over p in the ring of integers of Q(f), where Q(f) is the number…

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to…

In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ (IMRN, 1998), it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a…

Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell…

We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are…

Given a family $\varphi= (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $\rho$, we consider the mean value $$ M_{k, \rho} (\varphi, N) =…

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{F}_{q}[x]$ be a polynomial of degree $d > 0$. Denote the image set of this polynomial as $V_{f}=\{f(\alpha)\mid\alpha\in\mathbb{F}_{q}\}$ and denote the…

The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) +…