Related papers: Complete convergence of the Hilbert transform

Assume that a convergent series of real numbers $\sum\limits_{n=1}^\infty a_n$ has the property that there exists a set $A\subseteq \N$ such that the series $\sum\limits_{n \in A} a_n$ is conditionally convergent. We prove that for a given…

Motivated by Bourgain's work on pointwise ergodic theorems, and the work of Stein and Stein-Wainger on maximally modulated singular integrals without linear terms, we prove that the maximally monomially modulated discrete Hilbert transform,…

Let $h:\mathbb Z \to \mathbb Z_{\geq 0}$ be a nonzero function with $h(k)=0$ for $k\ll 0$. We define the Hilbert depth of $h$ by $\operatorname{hdepth}(h)=\max\{d\;:\; \sum_{j\leq k} (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq…

We prove some results on the behavior of infinite sums of the form $\Sigma f\circ T^n(x)\frac{1}{n}$, where $T:S^1\to S^1$ is an irrational circle rotation and $f$ is a mean-zero function on $S^1$. In particular, we show that for a certain…

Let $\{A_{i,n}\}$ be a triangular array of elements in a Banach algebra, whose norms do not grow too fast, and whose row averages converge to $A$. Let $\sigma \in S(n)$ be a permutation drawn uniformly at random. If the array only contains…

For any $s\in (1/2,1]$, the series$F_s(x)=\sum_{n=1}^{\infty} e^{i\pi n^2 x}/n^s$ converges almost everywhere on $[-1,1]$ by a result of Hardy-Littlewood, but not everywhere. However, there does not yet exist an intrinsic description of the…

For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take…

For a closed minimal immersed hypersurface $M$ in $\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\ge 2$, define a constant $\sigma_k=\dfrac{\int_M (|A|^2)^k}{|M|}$. We show that $\sigma_k \ge 2^k$ provided $n=2$ and…

The subspaces and subalgebras of B(H) which are hyperreflexive with constant 1 are completely classified. It is shown that there are 1-hyperreflexive subspaces for which the complete hyperreflexivity constant is strictly greater than 1. The…

Suppose $A$ is a pro-C*-algebra. Let $L_{A}(E)$ be the pro-C*-algebra of adjointable operators on a Hilbert $A$-module $E$ and let $K_{A}(E)$ be the closed two sided $*$-ideal of all compact operators on $E$. We prove that if $E$ be a full…

Let $A$ be a $m\times m$ complex matrix with zero trace. Then there are $m\times m$ matrices $B$ and $C$ such that $A=[B,C]$ and $\|B\|\|C\|_2\le (\log m+O(1))^{1/2}\|A\|_2$ where $\|D\|$ is the norm of $D$ as an operator on $\ell_2^m$ and…

We provide a new upper bound for sampling numbers $(g_n)_{n\in \mathbb{N}}$ associated to the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants…

If $(k,\ell)$ is an exponent pair such that $k+\ell<1$, then we have $$ \int_1^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^2dt \ll_\epsilon T^{1+\epsilon}\quad(\sigma > \min({5\over6},\max(\ell-k, {5k+\ell\over4k+1})), $$ while if $(k,\ell)$ is an…

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…

We study non-stationary averaging processes, where each term of a sequence is a weighted average of previous terms, namely $a_{n+1} = \sum_{j=1}^n p_n(j) a_j$. Our results extend classical theory in two distinct regimes. First, we prove a…

Given k>1, let a_n be the sequence defined by the recurrence a_n=c_1a_{n-1}+c_2a_{n-2}+...+c_ka_{n-k} for n>=k, with initial values a_0=a_1=...=a_{k-2}=0 and a_{k-1}= 1. We show under a couple of assumptions concerning the constants c_i…

We derive $p$-adic expansions for the generalized Harmonic numbers $H^{(j)}_{p-1}$ and $H^{(j)}_{\frac{p-1}{2}}$ involving the Bernoulli numbers $B_j$ and the the base-2 Fermat quotient $q_p$. While most of our results are not new, we…

A real sequence $(a_k)_{k=0}^\infty$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix $ \left\| a_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (here $a_k=0$ for $k<0$). In this paper, which continues our…

We investigate the convergence of signed null sequences of the form \[ \sum_{n=1}^\infty \varepsilon_n a_n, \quad \varepsilon_n \in \{-1,1\}, \] where $(a_n)$ tends to zero in $\mathbb{R}^d$. Our main result shows that for any such…

Neither the Euler-Mascheroni constant, $\gamma=0.577215\ldots$, nor the Euler-Gompertz constant, $\delta=0.596347\ldots$, is currently known to be irrational. However, it has been proved that these two numbers are disjunctively…