Gady Kozma, Alexander Olevskii
Inspired by Menshov's representation theorem, we prove that there exists a sequence of frequecies such that any measurable (complex valued) function on R can be represented as a sum of almost everywhere convergent trigonometric series with these frequencies.
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Trigonometric polynomials with frequencies in the set of cubes
该研究针对频率取自立方数集合的三角多项式，证明了其4范数与2范数的一个带绝对隐含常数的不等式，并证明特定区间内的立方数集合是Sidon集。
Using the multilinear theory of oscillatory integral operators, we establish a square function inequality with implications to moment inequalities for discrete exponential sums which frequencies lie on curved hyper-surfaces. In particular a new distributional estimate for toral eigenfunctions is established.
On closed forms of some trigonometric series
本文针对参数替换引发奇点的三角级数，利用Choi-Srivastava定理推导出以Hurwitz zeta函数导数表示的闭式公式。
We show that for every finite symetric set S of integer vectors, every real trigonometric polynomial on the d dimensional torus with spectrum in S has a zero in every closed ball of diameter D, where D is the sum over S of 1 over 4 times the L2 norm of the vector. We investigate tightness in some special cases.
Let $\{c_k\}$ be a nonincreasing sequence of positive numbers (more general classes of sequences are also considered), and $α>0$ be not an integer. We find necessary and sufficient conditions for the uniform convergence of the series $\sum_k c_k\sin k^αx$ and $\sum_k c_k\cos k^αx$ on the real line and its bounded subsets.
Trigonometric polynomials with frequencies in the set of squares
本文研究频率限制在平方数集合中的三角多项式，证明了当频率范围长度为 N^{γ0-ε} 时其 L4 范数可由 L2 范数控制，推进了 Cilleruelo-Cordoba 猜想，并揭示了该猜想与 Ruzsa 因数分布猜想之间的联系。
The generalized summation of divergent trigonometric series, namely by method of $σ_k(r,a)$-factors is considered in this paper. It is proved that such summation of Fourier series of periodical function $f(t)$ results in the convolution of this function with kernels $De(r,α,t)$; if the parameter $r$ is integer, these kernels are polynomial normalized basic $B$-splines of order $r-1$ $(r=1,2,\ldots)$. Also it is proved that the method of summation with $σ_k(r,a)$-factors is $F$-effective.
In 1936 Zygmunt Zalcwasser asked with respect to the trigonometric system that how "rare" can a sequence of strictly monotone increasing integers $(n_j)$ be such that the almost everywhere relation $\frac{1}{N}\sum_{j=1}^N S_{n_j}f \to f$ is fulfilled for each integrable function $f$. In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function $f$ and lacunary sequence $(n_j)$ of natural numbers.
Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series
针对Erdős提出的加法组合学长期问题，通过三角级数的$L^1$估计证明：任意$n$元整数集含大小至少为$n/3+c\boldsymbol{\text{log log}}n$的无和子集，解决了该领域经典难题。
We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (πn τ)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $τ$ a real quadratic irrationality and $s$ an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating the series to combinations of derivatives of Eichler integrals and polylogarithms.
From Ingham to Nazarov's inequality: a survey on some trigonometric inequalities
本文综述了调和与非调和三角多项式在 L^1 和 L^2 范数下的若干不等式，重点介绍 Ingham 不等式、Littlewood 猜想的证明及 Nazarov 的非调和对应结果，并补充了小区间上的定量估计与更强条件下的下界。
Lower bounds on the measure of the support of positive and negative parts of trigonometric polynomials
本文证明三角多项式正负部分支撑测度的最小值有下界，该下界由频率集合的组合特征决定，并可推广至多类函数。
We prove that for all p>1/2 there exists a constant $γ_p>0$ such that, for any symmetric measurable set of positive measure $E\subset \TT$ and for any $γ<γ_p$, there is an idempotent trigonometrical polynomial f satisfying $\int_E |f|^p > γ\int_{\TT} |f|^p$. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $γ_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $γ_p=1$ when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when $p\neq 2$. This shows str…
On L1-norms for non-harmonic trigonometric polynomials with sparse frequencies
本文研究非调和三角多项式的L1范数估计，证明当频率序列间隙趋于无穷时，相关不等式中的时间区间长度T可任意小，推广了已有经典结果，并将其应用于带移动传感器的薛定谔方程能观性分析。
Let $$ A_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j\choose 2j}\sum_{k=0}^{n-1} \sin(a+2kπ/n) \cos^{2j}(a+2kπ/n) $$ and $$ B_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j+1\choose 2j+1}\sum_{k=0}^{n-1} \sin(a+2kπ/n) \cos^{2j+1}(a+2kπ/n), $$ where $m\geq 0$ and $n\geq 1$ are integers and $a$ is a real number. We present two proofs for the following results: (i) If $2m+1 \equiv 0 \, (\mbox{mod} \, n)$, then $$ A_{m,n}(a)=(-1)^m n \sin((2m+1)a). $$ (ii) If $2m+1 \not\equiv 0 \, (\mbox{mod} \, n)$, then $A_{m,n}(a)=0$. (iii) If $2(m+1) \equiv 0 \, (\mbox{mod} \, n)$, then $$ B_{m,n}(a)=(-1)^m \frac{n}{2} \sin(2(m+1)a).…
One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n α)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $α$. Another important result of Weyl states that for every sequence $(n_k)_{k \geq 1}$ of distinct positive integers the sequence of fractional parts of $(n_k α)_{k \geq 1}$ is u.d. mod 1 for almost all $α$. However, in this general case it is usually extremely difficult to classify those $α$ for which uniform distribution occurs, and to measure the speed of convergence of the empirical …
Let $2s$ points $y_i=-π\le y_{2s}<\ldots<y_1<π$ be given. Using these points, we define the points $y_i$ for all integer indices $i$ by the equality $y_i=y_{i+2s}+2π$. We shall write $f\in\bigtriangleup^{(1)}(Y)$ if $f$ is a $2π$-periodic function and $f$ does not decrease on $[y_i, y_{i-1}]$ if $i$ is odd; and $f$ does not increase on $[y_i, y_{i-1}]$ if $i$ is even. We denote $E_n^{(1)}(f;Y)$ the value of the best uniform comonotone approximation. In this article the following counterexample of comonotone approximation is proved. Example. For each $k\in\Bbb N$, $k>3$, and $n\in\Bbb N$ there …
Lommel functions, Padé approximants and hypergeometric functions
研究半整数参数下Lommel函数的多项式与三角函数显式组合，揭示多项式构成三角函数的Padé逼近，并给出整数参数下的积分表示与超几何函数闭式公式。
The Littlewood problem and non-harmonic Fourier series
针对 Littlewood 问题，本文建立非整数频率三角多项式在大区间上的 $L^1$ 范数定量估计，推广并统一了多项经典结果。
Norm bounds on Fourier series with polynomial spectra and constrained coefficients
本文针对谱为多项式的三角多项式，通过系数约束建立了L⁴范数上界，补充了前人成果并解决了特定猜想，确定了对应范数增长的精确阶。
The function $ \tan(πx / 2) / (πx / 2) $ is expanded into a Laurent series of $ 1 - x^2 $, where the coefficients are given explicitly as combinations of zeta function of even integers. This is used to achieve a sequence of upper and lower bounds which are very precise even at the poles $ x = 1, -1 $. Similar results are obtained for other trigonometric functions with poles.
We have discovered three non-power infinite series representations for Bessel functions of the first kind of integer orders and real arguments. These series contain only elementary functions and are remarkably simple. Each series was derived as a Fourier series of a certain function that contains Bessel function. The series contain parameter $b$ by setting which to specific values one can change specific form of series. Truncated series retain qualitatively behaviour of Bessel functions at large $x$: they have sine-like shape with decreasing amplitude. Derived series allow to obtain new series…
In the framework of KAM theory, the persistence of invariant tori in quasi-integrable systems is proved by assuming a non-resonance condition on the frequencies, such as the standard Diophantine condition or the milder Bryuno condition. In the presence of dissipation, most of the quasi-periodic solutions disappear and one expects, at most, only a few of them to survive together with the periodic attractors. However, to prove that a quasi-periodic solution really exists, usually one assumes that the frequencies still satisfy a Diophantine condition and, furthermore, that some external parameter…
A Menshov spectrum is a subset of the integers that is sufficient for representing every measurable function as an almost-everywhere converging trigonometric (non-Fourier) sum. In this language the celebrated "Menshov representation theorem" states that Z is a Menshov spectrum. In this paper we construct 1) Menshov spectra that are almost exponentially sparse 2) that are almost squares. Then we show that the positive integers are not a Menshov spectrum but are a Menshov spectrum in measure, and repeat 1) and 2) in the analytic settings.
For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^γ<\infty$, $γ>4$ implies the almost everywhere convergence of $\sum_{k=1}^\infty c_k f(n_kx)$ for any increasing sequence $(n_k)$ of integers. We also construct an example showing that the previous condition is not sufficient for $γ<2$. Finally we give an a.e. bound for the growth of sums $\sum_{k=1}^N f(n_kx)$ differing from the corresponding optimal result for trigonometric sums by a loglog factor.
Double summation addition theorems for Jacobi functions of the first and second kind
本文推导并分析第一、二类 Jacobi 函数的双曲型与三角型双重求和加法定理，考察其与二次变换超几何函数的关系及 Olver 规范化，并应用于秩一对称空间基本解展开。
On the spectra of Cantor measures
本文研究了线上的康托测度，考虑了收缩因子为$N^{-1}=p^{-α}$（其中$p$为正素数，$α$为正整数）且$m$个正整数位位于不同余类模$N$的测度。文章对这类测度$μ$的极大正交指数集进行了完整刻画，并证明了在极大正交集中的频率的第$n+1$位在前$n$位确定的情况下有$m$种可能值，从而建立了$m$-均匀根树的标记与极大正交频率集之间的对应关系。
We show that a spectrum of frequencies obtained by a random perturbation of the integers allows one to represent any measurable function on R by an almost everywhere converging sum of harmonics almost surely.
It is well-known that for a quickly increasing sequence $(n_k)_{k \geq 1}$ the functions $(\cos 2 πn_k x)_{k \geq 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k)_{k \geq 1}$ is relaxed then this almost-independent behavior generally fails. Still, probabilistic constructions show that for \emph{some} very slowly increasing sequences $(n_k)_{k \geq 1}$ this almost-independence property is preserved. For example, there exists $(n_k)_{k \geq 1}$ having bounded gaps such that the normalized sums $\sum \cos 2 πn_k x$ satisfy the c…
In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $β\in(1,1.787\ldots)$ then every $x\in(0,\frac{1}{β-1})$ has a simply normal $β$-expansion. We also prove that if $β\in(1,\frac{1+\sqrt{5}}{2})$ then every $x\in(0,\frac{1}{β-1})$ has a $β$-expansion for which the digit frequency does not exist, and a $β$-expansion with limiting frequency of zeros $p$, where $p$ is any real number sufficiently close to $1/2$. For a class of planar self-affine sets we sho…
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