Related papers: On existence of Becker extension

Let $Y=(Y(t))_{t\geq0}$ be a zero-mean Gaussian stationary process with covariance function $\rho:\mathbb{R}\to\mathbb{R}$ satisfying $\rho(0)=1$. Let $f:\mathbb{R}\to\mathbb{R}$ be a square-integrable function with respect to the standard…

We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;\Omega]:=\int_{\Omega}F(\nabla u)\,\mathrm{d} x,\qquad u\colon\Omega\to\mathbb{R}^{N},…

The Fourier-Walsh expansion of a Boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ is its unique representation as a multilinear polynomial. The Kindler-Safra theorem (2002) asserts that if in the expansion of $f$, the total weight…

Let $f$ be a real function defined on the interval $[0,1]$ which is constant on $(a,b)\subset [0,1]$, and let $B_nf$ be its associated $n$th Bernstein polynomial. We prove that, for any $x\in (a,b)$, $|B_nf(x)-f(x)|$ converges to $0$ as…

In recent literature concerning integer partitions one can find many results related to both the Bessenrodt-Ono type inequalities and log-concavity property. In this note we offer some general approach to this type of problems. More…

The Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2. We show that (BC) holds for all graphs with n vertices if n…

In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in…

Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \[ \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq…

Let K be any compact set in the complex plane that has a connected complement, let A(K) be the uniforn algebra of all continuous complex functions on K that are holomorphic on the interior of K, let bK be the topological boundary of K, let…

We study the problem of extending a positive-definite operator-valued kernel, defined on words of a fixed finite length from a free semigroup, to a global kernel defined on all words. We show that if the initial kernel satisfies a natural…

We show there exists a constant $0<c_{0}<1$ such that the dimension of every measure on $[0,1]$, which makes the digits in the continued fraction expansion independent, is at most $1-c_{0}$. This extends a result of Kifer, Peres and Weiss…

R. K\"ustner proved in his 2002 paper that the function $w_{a,b,c}(z)=$ $F(a+1,b;c;z)/F(a,b;c;z)$ maps the unit disk $|z|<1$ onto a domain convex in the direction of the imaginary axis under some condition on the real parameters $a,b,c.$…

We prove that for a homogeneous linear partial differential operator $\mathcal A$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation…

We show that if a closed discrete subset $A \subseteq \mathbf{R}^d$ is denser than a certain critical threshold, then $A$ is a Fourier uniqueness set, while if $A$ is sparser, then uniqueness fails and one can prescribe arbitrary values for…

We prove general equidistribution statements (both conditional and unconditional) relating to the Fourier coefficients of arithmetically normalized holomorphic Hecke cusp forms $f_1,\ldots,f_k$ without complex multiplication, of equal…

We improve upon a result of Steinerberger (2024) by demonstrating that for any fixed $k \in \mathbb{N}$ and sufficiently large $n$, there exist integers $1 \leq a_1, \dots, a_k \leq n$ satisfying: \begin{align*} 0 < \left\| \sum_{j=1}^{k}…

If alpha is an irrational number, we define Yoccoz's Brjuno function Phi by Phi(alpha)=sum_{n geq 0} alpha_0*alpha_1*...*alpha_{n-1}*log(1/alpha_n), where alpha_0 is the fractional part of alpha and alpha_{n+1} is the fractional part of…

In the 1950s L. Schwartz proved his famous impossibility result: for every k in N there does not exist a differential algebra (A,+,*,D) in which the distributions can be embedded, where D is a linear operator that extends the distributional…

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no…

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times…