Related papers: On pointwise estimates of positive definite functi…

We study the following question: Given an open set $\Omega$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $\Omega$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied…

We study the following question posed by Turan. Suppose K is a convex body in Euclidean space which is symmetric with respect to the origin. Of all positive definite functions supported in K, and with value 1 at the origin, which one has…

The century old extremal problem, solved by Carath\'eodory and Fej\'er, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the…

We study the extremal solution for the problem $(-\Delta)^s u=\lambda f(u)$ in $\Omega$, $u\equiv0$ in $\R^n\setminus\Omega$, where $\lambda>0$ is a parameter and $s\in(0,1)$. We extend some well known results for the extremal solution when…

We study the problem $(-\Delta)^su=\lambda e^u$ in a bounded domain $\Omega\subset\mathbb R^n$, where $\lambda$ is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result…

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$,…

We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=& \lambda g(x) f(u) \qquad \text{in}\ \Omega \hfill u&=& 0 \qquad \qquad \qquad \text{on}\ \pOm, {array}. {eqnarray*} where $ \lambda >0$ is a parameter and $…

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $[-1,1]$ if they are bounded by $1$ on a subset of $[-1,1]$ of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev…

We shall study non-linear extremal problems in Bergman space $\mathcal{A}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials,…

We establish a new $W^{1,2\frac{n-1}{n-2}}$ estimate for the extremal solution of $-\Delta u=\lambda f(u)$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$, which is convex, for arbitrary positive and increasing nonlinearities $f\in…

We study two optimization problems for positive definite functions on Euclidean space with restrictions on their support and sign: the Turan problem and the Delsarte problem. These problems have been studied also for their connections to…

Under general multivariate regular variation conditions, the extreme Value-at-Risk of a portfolio can be expressed as an integral of a known kernel with respect to a generally unknown spectral measure supported on the unit simplex. The…

Let $\ell>0$ be arbitrary. We introduce the extremal quantities $$ G(\ell):=\frac{\sup_{f} \int_{-\ell}^{\ell} f\,dx}{\int_{-1}^1 f\,dx},\quad C(\ell):=\frac{\sup_{f} \sup_{a\in {\mathbb R}} \int_{a-\ell}^{a+\ell} f\,dx}{\int_{-1}^1 f\,dx},…

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we first…

There exists a positive function $\psi(t)${on}$t\geq0${, with fast decay at infinity, such that for every measurable set}$\Omega${in the Euclidean space and}$R>0${, there exist entire functions}$A(x) ${and}$B(x) ${of exponential type}$R${,…

Let $n\ge1$ and $B\ge2$. A real-valued function $f$ defined on the $n$-simplex $\Delta_n$ is approximately convex with respect to $\Delta_{B-1}$ iff f(\sum_{i=1}^B t_ix_i) \le \sum_{i=1}^B t_if(x_i) +1 for all $x_1,...,x_B \in \Delta_n$ and…

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

Let $\Omega$ be an unbounded domain in $\mathbb{R}\times\mathbb{R}^{d}.$ A positive harmonic function $u$ on $\Omega$ that vanishes on the boundary of $\Omega$ is called a Martin function. In this note, we show that, when $\Omega$ is…

Alexandrov's estimate states that if $\Omega$ is a bounded open convex domain in ${\mathbb R}^n$ and $u:\bar \Omega\to {\mathbb R}$ is a convex solution of the Monge-Ampere equation $\det D^2 u = f$ that vanishes on $\partial \Omega$, then…

We consider an extremum problem posed by Turan. The aim of this problem is to find a maximum mean value of 1-periodic continuous even function such that sum of Fourier coefficient modules for this function is equal to 1 and support of this…