Related papers: Optimal $C^\infty$-approximation of functions with…

We consider gradient flow/gradient descent and heavy ball/accelerated gradient descent optimization for convex objective functions. In the gradient flow case, we prove the following: 1. If $f$ does not have a minimizer, the convergence…

We prove the exponential law $\mathcal A(E \times F, G) \cong \mathcal A(E,\mathcal A(F,G))$ (bornological isomorphism) for the following classes $\mathcal A$ of test functions: $\mathcal B$ (globally bounded derivatives), $W^{\infty,p}$…

We investigate the behavior of sequences $(f(c_nx))$ for Lebesgue integrable functions $f:\mathbb R^d\to\mathbb R$. In particular, we give a~description of classes of multipliers $(c_n)$ and $(d_n)$ such that $f(c_nx)\to0$ or…

We propose a homogenized supremal functional rigorously derived via $L^p$-approximation by functionals of the type $\underset{x\in\Omega}{\mbox{ess-sup}}\hspace{0.03cm} f\left(\frac{x}{\varepsilon}, Du\right)$, when $\Omega$ is a bounded…

We show that for any uniformly elliptic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term one can find an approximating equation which has a unique continuous and having the second…

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply…

Let $(T,d)$ be a metric space and $\phi:\mathbb{R}_+\to \mathbb{R}$ an increasing, convex function with $\phi(0)=0$. We prove that if $m$ is a probability measure $m$ on $T$ which is majorizing with respect to $d,\phi$, that is,…

The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_\Omega F(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(\Omega , \mathbb R^N )\cap K$, where $K$ is a closed convex…

We obtained exact order estimates of approximation of the classes $S^{\boldsymbol{r}}_{1,\theta}B$ by entire functions of exponential type with supports of their Fourier transforms in step hyperbolic cross. The error of the approximation…

Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to…

We introduce a numerical method for the approximation of functions which are analytic on compact intervals, except at the endpoints. This method is based on variable transforms using particular parametrized exponential and…

In this paper we study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Walsh-Kaczmarz-Fourier series. In particular, it is proved that the Marcinkiewicz type of two-dimensional Walsh-Kaczmarz-Fourier…

Given an increasing function $H:[0,1)\to [0,\infty)$ and $$ A_n(H):=\inf_{\tau\in \mathcal{T}_n}(\sum_{i=1}^n \int_{t_{i-1}}^{t_i} (t_i-t)H^2(t)dt)^{{1/2}}, $$ where $\mathcal{T}_n:=\{\tau=(t_i)_{i=0}^n: 0=t_0<t_1<...<t_n=1\}$, we…

We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace--Beltrami operator and provides a global measure of…

We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…

In this paper, we study the properties of integral functionals induced on $L^1_E (S,\mu)$ by closed convex functions on a Euclidean space $E$. We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We…

Let $d \ge 1$, $p \ge d$, and let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\Omega$ such that…

We prove a $\mathsf{C}^1$-elliptic estimate of the form $ \sup_{B(x,r/2)} |\mathrm{grad} (\psi) | \leq C \left\{ \sup_{B(x,r)} |\Delta \psi| + \sup_{B(x,r)} |\psi| \right\}, $ valid on any complete Riemannian manifold $M$ and for any smooth…

We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to…

We propose a novel study of the stochastic proximal gradient method for minimizing the sum of two convex functions, one of which is smooth. Under suitable assumptions and without requiring any boundedness or control of the variance of the…