经典分析与常微分方程

Aichinger's equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger's equation characterizes polynomial functions…

The lower dimension $\dim_L$ is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu introduced the modified lower dimension $\dim_{ML}$ by making the lower dimension monotonic with the simple formula…

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of $\text{Homeo}([0,1])$ has length $2$. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in…

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The…

Strichartz inequality for the solutions of free Schr\"odinger equation associated with Dunkl Hermite operator $H_\kappa$ is generalized to any system of orthonormal functions with initial data. A relation between the kernels of…

J. Marcinkiewicz and A. Zygmund proved in 1936 that, for all functions $f$ and points $x$, the existence of the $n$th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of both $f_{(n-1)}(x)$ and the $n$th generalized Riemann…

For each positive integer $n$, function $f$, and point $c$, the GGR Theorem states that $f$ is $n$ times Peano differentiable at $c$ if and only if $f$ is $n-1$ times Peano differentiable at $c$ and the following $n$-th generalized…

For each positive integer $n$, function $f$, and point $x$, the 1998 conjecture by Ghinchev, Guerragio, and Rocca states that the existence of the $n$-th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of all $n(n+1)/2$…

We discuss boundedness properties of certain classes of discrete bilinear operators that are similar to those of the continuous bilinear pseudodifferential operators with symbols in the H\"ormander classes $BS^{\omega}_{1, 0}$. In…

The ternary Cantor set $C$, constructed by George Cantor in 1883, is probably the best-known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article will delve into the…

For the kernel $B_{\kappa,a}(x,y)$ of the $(\kappa,a)$-generalized Fourier transform $\mathcal{F}_{\kappa,a}$, acting in $L^{2}(\mathbb{R}^{d})$ with the weight $|x|^{a-2}v_{\kappa}(x)$, where $v_{\kappa}$ is the Dunkl weight, we study the…

The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the $d$-dimensional integer lattice, $\mathbb{Z}^d$. Under certain mild assumptions on the distribution, the…

In this paper we consider weighted Morrey spaces ${\mathcal M}_{\lambda, {\mathcal F}}^p(w)$ adapted to a family of cubes ${\mathcal F}$, with norm $$\|f\|_{{\mathcal M}_{\lambda, {\mathcal F}}^p(w)}:=\sup_{Q\in {\mathcal…

In 2011, Armentano, Beltr\'an and Shub obtained in \cite{ABS11} a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different…

A set of complex numbers $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ with multiple terms \[ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:= \{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},…

A study of the greatest possible ratio of the smallest absolute value of a higher derivative of some function, defined on a bounded interval, to the L p-norm of the function.

We prove mean convergence of the Fourier series in Akhiezer-Chebyshev polynomials in $L^p$, $p>1$, using a weighted inequality for the Hilbert transform in an arc of the unit circle.

We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cos\theta)\cos\phi,\,(t+s\cos\theta)\sin\phi,\,…

Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence…

We deal with a planar differential system of the form \begin{equation*} \begin{cases} \, u' = h(t,v), \\ \, v' = - \lambda a(t) g(u), \end{cases} \end{equation*} where $h$ is $T$-periodic in the first variable and strictly increasing in the…