经典分析与常微分方程

A nonnegative real function $f$ is said to be bell-shaped if it converges to zero at $\pm\infty$ and the $n$th derivative of $f$ changes sign $n$ times for every $n = 0, 1, 2, \ldots$ In a similar way, we may say that a nonnegative sequence…

This is a survey article on an old topic in classical analysis. We present some new developments in asymptotics in the last fifty years. We start with the classical method of Darboux and its generalizations, including an uniformity…

If $\{x_n\}_{n \in \mathbb{N}}$ is a frame for a Hilbert space $H,$ then there exists a canonical dual frame $\{\tilde{x_n}\}_{n \in \mathbb{N}}$ such that for every $x \in H$ we have $x = \sum \langle x, \tilde{x_n} \rangle \, x_n,$ with…

Let $C(a ),C(b)\subset \lbrack 0,1]$ be the central Cantor sets generated by sequences $ a,b \in (0,1)^{\mathbb{N}}$. The first main result of the paper gives a necessary and a sufficient condition for sequences $a$ and $b$ which inform…

Let $\Delta_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{it\Delta_{\mathbb{T}^m\times…

In this paper we consider the problem of estimation of oscillatory integrals with Mittag-Leffler functions in two variables. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study…

We consider two sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ with respect regular functionals ${\bf u}$ and ${\bf v}$, respectively. We assume that $$\sum_{j=1} ^{M} a_{j,n}\mathrm{D}_x ^k P_{k+n-j}…

In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are…

We consider the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$…

On the sets of $2\pi$-periodic functions $f$, which are defined with a help of $(\psi, \beta)$-integrals of the functions $\varphi$ from $L_{1}$, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier…

We present a summation rule using the Mellin transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the…

Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of…

Here we present a new approach to deal with first order ordinary differential equations (1ODEs), presenting functions. This method is an alternative to the one we have presented in [1]. In [2], we have establish the theoretical background…

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most…

We deal with multivariate Brass-Stancu-Kantorovich operators depending on a non-negative integer parameter and defined on the space of all Lebesgue integrable functions on a unit hypercube. We prove $L^{p}$-approximation and provide…

In the cases where there is no Sobolev-type or Gagliardo-Nirenberg-type fractional estimate involving $\lvert u\rvert_{W^{s,p}}$, we establish alternative estimates where the strong $L^p$ norms are replaced by Lorentz norms.

We study a limiting case of the Askey-Wilson polynomials when one of the parameters goes to infinity, namely continuous dual q-Hahn polynomials when q > 1. Solutions to the associated indeterminate moment problem by general theory are found…

In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete…

We show that any weighted geometric mean of Chebyshev polynomials is bounded from above by another Chebyshev polynomial. We also study a related homogeneous cyclic inequality $$ \left (\sum_{i=1}^n x_i^{(a+b+1)/2} \right )^2 \geq…

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its…