Related papers: On some positive definite functions
Systems of the form $x = (A x^s)^{1/s} + b$ arise in a range of economic, financial and control problems, where $A$ is a linear operator acting on a space of real-valued functions (or vectors) and $s$ is a nonzero real value. In these…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
In this article, we study the properties of a class of functional spaces which arise from the investigation of nonlinear differential equations. We establish some integral inequalities then by applying these inequalities, we prove some…
A classification of upper semicontinuous, translation and dually epi-translation invariant valuations is established on the space of convex Lipschitz function on $\mathbb{R}$ with compact domain.
After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner.
We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of…
The action of the finite Hilbert transform defined on $L^\infty(-1,1)$ and taking its values in the Zygmund space $L_{\textnormal{exp}}(-1,1)$ is studied in detail. This is a reciprocal situation to the investigation recently undertaken in…
We study upper and lower estimates as well as the asymptotic behavior of the sharp constant $C=C_n(U,V)$ in the doubling-type condition at the origin \[ \frac{1}{|V|}\int_{V}f(x)\,dx\le C\,\frac{1}{|U|}\int_{U}f(x)\,dx, \] where $U,V\subset…
We study some properties convex functions fulfill. Among the conclusions we obtain from such result, we are able to prove some nontrivial inequalities among real numbers, and we give an improvement of the reverse triangle inequality in the…
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…
In this study, Firstly, we will write two new convex functions for $-1<n-\alpha \leq 1\ $and two new lemmas. Then we will find the relevance of the two new lemmas to Caputo-left-sided derivatives under additional conditions and draw…
We present a characterization of spaces of strictly decreasing functions on trees in terms of bisequentiality. This characterization answers Questions 6.1 and 6.2 of "A filter on a collection of finite sets and Eberlein compacta" by T.…
We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic…
The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
Let ${\mathcal A}$ be the class of functions $f$ that are analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. Let $0<\lambda\le1$ and \[ {\mathcal U}(\lambda) = \left\{ f\in{\mathcal A}: \left…
Let $X$ be a rearrangement-invariant space over a non-atomic $\sigma$-finite measure space $(\mathscr{R},\mu)$ and let $\alpha\in(0,\infty)$. We define the functional \begin{equation*} \|f\|_{X^{\langle \alpha \rangle}} =…
Recursive domain equations have natural solutions. In particular there are domains defined by strictly positive induction. The class of countably based domains gives a computability theory for possibly non-countably based topological…
In this paper, we consider the existence (and nonexistence) of solutions to \[ -\mathcal{M}_{\lambda,\Lambda}^\pm (u'') + V(x) u = f(u) \quad {\rm in} \ \mathbf{R} \] where $\mathcal{M}_{\lambda,\Lambda}^+$ and…
Assume that $f$ is a real $\rho$-harmonic function of the unit disk $\mathbb{D}$ onto the interval $(-1,1)$, where $\rho(u,v)=R(u)$ is a metric defined in the infinite strip $(-1,1)\times \mathbb{R}$. Then we prove that $|\nabla…