We extend the property (N) introduced by Jameson for closed convex cones to normal a finite collection of sets in Hilbert space. Variations property, such as weak and uniform are also introduced. A dual form is derived. When applied cones, (G) Jameson. Normality provides new perspective on relationship between strong conical hull intersection (strong CHIP) various regularity properties. In particular, we prove that characterization CHIP, linear regularity. Moreover, equivalent fact normality...

It has been proved that algebraic polynomials $\mathcal {P}$ are dense in the space $L^{p}({\mathbb {R}},d\mu )$, $p\in (0, \infty iff measure $\mu$ is representable as $d\mu =w^p d\nu$ with a finite non-negative Borel $\nu$ and an upper semi-continuous function $w:\mathbb {R}\to \mathbb {R}^+: =[0,\infty )$ such subset of $C^0_w : = \{f\in C(\mathbb {R}) w(x)f (x)\to 0 \mbox {as} |x|\to \}$ equipped seminorm $\| f \|_{w}:= \sup _{x \in {\mathbb {R}}} w(x)|f(x)|$. The similar representation...

A deep result of J. Lewis (1983) shows that the polylogarithms $Li_\alpha (z)$ $:=$ $\sum _{k=1}^{\infty }z^k/k^\alpha$ map open unit disk $\mathbb {D}$ centered at origin one-to-one onto convex domains for all $\alpha \geq 0$. In present paper this is generalized to so-called universal convexity and starlikeness (with respect origin) in slit-domain $\Lambda := \mathbb {C}\setminus [1,\infty )$, introduced by S. Ruscheweyh, L. Salinas T. Sugawa (2009). This settles a conjecture made work...

In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions ([Formula: see text]), which a function could be uniquely reconstructed values it and its transform on discrete set, with striking application resolving sphere packing problem [Formula: text] this short note, we present an alternative approach such results, viable even dimensions, based instead for Klein-Gordon equation. Since existing method is study iterations...

We give characterisations of certain positive finite Borel measures with unbounded support on the real axis so that algebraic polynomials are dense in all spaces Lp(R, dμ), p≥1. These conditions apply, particular, to satisfying classical Carleman conditions.

In an effort to extend classical Fourier theory, Hedenmalm and Montes-Rodr\'{\i}guez (2011) found that the function system \[ e_m(x)=e^{i\pi mx},\quad e_n^\dagger(x)=e_n(-1/x)=e^{-i\pi n/x} \] is weak-star complete in $L^{\infty}(\mathbb{R})$ when $m,n$ range over integers with $n\ne0$. It turns out can be used provide unique representation of functions more generally distributions on real line $\mathbb{R}$. For instance, we may represent uniquely unit point mass at a $x\in\mathbb{R}$:...

We describe all zero-diminishing sequences (over the real-valued polynomials on $\mathbb {R}$) which additionally satisfy a Carleman condition and show that they are of same kind as those in E. Laguerre’s theorem from 1884.

A calculation formula is established for the codimension of polynomial subspace in $L_2 ({\mathbb {R}}, d \mu )$ with discrete indeterminate measure $\mu$. We clarify how much masspoint $n$-canonical solution an Hamburger moment problem differs from corresponding $N$-extremal at a given point real axis.

In this article we explain the essence of interrelation described in [PNAS 118, 15 (2021)] on how to write explicit interpolation formula for solutions Klein-Gordon equation by using recent Fourier pair Viazovska and Radchenko from [Publ Math-Paris 129, 1 (2019)]. We construct explicitly sequence $L^1 (\mathbb{R} )$ which is biorthogonal system $1$, $\exp ( i \pi n x)$, n/ $n \in \mathbb{Z} \setminus \{0\}$, show that it complete (\mathbb{R})$. associate with each $f L^1 (\mathbb{R},...