Volume 8, Issue 1

In this paper we shall find some sufficient conditions for a uniqueness polynomial to be a strong uniqueness polynomial, as this type of problem was never investigated by researchers earlier. We also exhibit some examples to substantiate our theorems.

Two discrete frames {ϕi}i∈I${\{\phi_{i}\}_{i\in I}}$ and {ψi}i∈I${\{\psi_{i}\}_{i\in I}}$ for a separable Hilbert space ℍ${\mathbb{H}}$ are said to be woven if there are universal positive constants A and B such that for every subset σ⊂I${\sigma\subset I}$, the family {ϕi}i∈σ∪{ψi}i∈σc${\{\phi_{i}\}_{i\in\sigma}\cup\{\psi_{i}\}_{i\in\sigma^{c}}}$ is a frame for ℍ${\mathbb{H}}$ with lower and upper frame bounds A and B , respectively. Weaving frames are powerful tools in pre-processing signals and distributed data processing. Motivated by the recent work of Bemrose–Casazza–Gröchenig–Lammers–Lynch and Casazza–Lynch on weaving frames for separable Hilbert spaces, we study continuous weaving frames for Hilbert spaces with respect to a measure space. In this paper, first we present two necessary conditions in terms of frame bounds for continuous weaving frames. A sufficient condition for continuous weaving frames is given. We provide an estimate of a series associated with the metric operators of continuous weaving frames. Finally, two Paley–Wiener-type perturbation results for continuous weaving frames are given.

In this paper, we study a transmission problem given by a system of two nonlinear equations of p⁢(x)${p(x)}$-Kirchhoff type with nonstandard growth conditions. Using a variational approach, we establish at least one nontrivial weak solution.

In a previous paper we introduced a family of differential-reflection operators ΛA,ε${\Lambda_{A,\varepsilon}}$ acting on smooth functions defined on ℝ${{\mathbb{R}}}$, where the spectral problem for the operators ΛA,ε${\Lambda_{A,\varepsilon}}$ has been studied. Here A is a Sturm–Liouville function with additional hypotheses and -1≤ε≤1${-1\leq\varepsilon\leq 1}$. Via the eigenfunctions of ΛA,ε${\Lambda_{A,\varepsilon}}$, we introduce in this paper a generalized Fourier transform ℱA,ε${\mathscr{F}_{A,\varepsilon}}$. An Lp${L^{p}}$-harmonic analysis for ℱA,ε${\mathscr{F}_{A,\varepsilon}}$ is developed when 0<p≤21+1-ε2${0<p\leq{2\over{1+\sqrt{1-\varepsilon^{2}}}}}$. In particular, an Lp${L^{p}}$-Schwartz space isomorphism theorem for ℱA,ε${\mathscr{F}_{A,\varepsilon}}$ is proved.

Our aim is to investigate through a special family of Laguerre–Hahn forms of class one verifying the three-term recurrence relation with βn=(-1)n⁢β0${\beta_{n}=(-1)^{n}\beta_{0}}$, n≥0${n\geq 0}$, by using quadratic decomposition. The corresponding characteristic elements of the three-term recurrence relation are given explicitly.