arXiv subjects
Publications and source records attributed to Saulius Norvidas.
In this paper, we extend the large sieve type estimates to sums involving pth powers of trigonometric polynomials. An approach to such estimates that does not rely on the usual L^2-technique is given. Our method is based on comparing the norm and the spectral radius of convolution operators on a normed space of trigonometric polynomials.
Let $PD(\mathbb{R})$ be the family of continuous positive definite functions on $\mathbb{R}$. For an integer $n>1$, a $f\in PD(\mathbb{R})$ is called $n$-divisible if there is $g\in PD(\mathbb{R})$ such that $g^n=f$. Some properties of infinite-divisible and $n$-divisible functions may differ in essence. Indeed, if $f$ is infinite-divisible, then for each integer $n>1$, there is an unique $g$ such that $g^n=f$, but there is a $n$-divisible $f$ such that the factor $g$ in $g^n=f$ is generally not unique. In this paper, we discuss about how rich can be the class $\{g\in PD(\mathbb{R}): g^n=f\}$ for $n$-divisible $f\in PD(\mathbb{R})$ and obtain precise estimate for the cardinality of this class.
We propose a necessary and sufficient condition for a real-valued function on the real line to be a characteristic function of a probability measures. The statement is given in terms of harmonic functions and completely monotonic functions.
We derive necessary and sufficient conditions for a continuous bounded function $f: R\to C$ to be a characteristic function of a probability measure. The Cauchy transform $K_f$ of $f$ is used as analytic continuation of $f$ to the upper and lower half-planes in $C$. The conditions depend on the behavior of $K_f(z)$ and its derivatives on the imaginary axis in $C$. The main results are given in terms of completely monotonic and absolutely monotonic functions.
Let $f:R\to C$ be the characteristic function of a probability measure. We study the following question: Is it true that for any closed interval $I$ on $R$, which does not contain the origin, there exists a characteristic function $g$ such that $g$ coincides with $f$ on $I$ but $g \not\equiv f$ on $R$?
Assume that $f$ is the characteristic function of a probability measure $μ_f$ on $R^n$. Let $σ>0$. We study the following extrapolation problem: under what conditions on the neighborhood of infinity $V_σ=\{x\in R^n: |x_k|>σ, \ k=1,\dots, n\}$ in $R^n$ does there exist a characteristic function $g$ on $R^n$ such that $g=f$ on $V_σ$, but $g\not\equiv f$? Let $μ_f$ have a nonzero absolutely continuous part with continuous density $φ$. In this paper certain sufficient conditions on $φ$ and $V_σ$ are given under which the latter question has an affirmative answer. We also address the optimality of these conditions. Our results indicate that not only does the size of both $V_σ$ and the support ${\text{\,supp}\,}φ$ matter, but also certain arithmetic properties of ${\text{\,supp}\,}φ$.
For $σ>0$, the Bernstein space \ $B^1_σ$ consists of those $L^1(R)$\ functions whose Fourier transforms are supported by $[-σ,σ]$. Since $B^1_σ$ is separable and dual to some Banach space, the closed unit ball $D(B^1_σ)$ of $B^1_σ$\ has sufficiently large sets of both exposed and strongly exposed points. Moreover, $D(B^1_σ)$ coincides with the closed convex hull of its strongly exposed points. We investigate some properties of exposed points, construct several examples and obtain as corollaries the relations between the sets of exposed, strongly exposed, weak$^{\ast}$ exposed, and weak$^{\ast}$ strongly exposed points of $D(B^1_σ)$.
Let $σ>0$. For $1\le p\le \infty$, the Bernstein space $B^p_σ$ is a Banach space of all $f\in L^p(R)$ such that $f$ is bandlimited to $σ$; that is, the distributional Fourier transform of $f$ is supported in $[-σ, σ]$. We study the approximation of\ $f\in B^p_σ by finite trigonometric sums \[ P_τ(x)=χ_τ(x) \sum_{|k|\le στ/π}c_{k,τ} e^{i\fracπτk x } \] in $L^p$ norm on $R$ as\ $τ\to\infty$,\ where\ $χ_τ$ denotes the indicator function of $[-τ, τ]$.
Suppose that $f$ is the characteristic function of a probability measure on the real line $\R$. In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that $f$ is never determined by its imaginary part $\Im f$? In other words, is it true that for any characteristic function $f$ there exists a characteristic function $g$ such that $\Im f\equiv \Im g$ but $ f\not\equiv g$? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts.
Let $ A$ be a complex unital Banach algebra. An element $a \in A$ is said to be Hermitian, if $ \| \exp (ita) \| =1$ for all $t\in R$. In the case of the algebra of bounded linear operators in a Hilbert space this Hermitian property agrees with the ordinary selfadjointness. If $a \in A$ is Hermitian, then $|a|=||a||$, where $|a|$ denotes the spectral radius of $a$. A function $F: R\to \C$ is called the universal symbol if $ \|| F(a)||= |F(a)|$\ for each $ A$ and all Hermitian $a\in A$. We characterize universal symbols in terms of positive definite functions.
Let $CH(R)$ denote the family of characteristic functions of probability measures (distributions) on the real line $R$. We study the following question: given an integer $n>1$, do there exist two different $f, g\in CH(R)$ such that $ f^n\equiv g^n$? For positive even $n$, well-known examples answer this question in the affirmative. It turns out that the same is true also for any odd $n>1$. For $f\in CH(R)$ and integer $n>1$, set $C_n(f)=\{g\in CH(R): g^n\equiv f^n\}$. In this paper, we give an estimate for cardinality (or cardinal number) of $C_n(f)$. In addition, we describe such $f$ for which our estimate is sharp.
We provide necessary and sufficient conditions for a tempered distribution $F\in S'(R)$ to be positive definite. A generalized Cauchy transform $\widetilde{F}$ of $F$ is used as a numerical continuation of $F$ to the open upper and lower complex half-planes in $C$. In fact, our necessary and sufficient conditions for $F$ are determined completely by the properties of the restriction of $\widetilde{F}$ to the imaginary axis in $C$. The main result is given in terms of completely monotonic and absolutely monotonic functions.
Let $B^p_σ$, $1\le p<\infty$, $σ>0$, denote the space of all $f\in L^p(\mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-σ,σ]$. The classical sampling theorem states that each $f\in B^p_σ$ may be reconstructed exactly from its sample values at equispaced sampling points $\{πm/σ\}_{m\in\mathbb{Z}} $ spaced by $π/σ$. Reconstruction is also possible from sample values at sampling points $\{πθm/σ\}_m $ with certain $1< θ\le 2$ if we know $f(θπm/σ) $ and $f'(θπm/σ)$, $m\in\mathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives.