In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers $p< q$ do the equalities $T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$ , $k=p, q$ , imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^*$ -algebras is also provided.

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in $\mathbb{R}^d$ of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.

Multi-compartment models described by systems of linear ordinary differential equations are considered. Catenary models are a particular class where the compartments are arranged in a chain. A unified methodology based on the Laplace transform is utilised to solve direct and inverse problems for multi-compartment models. Explicit formulas for the parameters in a catenary model are obtained in terms of the roots of elementary symmetric polynomials. A method to estimate parameters for a general multi-compartment model is also provided. Results of numerical simulations are presented to illustrate the effectiveness of the approach.

The goal of the paper is to obtain analogs of the sampling theorems and of the Riesz–Boas interpolation formulas which are relevant to the discrete Hilbert and Kak–Hilbert transforms in $l^{2}$ .

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

This paper considers logarithmic asymptotics of tails of randomly stopped sums. The stopping is assumed to be independent of the underlying random walk. First, finiteness of ordinary moments is revisited. Then the study is expanded to more general asymptotic analysis. Results are applicable to a large class of heavy-tailed random variables. The main result enables one to identify if the asymptotic behaviour of a stopped sum is dominated by its increments or the stopping variable. As a consequence, new sufficient conditions for the moment determinacy of compounded sums are obtained.

We consider nonlocal equations of the general form

\begin{equation} \left(a*u''\right)(\cdot)+\lambda f\big(\cdot,u(\cdot)\big)=0.\nonumber \end{equation}

We investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.

Truncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate functions and the adjoint to that Hausdorff operator of the given function. We find estimates for the rate of approximation in various metrics in terms of the parameter of truncation and the components of the Hausdorff operator. Explicit rates of approximation of functions and comparison with approximate identities are given in the case of continuous functions from the class $\text {Lip }\alpha $ .

Let M(u), H(u) be the maximal operator and Hilbert transform along the parabola (t, ut2). For U ⊂ (0, ∞) we consider Lp estimates for the maximal functions sup u∈U|M(u)f| and sup u∈U|H(u)f|, when 1 < p ≤ 2. The parabolas can be replaced by more general non-flat homogeneous curves.

The longest gap $L(t)$ up to time $t$ in a homogeneous Poisson process is the maximal time subinterval between epochs of arrival times up to time $t$; it has applications in the theory of reliability. We study the Laplace transform asymptotics for $L(t)$ as $t\rightarrow \infty$ and derive two natural and different large-deviation principles for $L(t)$ with two distinct rate functions and speeds.

We investigate convergence in the cone of completely monotone functions. Particular attention is paid to the approximation of and by exponentials and stretched exponentials. The need for such an analysis is a consequence of the fact that although stretched exponentials can be approximated by sums of exponentials, exponentials cannot in general be approximated by sums of stretched exponentials.

This paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

We give an upper estimate for the order of the entire functions in the Nevanlinna parameterization of the solutions of an indeterminate Hamburger moment problem. Under a regularity condition this estimate becomes explicit and takes the form of a convergence exponent. Proofs are based on transformations of canonical systems and I.S.Kac' formula for the spectral asymptotics of a string. Combining with a lower estimate from previous work, we obtain a class of moment problems for which order can be computed. This generalizes a theorem of Yu.M.Berezanskii about spectral asymptotics of a Jacobi matrix (in the case that order is ⩽ 1/2).

We establish the mapping properties of Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. In particular, we have the mapping properties of the Laplace transform, the Hankel transforms, the Kontorovich-Lebedev transform and some oscillatory integral operators. We achieve these mapping properties by using an interpolation functor that can explicitly generate a given rearrangement-invariant quasi-Banach function space via Lebesgue spaces.

We characterize those non-negative, measurable functions ψ on [0, 1] and positive, continuous functions ω1 and ω2 on ℝ+ for which the generalized Hardy–Cesàro operator

$$\begin{equation*}(U_{\psi}f)(x)=\int_0^1 f(tx)\psi(t)\,dt\end{equation*}$$

We investigate conditions in order to decide whether a given sequence of real numbers represents expected maxima or expected ranges. The main result provides a novel necessary and sufficient condition, relating an expected maxima sequence to a translation of a Bernstein function through its Lévy–Khintchine representation.

The valuation of perpetual timer options under the Hull–White stochastic volatility model is discussed here. By exploring the connection between the Hull–White model and the Bessel process and using time-change techniques, the triple joint distribution for the instantaneous volatility, the cumulative reciprocal volatility and the cumulative realized variance is obtained. An explicit analytical solution for the price of perpetual timer call options is derived as a Black–Scholes–Merton-type formula.

We obtain explicit inversion formulas for the Radon-like transform that assigns to a function on the unit sphere the integrals of that function over hemispheres lying in lower-dimensional central cross-sections. The results are applied to the determination of star bodies from the volumes of their central half-sections.