Summable Coefficients Research Articles

On Solutions of Systems of Differential Equations on Half-Line with Summable Coefficients

On Solutions of Systems of Differential Equations on Half-Line with Summable Coefficients

Composition operators and generalized primes

We study composition operators on the Hardy space H 2 \mathcal {H}^2 of Dirichlet series with square summable coefficients. Our main result is a necessary condition, in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on H 2 \mathcal {H}^2 . To do that we extend our notions to a Hardy space H Λ 2 \mathcal {H}_{\Lambda }^2 of generalized Dirichlet series, induced in a natural way by a sequence of Beurling’s primes.

On Convergence of Orthogonal Expansion of a Function From the Class in the Eigenfunctions of a Differential Operator of the Third Order

We consider a third-order ordinary differential operator with summable coefficients. The absolute and uniform convergence of the orthogonal expansion of a function from the class in the eigenfunctionsof this operator is studied and the rate of uniform convergence of these expansions on is estimated

Linear combinations of composition operators with linear symbols on a Hilbert space of Dirichlet series

On the Hilbert space of Dirichlet series with square summable coefficients, we show that the linear combinations of two composition operators induced by linear symbols are compact only when each one of them is compact. Moreover, such rigid behavior holds partially for some more general symbols.

On Bessel property of the system of root functions of the second order differential operator

In this paper studied the Bessel property of the system of root functions of a second order differential operator with summable coefficients. Established the criterion of Bessel property in L 2 (G), G = (0,1) of the system of root functions of the given operator.

Consistency Results for Stationary Autoregressive Processes With Constrained Coefficients

We consider stationary autoregressive processes with coefficients restricted to an ellipsoid. These are included in the family of autoregressive processes with absolutely summable coefficients. We provide consistency results under different norms for the estimation of such processes using constrained and penalized estimators. As an application, we show a weak form of universal consistency. Simulations show that directly including the constraint in the estimation can lead to results that are more robust.

Weighted composition operators on the Hilbert space of Dirichlet series

In this paper, we study weighted composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The Hermitianness, Fredholmness and invertibility of such operators are characterized, and the spectra of compact and invertible weighted composition operators are also described.

Estimate for Norm of a Composition Operator on the Hardy–Dirichlet Space

By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on $$\mathcal {H}^2$$ , the space of Dirichlet series with square summable coefficients, for the inducing symbol $$\varphi (s)=c_1+c_{q}q^{-s}$$ where $$q\ge 2$$ is a fixed integer. We also give an estimate on the approximation numbers of such an operator.

Sharp norm estimates for composition operators and Hilbert-type inequalities

Let $\mathscr{H}^2$ denote the Hardy space of Dirichlet series $f(s) = \sum_{n\geq1} a_n n^{-s}$ with square summable coefficients and suppose that $\varphi$ is a symbol generating a composition operator on $\mathscr{H}^2$ by $\mathscr{C}_\varphi(f) = f \circ \varphi$. Let $\zeta$ denote the Riemann zeta function and $\alpha_0=1.48\ldots$ the unique positive solution of the equation $\alpha\zeta(1+\alpha)=2$. We obtain sharp upper bounds for the norm of $\mathscr{C}_\varphi$ on $\mathscr{H}^2$ when $0<\operatorname{Re}\varphi(+\infty)-1/2 \leq \alpha_0$, by relating such sharp upper bounds to the best constant in a family of discrete Hilbert-type inequalities.

Toeplitzness of composition operators on a Hilbert space of Dirichlet series

In this paper, we investigate the -Toeplitzness and asymptotic -Toeplitzness of composition operators and their adjoints on the Hilbert space of Dirichlet series with square summable coefficients. The operator is a shift specific to that space. We obtain characterizations of -Toeplitz and -asymptotically Toeplitz composition operators, where the asymptotic concepts are relative to the uniform, strong, respectively, and weak operator topologies. A characterization of strong asymptotic -Toeplitzness of adjoints of composition operators is also obtained.

Complex symmetric composition operators on a Hilbert space of Dirichlet series

On the Hilbert space of Dirichlet series with square summable coefficients, there are no non-normal complex symmetric composition operators induced by non-constant symbols. This is in sharp contrast with the phenomenon on the classical Hardy space over the unit disk.

On Locally Convex Curves

We introduce the definition of locally convex curves and establish some properties of such curves. In the section 1, we consider the curve \(K\) allowing the parametric representation \(x = u(t),\, y = v(t), \, (a \leqslant t \leqslant b)\), where \(u(t)\), \(v(t)\) are continuously differentiable on \([a,b]\) functions such that \(|u'(t)| + |v'(t)| > 0 \,\forall t \in [a,b]\). A continuous on \([a,b]\) function \(\theta(t)\) is called it the angle function of the curve \(K\) if the following conditions hold: \(u'(t) = \sqrt{(u'(t))^2 + (v'(t))^2}\, \cos \theta(t), \quad v'(t) = \sqrt{(u'(t))^2 + (v'(t))^2}\, \sin \theta(t)\). The curve \(K\) is called it locally convex if its angle function \(\theta(t)\) is strictly monotonous on \([a,b]\). For a closed curve \(K\) the number \(deg K= \cfrac{\theta(b)- \theta(a)}{2 \pi}\) is whole. This number is equal to the number of rotations that the speed vector \((u'(t),v'(t))\) performs around the origin. The main result of the first section is the statement: if the curve \(K\) is locally convex, then for any straight line \(G\) the number \(N(K;G)\) of intersections of \(K\) and \(G\) is finite and the estimate \(N(K;G) \leqslant 2 |deg K|\) holds. We discuss versions of this estimate for closed and non-closed curves. In the sections 2 and 3, we consider curves arising in the investigation of a linear homogeneous differential equation of the form \(L(x) \equiv x^{(n)} + p_1(t) x^{(n-1)} + \cdots p_n(t) x = 0 \) with locally summable coefficients \(p_i(t)\, (i = 1, \cdots,n)\). We demonstrate how conditions of disconjugacy of the differential operator \(L\) that were established in works of G.A. Bessmertnyh and A.Yu.Levin, can be applied.

Weak Product Spaces of Dirichlet Series

Let $\mathscr{H}^2$ denote the space of ordinary Dirichlet series with square summable coefficients, and let $\mathscr{H}^2_0$ denote its subspace consisting of series vanishing at $+\infty$. We investigate the weak product spaces $\mathscr{H}^2\odot\mathscr{H}^2$ and $\mathscr{H}^2_0\odot\mathscr{H}^2_0$, finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that $\mathscr{H}^2_0\odot\mathscr{H}^2_0$ does not contain the infinite-dimensional subspace of $\mathscr{H}^2$ of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of multiplicative Hankel forms. We show that there are bounded, even Schatten class, multiplicative Hankel forms on $\mathscr{H}^2_0 \times \mathscr{H}^2_0$ whose analytic symbols are not in $\mathscr{H}^2$. Based on this result we examine Nehari's theorem for such Hankel forms. We define also the skew product spaces associated with $\mathscr{H}^2\odot\mathscr{H}^2$ and $\mathscr{H}^2_0\odot\mathscr{H}^2_0$, with respect to both half-plane and polydisc differentiation, the latter arising from Bohr's point of view. In the process we supply square function characterizations of the Hardy spaces $\mathscr{H}^p$, for $0 < p < \infty$, from the viewpoints of both types of differentiation. Finally we compare the skew product spaces to the weak product spaces, leading naturally to an interesting Schur multiplier problem.

Invariant subspaces of composition operators on a Hilbert space of Dirichlet series

In this paper, we study invariant subspaces of composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The structure of invariant subspaces of a composition operator is characterized, and the strongly closed algebras generated by some composition operators with irrational symbols are shown to be reflexive. As an application, we provide a criterion for composition operators with certain symbols not to be algebraic.

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an \(\alpha \)-stable Levy Motion are given. The conditions lead to new, tractable sufficient conditions in the case \(\alpha \le 1\). In the functional setting, we complement the existing results on \(M_1\)-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the \(S\) topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).

Conservation laws of generalized billiards that are polynomial in momenta

This paper deals with dynamics particles moving on a Euclidean n-dimensional torus or in an n-dimensional parallelepiped box in a force field whose potential is proportional to the characteristic function of the region D with a regular boundary. After reaching this region, the trajectory of the particle is refracted according to the law which resembles the Snell -Descartes law from geometrical optics. When the energies are small, the particle does not reach the region D and elastically bounces off its boundary. In this case, we obtain a dynamical system of billiard type (which was intensely studied with respect to strictly convex regions). In addition, the paper discusses the problem of the existence of nontrivial first integrals that are polynomials in momenta with summable coefficients and are functionally independent with the energy integral. Conditions for the geometry of the boundary of the region D under which the problem does not admit nontrivial polynomial first integrals are found. Examples of nonconvex regions are given; for these regions the corresponding dynamical system is obviously nonergodic for fixed energy values (including small ones), however, it does not admit polynomial conservation laws independent of the energy integral.

Local properties of Hilbert spaces of Dirichlet series

We show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the Hilbert space of Dirichlet series defined using these as weights. This extends results recently obtained describing the local behavior of Dirichlet series with square summable coefficients in terms of local integrability, boundary behavior, Carleson measures and interpolating sequences. As these spaces can be identified with functions spaces on the infinite-dimensional polydisk, this gives new results on the Dirichlet and Bergman spaces on the infinite-dimensional polydisk, as well as the scale of Besov–Sobolev spaces containing the Drury–Arveson space on the infinite-dimensional unit ball. We use both techniques from the theory of sampling in Paley–Wiener spaces, and classical results from analytic number theory.

On the basis property of the root functions of differential operators with matrix coefficients

Abstract We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.

The asymptotics of the eigenvalues of a fourth order differential operator with summable coefficients

A fourth order differential operator with summable coefficients and some boundary conditions is considered. Asymptotics of solutions to a fourth order differential equation is studied. The equation for eigenvalues is also studied and an asymptotics of the eigenvalues of the considered boundary value problem is obtained.

On the Differential Operators with Periodic Matrix Coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then by using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self‐adjoint differential operator with the periodic matrix coefficients is finite.