Operators Of Infinite Order Research Articles

The integro‐differential operators of infinite order on the half line related to the Legendre operator

In this paper, we introduce certain integro‐differential operators of Legendre type of infinite order. Some properties of these operators are obtained. Moreover, we establish the necessary and sufficient conditions for a class of integro‐differential operators of infinite order being unitary on . Some classes of related integro‐differential equations are also considered.

Quantizations and Global Hypoellipticity for Pseudodifferential Operators of Infinite Order in Classes of Ultradifferentiable Functions

We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global \(\omega \)-hypoellipticity and global \(\omega \)-regularity of these classes of pseudodifferential operators.

Global Wave Front Sets in Ultradifferentiable Classes

We introduce a global wave front set using Weyl quantizations of pseudodifferential operators of infinite order in the ultradifferentiable setting. We see that in many cases it coincides with the Gabor wave front set already studied by the last three authors of the present work. In this sense, we also extend, to the ultradifferentiable setting, previous work by Rodino and Wahlberg. Finally, we give applications to the study of propagation of singularities of pseudodifferential operators.

Global well-posedness of a class of strictly hyperbolic Cauchy problems with coefficients non-absolutely continuous in time

We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on [0,T]×Rn in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in x and with their t-derivative of order O(t−q), where q∈[1,32). For this purpose, an appropriate generalized symbol class based on the metric is defined and the associated Planck function is used to define a new class of infinite order operators to perform conjugation. We demonstrate that the solution not only experiences a loss of regularity (usually observed for the case of coefficients bounded in x) but also a decay in relation to the initial datum defined in a Sobolev space tailored to the generalized symbol class. Further, we observe that a precise behavior of the solution could be obtained by making an optimal choice of the metric in relation to the coefficients of the given equation. We also derive the cone conditions in the global setting.

Infinite order $$\Psi $$DOs: composition with entire functions, new Shubin-Sobolev spaces, and index theorem

We study global regularity and spectral properties of power series of the Weyl quantisation \(a^w\), where \(a(x,\xi ) \) is a classical elliptic Shubin polynomial. For a suitable entire function P, we associate two natural infinite order operators to \(a^{w}\), \(P(a^w)\) and \((P\circ a)^{w},\) and prove that these operators and their lower order perturbations are globally Gelfand–Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to \(\infty \) for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of f-\(\Gamma ^{*,\infty }_{A_p,\rho }\)-elliptic symbols, where f is a function of ultrapolynomial growth and \(\Gamma ^{*,\infty }_{A_p,\rho }\) is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov–Hörmander integral formula.

Boundary Control Problems for 2 × 2 Cooperative Hyperbolic Systems with Infinite Order Operators

In this study, boundary control problems with Neumann conditions for 2 × 2 cooperative hyperbolic systems involving infinite order operators are considered. The existence and uniqueness of the states of these systems are proved, and the formulation of the control problem for different observation functions is discussed.

On the Generalized Cauchy Problem for One Class of Differential Equations of Infinite Order

We establish the solvability of a nonlocal multipoint (in time) problem regarded as a generalization of the Cauchy problem for the evolution equation with pseudodifferential operator (differentiation operator of infinite order) with initial conditions from the space of generalized functions of the ultradistribution type.

Про узагальнену задачу Коші для одного класу диференціальних рівнянь нескінченного порядку

УДК 517.98 Встановлено розв’язнiсть нелокальної багатоточкової за часом задачi (яка трактується як певне узагальнення задачi Кошi) для еволюцiйного рiвняння з псевдодиференцiальним оператором (оператором диференцiювання нескiнченного порядку) з початковою умовою у просторi узагальнених функцiй типу ультрарозподiлiв.

A Nonlocal in Time Problem for Evolutionary Singular Equations in Generalized Spaces of Type S∘

In this paper, we establish the correct solvability of a nonlocal multipoint in time problem for the evolutionary equation of a parabolic type with the Bessel operator of infinite order in the case where the initial function is an element of the space of generalized functions of type S∘′.

Characterization of continuous endomorphisms of the space of entire functions of a given order

In this paper we show that continuous endomorphisms on the space of entire functions with a given order can be expressed as differential operators of infinite order satisfying suitable growth conditions.

Nonlocal Multipoint (in Time) Problem for Evolutionary Pseudodifferential Equations with Analytic Symbols in the Spaces of Type W

UDC 517.956 The correct solvability of a nonlocal multipoint (in time) problem for the evolution equations with differentiation operators of infinite order is established for an infinite time interval and an initial function, which is an element of the space of generalized functions of the type $ W'$. The properties of the fundamental solution and the behavior of the solution as $ t \to + \infty $ are investigated.

Distributed Control for <i>n</i> × <i>n</i> Cooperative Systems Governed by Hyperbolic Operator of Infinite Order

In this study, a distributed optimal control problem for n × n cooperative hyperbolic systems with infinite order operators and Dirichlet conditions are considered. The existence and uniqueness of the state of these systems are proved. The necessary and sufficient conditions for optimality of distributed control with constraints are found, and the set of equations and inequalities that defining the optimal control of these systems is also obtained. Finally, some examples for the control problem without constraints are given.

Invertibility of matrix type operators of infinite order with exponential off-diagonal decay

We analyse various exponential off-diagonal decay rates of the elements of infinite matrices and their inverses. It is known that such decay of the elements of an infinite matrix does not imply inverse–closeness, i.e. the inverse, if exists, does not have the same order of decay. We discuss some consequences and extensions of this result.

Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

We develop a theory of pseudodifferential operators of infinite order for the global classes $\mathcal{S}_{\omega}$ of ultradifferentiable functions in the sense of Bj\"orck, following the previous ideas given by Prangoski for ultradifferentiable classes in the sense of Komatsu. We study the composition and the transpose of such operators with symbolic calculus and provide several examples.

Differential Operators of Infinite Order in the Space of Formal Laurent Series and in the Ring of Power Series with Integer Coefficients

We study the Hurwitz product (convolution) in the space of formal Laurent series over an arbitrary field of zero characteristic. We obtain the convolution equation which is satisfied by the Euler series. We find the convolution representation for an arbitrary differential operator of infinite order in the space of formal Laurent series and describe translation invariant operators in this space. Using the p-adic topology in the ring of integers, we show that any differential operator of infinite order with integer coefficients is well defined as an operator from ℤ[[z]] to ℤp[[z]].

On the possibility of constructing relativistic quantum mechanics on the basis of the definition of the functions of differential operators

A new definition of the function of a differential operator that leads to local operators of infinite order is proposed. It allows one to obtain an expression for the square root of the differential operator (the Hamiltonian of a free spinless particle) and to determine the relativistic Schrödinger equation as a close analog of the non-relativistic Schrödinger equation. It is shown that this equation does not lead to difficulties of the Klein-Gordon equation. Boundary conditions that lead to self-adjoint boundary problems similar to Sturm-Liouville problems, periodic boundary value problems, and singular boundary value problems are determined. Some problems of relativistic quantum mechanics are solved.

The Gabor wave front set in spaces of ultradifferentiable functions

We consider the spaces of ultradifferentiable functions $$\mathcal {S}_\omega $$ as introduced by Bjorck (and its dual $$\mathcal {S}'_\omega $$ ) and we use time-frequency analysis to define a suitable wave front set in this setting and obtain several applications: global regularity properties of pseudodifferential operators of infinite order and the micro-pseudolocal behaviour of partial differential operators with polynomial coefficients and of localization operators with symbols of exponential growth. Moreover, we prove that the new wave front set, defined in terms of the Gabor transform, can be described using only Gabor frames. Finally, some examples show the convenience of the use of weight functions to describe more precisely the global regularity of (ultra)distributions.

Stabilization of Solutions of a Nonlocal Problem Multipoint in Time for One Class of Evolutionary Pseudodifferential Equations

We show that the solution of a nonlocal problem multipoint in time for an evolutionary equation with differential operator of infinite order is stabilized to zero as t → + ∞ in the space of generalized functions of the form S′:

Ultradifferential operators in the study of Gevrey solvability and regularity

AbstractIn this work we present a new representation formula for ultradistributions using the so‐called ultradifferential operators. The main difference between our representation result and other works is that here we do not break the duality of Gevrey functions of other s and their ultradistributions, i.e., we locally represent an element of by an infinite order operator acting on a function of class . Our main application was in the local solvability of the differential complex associated to a locally integrable structure in a Gevrey environment.

Unconditionally marginal stability of harmonic electron hole equilibria in current-driven plasmas

Two forms of the linearized eigenvalue problem with respect to linear perturbations of a privileged cnoidal electron hole as a structural nonlinear equilibrium element are established. Whereas its integral form involves integrations along the characteristics or unperturbed particle orbits, the differential form has to cope with a differential operator of infinite order. Both are hence faced with difficulties to obtain a solution. A first successful attempt is, however, made by addressing a single harmonic wave as a nonlinear equilibrium structure. By this microscopic nonlinear approach, its marginal stability against linear perturbations in both linear stability regimes, the sub- and super-critical one, is shown independent of the mobility of ions and in favor with recent observations. Responsible for vanishing damping (growth) is the microscopic distortion of the resonant distribution function. The macroscopic form of the trapping nonlinearity—the 3/2 power term of the electrostatic potential in the density—which disappears in the monochromatic harmonic wave limit is consequently necessary for the occurrence of a nonlinear plasma instability in the sub-critical regime.