Differential Operators Of Infinite Order Research Articles

Finite and Infinite Order Differential Relations for Theta Functions

We give different linear and nonlinear differential relations on Jacobi theta functions with more emphasis on the nonlinear differential equation of the third order of Jacobi. We present different points of view with a special attention to the role played by the second order linear differential equations, and their link to the Riccati equation and the Schwarzian equation. We also study an identity for theta functions resulting from the action of certain infinite order differential operators.

On the Singularity Structure of WKB Solution of the Boosted Whittaker Equation: its Relevance to Resurgent Functions with Essential Singularities

Inspired by the symbol calculus of linear differential operators of infinite order applied to the Borel transformed WKB solutions of simple-pole type equation [Kamimoto et al. (RIMS Kokyuroku Bessatsu B 52:127–146, 2014)], which is summarized in Section 1, we introduce in Section 2 the space of simple resurgent functions depending on a parameter with an infra-exponential type growth order, and then we define the assigning operator \({\mathscr{A}}\) which acts on the space and produces resurgent functions with essential singularities. In Section 3, we apply the operator \({\mathscr{A}}\) to the Borel transforms of the Voros coefficient and its exponentiation for the Whittaker equation with a large parameter so that we may find the Borel transforms of the Voros coefficient and its exponentiation for the boosted Whittaker equation with a large parameter. In Section 4, we use these results to find the explicit form of the alien derivatives of the Borel transformed WKB solutions of the boosted Whittaker equation with a large parameter. The results in this paper manifest the importance of resurgent functions with essential singularities in developing the exact WKB analysis, the WKB analysis based on the resurgent function theory. It is also worth emphasizing that the concrete form of essential singularities we encounter is expressed by the linear differential operators of infinite order.

Nonlocal Problem Multipoint in Time for a Class of Partial Differential Equations of Infinite Order

We establish the correct solvability of a nonlocal problem multipoint in time for the evolutionary equation with differential operator of infinite order.

On some operators varying the dimensional parameters of d-orthogonality

ABSTRACTIn this work, we solve a characterization problem which consists to find all d-orthogonal polynomial sets (d-OPS) of Sheffer type. Then we consider its corresponding hierarchy problem. That allows us to obtain several couples where P is a d-OPS and T is an operator acting on polynomials for which TP is a -orthogonal polynomial set (-OPS), . We express some of these operators as infinite order differential operators. Then, we state some of their properties.

Differential operators of infinite order in the space of rapidly decreasing sequences

We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ(D)=∑k=0∞φkDk. φk constant numbers an a power of D. Dn, meaning, is there a isomorphism X (from s onto s) such that Xφ(D) = Dn X?. We prove that if φ(D) is equivalent to Dn, then φ(D) is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n =1.

The Euler-Maclaurin Formula and Differential Operators of Infinite Order

The Euler-Maclaurin Formula and Differential Operators of Infinite Order

Does the Riemann zeta function satisfy a differential equation?

In Hilbert's 1900 address at the International Congress of Mathematicians, it was stated that the Riemann zeta function is the solution of no algebraic ordinary differential equation on its region of analyticity. It is natural, then, to inquire as to whether ζ(z) satisfies any non-algebraic differential equation. In the present paper, an elementary proof that ζ(z) formally satisfies an infinite order linear differential equation with analytic coefficients, T[ζ−1]=1/(z−1), is given. We also show that this infinite order differential operator T may be inverted, and through inversion of T we obtain a series representation for ζ(z) which coincides exactly with the Euler–MacLauren summation formula for ζ(z). Relations to certain known results and specific values of ζ(z) are discussed.

On the continuous linear right inverse for convolution operators in spaces of ultradifferentiable functions

A surjective convolution operator is considered in spaces of complex-valued Beurling ultradifferentiable functions of normal type on a finite interval. A complete description of characteristic functions for which this operator has a continuous linear right inverse is obtained. The finiteand infinite-order differential operators with constant coefficients are studied as a special case.

Integral Cauchy Type Representations of Infinite Order Differential Operators in Spaces of Exponential Type Entire Functions Bounded on the Real Axis

We obtain an integral Cauchy type representation of a vector differential operator of infinite order in the space of entire functions of exponential type that are bounded on the real axis. This representation is used for obtaining integral representations of entire solutions to explicit and implicit linear inhomogeneous differential equations in a Banach space. Bibliography: 5 titles.

On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms

We deal with several classes of integral transformations of the form $$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$ , where D is an operator. In case D is the identity operator, we obtain several operator properties on Lp(ℝ+) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L2(ℝ+) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type.

Hadamard multipliers on spaces of real analytic functions

We consider multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We characterize sequences of complex numbers which are sequences of eigenvalues for some multiplier. We characterize invertible multipliers, in particular, we find which Euler differential operators of infinite order have global analytic solutions on the real line. We present a number of examples where our theory applies. In some cases we give algorithms for solving the respective equations. Perturbation results for solvability are presented.

The explicit equivalence between the standard and the logarithmic star product for Lie algebras, I

The purpose of this note is to establish an explicit equivalence between two star products ⋆ and ⋆log on the symmetric algebra S(g) of a finite-dimensional Lie algebra g over a field K⊃C associated with the standard angular propagator and the logarithmic one respectively: the differential operator of infinite order with constant coefficients realizing the equivalence is related to the incarnation of the Grothendieck–Teichmüller group considered by Kontsevich (1999) in [5, Theorem 7]. We present in the first part the main result, and devote the second part to its proof.

Flatness-based trajectory planning for diffusion–reaction systems in a parallelepipedon—A spectral approach

Spectral analysis is considered for the flatness-based solution of the trajectory planning problem for a boundary controlled diffusion–reaction system defined on a 1 ≤ r -dimensional parallelepipedon. By exploiting the Riesz spectral properties of the system operator, it is shown that a suitable reformulation of the resolvent operator allows a systematic introduction of a basic output, which yields a parametrization of both the system state and the boundary input in terms of differential operators of infinite order. Their convergence is verified for both infinite-dimensional and finite-dimensional actuator configurations by restricting the basic output to certain Gevrey classes involving non-analytic functions. With this, a systematic approach is introduced for basic output trajectory assignment and feedforward tracking control towards the realization of finite-time transitions between stationary profiles.

Behavior of lacunary series at the natural boundary

We develop a local theory of lacunary Dirichlet series of the form ∑ k = 1 ∞ c k exp ( − z g ( k ) ) , Re ( z ) > 0 as z approaches the boundary i R , under the assumption g ′ → ∞ and further assumptions on c k . These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with c k = 1 , the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at z = 0 . When sufficient analyticity information on g exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier. The singular behavior has remarkable universality and self-similarity features. If g ( k ) = k b , c k = 1 , b = n or b = ( n + 1 ) / n , n ∈ N , behavior near the boundary is roughly of the standard form Re ( z ) − b ′ Q ( x ) where Q ( x ) = 1 / q if x = p / q ∈ Q and zero otherwise. The Bötcher map at infinity of polynomial iterations of the form x n + 1 = λ P ( x n ) , | λ | < λ 0 ( P ) , turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map P ( x ) = x − x 2 , λ 0 = 1 , and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.

Holomorphic Operators Generating Dense Images

The existence of infinite dimensional closed linear spaces of holomorphic functions f on a domain G in the complex plane such that Tf has dense images on certain subsets of G, where T is a continuous linear operator, is analyzed. Necessary and sufficient conditions for T to have the latter property are provided and applied to obtain a number of concrete examples: infinite order differential operators, composition operators and multiplication operators, among others.

Cauchy problem for evolution equations with an infinite-order differential operator: II

Cauchy problem for evolution equations with an infinite-order differential operator: II

Local Euler–Maclaurin Formula for Polytopes

We give a local Euler-Maclaurin formula for rational convex polytopes in a rational euclidean space . For every affine rational polyhedral cone C in a rational euclidean space W, we construct a differential operator of infinite order D(C) on W with constant rational coefficients, which is unchanged when C is translated by an integral vector. Then for every convex rational polytope P in a rational euclidean space V and every polynomial function f (x) on V, the sum of the values of f(x) at the integral points of P is equal to the sum, for all faces F of P, of the integral over F of the function D(N(F)).f , where we denote by N(F) the normal cone to P along F.

Entire Functions on Infinite Dimensional Spaces

In this work we study partial differential operators of infinite order on spaces of entire functions, generalizing some of the results presented in [2] and [5] to the new spaces introduced in [3].

Infinite-order symmetries for quantum separable systems

We develop a calculus to describe the (in general) infinite-order differential operator symmetries of a nonrelativistic Schrodinger eigenvalue equation that admits an orthogonal separation of variables in Riemannian n space. The infinite-order calculus exhibits structure not apparent when one studies only finite-order symmetries. The search for finite-order symmetries can then be reposed as one of looking for solutions of a coupled system of PDEs that are polynomial in certain parameters. Among the simple consequences of the calculus is that one can generate algorithmically a canonical basis for the space. Similarly, we can develop a calculus for conformal symmetries of the time-dependent Schrodinger equation if it admits R separation in some coordinate system. This leads to energy-shifting symmetries.

Universality of holomorphic functions bounded on closed sets

In this note, the existence of translation-universal entire functions which are bounded on certain closed subsets is characterized in terms of topological and geometrical properties of such subsets. Corresponding results are also stated in the space of holomorphic functions on the unit disk and in the space of harmonic functions on the plane. Moreover, it is shown the existence of entire functions which are bounded on many rays and, simultaneously, are universal with respect to a prescribed infinite-order differential operator.