Differential Operators Of Infinite Order Research Articles

Preservation of the completely regular growth by a differential operator of infinite order

Preservation of the completely regular growth by a differential operator of infinite order

Partial differential operators of infinite order with constant coefficients on the space of analytic functions on the polydisc

Partial differential operators of infinite order with constant coefficients on the space of analytic functions on the polydisc

A question of Hille on inversion of differential operators of infinite order

The inversion problem for differential equations of the form G( D z ) = F( z) is studied. Here D z is a linear differential operator of order n having entire coefficients with the leading coefficient a polynomial whose zeros are suitably restricted. The function G is entire of order at most 1 n , minimal type, and the function F is analytic in a neighborhood of an interval on the real axis. Conditions for obtaining an analytic solution W of the above equation are given.

Generalised impedance boundary conditions for EM scattering problems

Generalised impedance boundary conditions (GIBC) are derived for a planar, homogeneous, magnetic dielectric slab grounded by a perfect electric conducting plane. These boundary conditions, which are expressed in terms of linear differential operators of infinite order, reduce to the Weinstein boundary conditions in the limiting case of small thickness of the dielectric slab.

On a class of linear differential operators of infinite order with finite index

The recent progress of the theory of microdifferential operators of infinite order [l-4] and their usefulness in several branches of analysis [IS, 11, 16-241 have enhanced the interest in such operators. However, our knowledge on such operators is still quite limited, particularly compared with our knowledge on operators of finite order. For example, we do not know much about such a basic problem as characterizing a linear differen- tial operator of infinite order which is of closed range as a map from germs of holomorphic functions to germs of holomorphic functions. To the best of our knowledge, Ishimura’s recent work [7] is the only result that has some general applicability, although it is still far from the final goal. The purpose of this paper is to prove some theorems which we hope to contribute toward the progress of the study of such operators. To be con- crete, we present some conditions on a linear differential operator P which guarantees the finiteness of the dimensions of the kernel and the cokernel of the map P: Ccn,o -+ O,,,,, where Ocn,O denotes germ at origin of sheaf 0,. of holomorphic functions on C:“. In particular, under those con- ditions the map has finite index, and it is of closed range (Theorem 4 and Theorem 5). Our results are a natural generalization of Bony and Schapira [S, Thtoreme 5.11 to the infinite-order case. It is also quite akin to Kashiwara, Kawai, and Sjostrand [12] in its philosophy in the sense that some ellipticity condition for a suitably chosen tangential system lies 155

Linear differential equations of infinite order and theta functions

The purpose of this article is to show that some finiteness theorem (= finite dimensionality of the space of solutions) holds for a class of systems of linear differential equations of infinite order. Although finiteness theorems for holonomic systems of (micro-)differential equations of finite order have recently become quite popular, the character of the theorems which we present here is different from the results for equations of finite order. Hence, in this introduction, we discuss a simple and instructive example so that it may help the reader’s understanding of the character of the results in this article. As the example will indicate, our results have close connection with the celebrated result of Hamburger on the characterization of the c-function of Riemann, although we deal with theta functions (Hamburger [2], Hecke [3], and Weil [8]; see also Ehrenpreis and Kawai [ 11). This connection was pointed out to one of us (T.K.) by Professor L. Ehrenpreis. Concerning the basic properties of linear differential operators of infinite order, we refer the reader to Sato-Kawai-Kashiwara [6, Chap. II]’ (hereafter referred to as S-K-K). Here we only emphasize that a linear differential operator of infinite order acts upon the sheaf of holomorphic functions as a sheaf homomorphism. Hence our main result (Theorem 2.14 in Section 2) is of local character. This forms a striking contrast to the hitherto known way of characterizing theta functions through their automorphic properties. Now, in order to provide an example of our results, let us show how the theta zero-value (Nullwerte) is related to a system of linear differential equations of infinite order. In order to fix the notations, let us consider

An example of a complex of linear differential operators of infinite order

An example of a complex of linear differential operators of infinite order

Invertibility for Microdifferential Operators of Infinite Order

For each holomorphic microlocal operator its symbol ib defined as a holomorphic function which satisfies a growth condition. A differential or micro (=pbeudo-) differential operator of infinite order is naturally regarded as a holomorphic microlocal operator and its symbol coincides with the usual total symbol. Some sufficient conditions of invertibility for holomorphic microlocal operators in terms of symbols are given.

Structure theorems for vector valued ultradistributions

It was proved by Komatsu that both Roumieu and Beurling ultradistributions can be locally expressed in the form P( D) f, where f is a continuous function and P is a differential operator of infinite order. We prove here a more general result (valid for ultradistributions taking values in arbitrary normed spaces, not necessarily reflexive) based on an elementary probabilistic construction. Several extensions (to more general range spaces) are indicated.

Whittaker vectors and conical vectors

A holomorphic family of differential operators of infinite order is constructed that transforms conical vectors for principal series representations of quasi-split, linear, semi-simple Lie groups into Whittaker vectors. Using this transform, it is shown that algebraic Whittaker vectors (as studied by Kostant) extend to ultradistributions of Gevrey type on principal series representations. For each element of the small Weyl group, a meromorphic family of Whittaker vectors is constructed from this transform and the Kunze-Stein intertwining integrals. An explict formula is derived for the smooth Whittaker vector (discovered by Jacquet), in terms of these families of ultradistribution Whittaker vectors. In particular, this gives new proofs of Jacquet's analytic continuation of the smooth Whittaker vector and its functional equation (Jacquet and Schiffman). Applications of the transform are also given to the theory of Verma modules.

Differential Operators of infinite order

SynopsisThe differential operators in question are of the form G(DZ) where G(w)is an entire function of order at most 1/n and minimal type while Dz is a linear differential operator of order n with coefficients which are entire ( = integral) functions of z, usually polynomials. This class of operators form a natural generalization of the class G(d/dz) studied during the first half of the century Muggli, Polya, Ritt and others. The class G(DZ) was introduced by the present author and his pupils in the 1940s. In fact, the present paper is partly based on a MS from that period, mostly devoted to the special casebut also containing generalizations, some of which were later worked out by Klimczak. A basic tool in this paper is the characteristic seriesExamples are given showing that the domain of absolute convergence of such a series need neither be convex nor of finite connectivity, a question which has puzzled the author for forty odd years. Characteristic series arising from regular or singular boundary value problems for the operator Dz are used to study the inversion problemfor given F(z). In particular it is shown that exp (Dx)[W(z)] = 0 has the unique solution W(z) ≡ 0. Some singular boundary value problems are considered briefly.

Holomorphic representations of nilpotent Lie groups

Let G be a connected, simply-connected complex nilpotent Lie group, and G r ⊂( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of G R to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of G R, we study the convex cone of entire functions on G whose restrictions to G R are positive-definite, and determine the minimal order of growth at infinity of such functions.

An inversion formula for a class of integral transforms, II

In this note we give a procedure for inverting the integral transform f( x) = ∫ 0 ∞ k( xt) φ( t) dt, where the functions f( x) and k( x) are known and φ( x) is to be found. The inversion is accomplished in two steps: by first defining a transforming function, which is an integral, followed by the application of an infinite order differential operator.

Applicability of partial differential operators of infinite order

Applicability of partial differential operators of infinite order

Representation of linear operators as differential operators of infinite order

Representation of linear operators as differential operators of infinite order

On the equivalence of differential operators of infinite order in analytic spaces

It is shown that in the spacesAR (0 <R ⩽ ∞) of all functions which are single-valued and analytic in the disk ¦z¦ < R with the topology of compact convergence, the differential operator of infinite order with constant coefficients\(\varphi (D) = \sum\nolimits_{k = 0}^\infty \varphi _k D^k\) is equivalent to the operator Dn (n is a fixed natural number) if and only if\(\varphi (D) = \sum\nolimits_{k = 0}^n \varphi _k D^k\) and ¦ϕn¦ = 1 for R < ∞ or ϕn ≠ 0 for R = ∞. Also the equivalence of two shift operators in the space A∞ is investigated.

Bending of thick plates under an arbitrary load: PMM vol. 41, n≗ 5, 1977, pp. 909–914

A problem of bending of a plate under an arbitrary load is studied. A method proposed by Lur'e [1] is employed, which reduces the three-dimensional problems of the plate theory to the two-dimensional problems using the infinite order differential operators of the displacements and rotations of the middle plane of the plate. Three stress functions are introduced, and these make it possible to consider the effects of the normal and tangential loads separately. Approximate equations are constructed using the method of homogeneous solutions [2]. a problem of axisymmetric bending of a plate under the action of tangential forces is solved using the first approximation equations.

Representation of linear operators in the form of differential operators of infinite order

Representation of linear operators in the form of differential operators of infinite order

An inversion formula for a class of integral transforms

Abstract In this note we give a procedure for inverting the integral transform f(x) = ∫0∞ k(xt) φ(t) dt, where the functions f(x) and k(x) are known and φ(x) is to be found. The inversion is accomplished in two steps: by first defining a transforming function, which is an integral, followed by the application of an infinite order differential operator.

On differential operators of infinite order

On differential operators of infinite order