Equation of infinite order

in the complex domain

A differential equation of the form

$$ \sum _ {n = 0 } ^ \infty a _ {n} ( z) y ^ {( n)} ( z) = \ f ( z), $$

where $ y( z) $ is the unknown function of the complex variable $ z $ and $ a _ {n} ( z) $, $ f( z) $ are given functions. Equations of infinite order which have been most thoroughly studied are those with constant coefficients:

$$ Ly \equiv \ \sum _ {n = 0 } ^ \infty a _ {n} y ^ {( n)} ( z) = \ f ( z). $$

If the characteristic function

$$ \phi ( \lambda ) = \ \sum _ {n = 0 } ^ \infty a _ {n} \lambda ^ {n} $$

is an entire function of exponential type $ \sigma $, the left-hand side $ Ly $ makes sense for $ z = z _ {0} $ if $ y( z) $ is an analytic function in the disc $ | z - z _ {0} | < R $, $ R > \sigma $. If $ \sigma = \infty $, one must assume that $ y( z) $ is an entire function. A difference from an equation of finite order consists already in the fact that the solution $ y( z) $ may have singular points even if $ f( z) $ is an entire function. If $ \sigma = 0 $ and $ f( z) $ is an entire function, the domain of existence of a solution is convex [1]. The general solution consists of a particular solution and the general solution of the corresponding homogeneous equation. Let $ \lambda _ {1} , \lambda _ {2} ,\dots $ be the roots of the characteristic equation $ \phi ( \lambda ) = 0 $ and let $ m _ {1} , m _ {2} ,\dots $ be their respective multiplicities. The homogeneous equation has elementary particular solutions $ z ^ {k} e ^ {\lambda _ {n} z } $( $ k = 0 \dots m _ {n} - 1 $; $ n = 1, 2 ,\dots $). The solution of the homogeneous equation can be written as a series of elementary particular solutions, formed according to a definite rule. If the characteristic function $ \phi ( \lambda ) $ has regular growth (in a certain sense), it is possible to find a subsequence of the partial sums of this series converging to $ y( z) $[4]. In the general case the function $ y( z) $ may be approximated, as accurately as one pleases, by finite linear combinations of elementary solutions [5]. If $ \sigma = 0 $, an equation of infinite order may have non-analytic solutions [2]. Under certain conditions these solutions form a quasi-analytic class with weaker bounds on the growth of the derivatives than in the classical Denjoy–Carleman theorem (cf. Quasi-analytic class).

Equations of infinite order have various applications. They are used in the study of sequences of Dirichlet polynomials, completeness of systems of analytic functions, uniqueness of analytic and harmonic functions, and in solvability questions of analytic problems such as the generalized quasi-analyticity problem, the generalized uniqueness problem of moments, etc.

References

[1]	G. Pólya, "Eine Verallgemeinung des Fabryschen Lückensatzes" Nachr. Ges. Wiss. Göttingen (1927) pp. 187–195
[2]	G. Valiron, "Sur les solutions des équations différentielles lineaires d'ordre infini et à coefficients constants" Ann. Sci. École Norm. Sup. (3) , 46 : 1 (1929) pp. 25–53
[3]	A.F. Leont'ev, "Differential equations of infinite order and their applications" , Proc. 4-th All-Union Math. Congress (1961) , Leningrad (1964) pp. 648–660 (In Russian)
[4]	A.F. Leont'ev, "On representation of functions by sequences of Dirichlet polynomials" Mat. Sb. , 70 (112) : 1 (1966) pp. 132–144 (In Russian)
[5]	I.F. Krasichkov-Ternovskii, "Invariant subspaces of analytic functions III. On the extension of spectral synthesis" Math. USSR-Sb. , 17 : 3 (1972) pp. 327–348 Mat. Sb. , 88 : 3 (1972) pp. 331–352

References

[a1]	I.I. Hirschmann, "The convolution transform" , Princeton Univ. Press (1955)
[a2]	L. Ehrenpreis, "Theory of infinite derivatives" Amer. J. Math , 81 (1959) pp. 799–845
[a3]	L. Ehrenpreis, "Fourier analysis in several complex variables" , Wiley (Interscience) (1970)