\(k\)-summable series and its applications.
(Les sĂ©ries \(k\)-sommables et leurs applications.)

*(French)*Zbl 1251.32008
Iagolnitzer, D. (ed.), Complex analysis, microlocal calculus and relativistic quantum theory. Proceedings of the colloquium held at Les Houches, Centre de Physique, September 1979. (ISBN 3-540-09996-4). 178-199 (1980).

The author formulates the basic facts of the theory of Gevrey asymptotic expansions, whose more detailed presentation he plans to give in future publications. He considers applications of the theory to differential equations.
The author first gives a definition of a formal Gevrey series \(\hat f\in C[[x]]\) and a function \(f\) of Gevrey type of order \(s\), where \(f\) is holomorphic in some sector \(V\) of the universal covering of \(C^\ast\), \(s\in(1,\infty)\), and gives known results on the relation between these two concepts. This relation is made more precise in considering \(k\)-summable series. A series that is \(k\)-summable in the direction \(\alpha\) is a formal series that is a Gavrey asymptotic expansion of some function of Gevrey type and of order \(S=1+1/k\), holomorphic in the sector \(V\) with bisectrix \(\alpha\) and opening \(\pi/k\). The support of the singularity \(\sum\hat f\) of the series \(\hat f\) is a finite set of directions \(\alpha_i\), along which \(\hat f\) is not \(k\)-summable. The author proves that a series with singular support \(\{\alpha_1,\dots,\alpha_l\}\) is represented as the sum \(\hat f=\hat f_0+\dots+\hat f_l\), where \(\hat f_0\) is \(k\)-summable in any direction, and \(\sum\hat f_i=\alpha_i\) for \(i=1,2,\dots l\). He gives a formula for the \(k\)-sum of a series using the Leroy transform. Using the Cauchy-Heine transform, he refines Malgrange’s cohomological description of asymptotic series with both numerical and matrix coefficients.

In the second part the author studies the effect of linear and nonlinear differential operators on formal series and functions of Gevrey type. He proves a fundamental theorem on Gevrey asymptotic expansions, affirming the exactness of the sequence \(\mathfrak A_{0,s}(V)\overset {D}\rightarrow\mathfrak A_{0,s}(V)\rightarrow 0\), where \(V\) is a sufficiently small sector, \(\mathfrak A_{0,s}(V)\) is the class of functions of Gevrey type of order \(s\), decreasing at zero exponentially of order \(k\;(s=1+1/k)\), and \(D\) is a linear differential operator. If the opening of the sector \(V\) exceeds \(\pi/k_1\), where \(k_1\) is the Katz invariant of the operator \(D\), then the exact sequence has another form \(0\rightarrow\mathfrak A_{0,s_1}(V)\overset {D}\rightarrow\mathfrak A_{0,s_1}(V)\), where \(s_1=1+1/k_1\). For a nonlinear differential equation he proves that its formal solution of Gevrey type is an asymptotic expansion of some function of Gevrey type; moreover for this case the fundamental theorem given above is generalized. The author gives theorems on the canonical form of the solution of the linear differential operator \(x^{k_1+1}(d/dx)-A\), where \(A\) is an \(m\times m\) matrix with holomorphic coefficients, as well as theorems on algebraization of such an equation.

For the entire collection see [Zbl 0428.00029].

In the second part the author studies the effect of linear and nonlinear differential operators on formal series and functions of Gevrey type. He proves a fundamental theorem on Gevrey asymptotic expansions, affirming the exactness of the sequence \(\mathfrak A_{0,s}(V)\overset {D}\rightarrow\mathfrak A_{0,s}(V)\rightarrow 0\), where \(V\) is a sufficiently small sector, \(\mathfrak A_{0,s}(V)\) is the class of functions of Gevrey type of order \(s\), decreasing at zero exponentially of order \(k\;(s=1+1/k)\), and \(D\) is a linear differential operator. If the opening of the sector \(V\) exceeds \(\pi/k_1\), where \(k_1\) is the Katz invariant of the operator \(D\), then the exact sequence has another form \(0\rightarrow\mathfrak A_{0,s_1}(V)\overset {D}\rightarrow\mathfrak A_{0,s_1}(V)\), where \(s_1=1+1/k_1\). For a nonlinear differential equation he proves that its formal solution of Gevrey type is an asymptotic expansion of some function of Gevrey type; moreover for this case the fundamental theorem given above is generalized. The author gives theorems on the canonical form of the solution of the linear differential operator \(x^{k_1+1}(d/dx)-A\), where \(A\) is an \(m\times m\) matrix with holomorphic coefficients, as well as theorems on algebraization of such an equation.

For the entire collection see [Zbl 0428.00029].

##### MSC:

32C36 | Local cohomology of analytic spaces |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |