Vol. 16, No. 1, 1966

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On the stability of the set of exponents of a Cauchy exponential series

S. Verblunsky

Vol. 16 (1966), No. 1, 175–188

If f L(D,D) and Q(z) is a meromorphic function whose poles, all simple, forms a sub-set of the set {λν}(ν = 0,±1,±2,), then the C.E.S. (Cauchy exponential series) of f with respect to Q(z) is cνeλνx, where

   λxν         ∫ D     z(x−t)
cνe  = rλeνsQ(z) −Df (t)e     dt.

Suppose we are given a class A of functions f each of which can be ‘represented’ in (D,D) by its C.E.S. with respect to Q(z). We define a set of neighbourhoods U of {λν}. Then {λν} is stable if there is a U such that to each {κν}∈ U there corresponds a meromorphic function q(z) whose poles, all simple, form a sub-set of {κν} and which is such that each f A can be represented in (D,D) by its C.E.S. with respect to q(z); and {λν} is unstable if there is no such neighbourhood.

The case in which λν = , A is BV [D,D], ‘representation of f in (D,D)’ means ‘ |ν|ncνeλνx 12(f(x+) + f(x)) boundedly within (D,D)’ is considered. It is shown, in particular, that with reasonable conditions on the set of neighbourhoods U, {} is unstable if D > 12π, and stable if D = 12π.

Mathematical Subject Classification
Primary: 30.60
Received: 27 July 1964
Published: 1 January 1966
S. Verblunsky