30E05 — 1 articles found.

Cauchy Type Integrals and a $D$-Moment Problem

C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (4) 2007, pp. 115–122
Vol.29 (4) 2007
V.A. Kisunko Details
(Received: 2007-03-10 , Revised: 2008-04-17 )
(Received: 2007-03-10 , Revised: 2008-04-17 )

V.A. Kisunko, Department of Mathematics, University of Toronto, Toronto, ON M5S 1A1; e-mail: vkisunko@math.toronto.edu


We consider a Cauchy-type integral \(F(z)= \int_{\Gamma} \frac {g(\xi)\,\d \xi}{\xi-z}\), where \(g\) is a piecewise analytic function satisfying an \(n\)-th order linear homogeneous differential equation \(Ly=\frac{\d^n y}{\d z^n} + c_{n-1}\frac{\d^{n-1}}{\d z^{n-1}} +\dots+ c_0y=0\) with coefficients \(c_k \in \C(z)\) rational functions. Our main theorem asserts that the function \(F\) satisfies a linear non-homogeneous equation \(Ly=R\) with \(R\) a rational function. The precise description of \(R\) leads to the solution of a vanishing problem and to the solution of a moment-type problem, which we call D-moment problem.

On considère une integrale du type Cauchy \(F(z)= \int_{\Gamma} \frac {g(\xi)\d \xi}{\xi-z}\), où \(g\) est une fonction analytique par morceaux satisfaisant une équation différentielle linéaire homogène d’ordre \(n\), \(Ly=\frac{\d^n y}{\d z^n} + c_{n-1}\frac{\d^{n-1}}{\d z^{n-1}} +\dots+ c_0y=0\), aux coefficients \(c_k\in \C(z)\) rationnels. Notre théorème principal affirme que la fonction \(F\) satisfait une équation linéaire non-homogène \(Ly=R\) avec \(R\) rationnelle. La description précise de \(R\) mène à la solution du problème d’évanescence et à la solution d’un problème du type moment que nous appelons problème de D-moment.

Keywords: Cauchy type integral, D-moment problem, piecewise polynomial moment problem, vanishing problem

AMS Subject Classification: Moment problems; interpolation problems 30E05

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