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September 2004 Integral representations on equipotential and harmonic sets
Francisco Marcellán, Franciszek Hugon Szafraniec
Bull. Belg. Math. Soc. Simon Stevin 11(3): 457-468 (September 2004). DOI: 10.36045/bbms/1093351384


The sets we are going to consider here are of the form ${z\in\mathbb C \mid |A(z)|=1}$ (equipotential) and ${z\in\mathbb C \mid IM A(z)=0}$ (harmonic) with $A$ being a polynomial with complex coefficients. There are two themes which we want to focus on and which come out from invariance property of inner products on $\mathbb C[Z]$ related to the aforesaid sets. First, we formalize the construction of integral representation of the inner products in question with respect to matrix measure. Then we show that these inner products when represented in a Sobolev way are precisely those with discrete measures in the higher order terms of the representation. In this way we fill up the case already considered in [3] by extending it from the real line to harmonic sets on the complex plane as well as we describe completely what happens in this matter on equipotential sets. As a kind of smooth introduction to the above we are giving an account of standard integral representations on the complex plane in general and of those supported by these two kinds of real algebraic sets.


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Francisco Marcellán. Franciszek Hugon Szafraniec. "Integral representations on equipotential and harmonic sets." Bull. Belg. Math. Soc. Simon Stevin 11 (3) 457 - 468, September 2004.


Published: September 2004
First available in Project Euclid: 24 August 2004

zbMATH: 1082.46026
MathSciNet: MR2098419
Digital Object Identifier: 10.36045/bbms/1093351384

Primary: ‎46E20‎ , 46E35 , 46E39
Secondary: 43A35 , 44A60

Keywords: equipotential and harmonic sets , inner product on the space of polynomials , matrix integration , Moment problems , recurrence relation , Sobolev inner product

Rights: Copyright © 2004 The Belgian Mathematical Society


Vol.11 • No. 3 • September 2004
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