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Sobolev formal orthogonality on algebraic curves and extensions of favard theorem. (English) Zbl 1260.42018

Summary: Sobolev formal orthogonality on harmonic algebraic curves (\(\text{Im}(h(z))= 0\), \(h(z)\in\mathbb{C}[z])\) and equipotential curves \((|h(z)|= 1\), \(h(z)\in\mathbb{C}[z])\) is defined and studied. In each case, such a study is done through the characterization of bilinear forms whose associated functional annihilates at the multiples of \((h(z)- \overline{h(z)})^{2n+1}\) for harmonic algebraic curves, or the characterization of bilinear forms whose associated functional annihilates at the multiples of \((h(z)\overline{h(z)}-1)^{2n+1}\) for equipotential curves, \(n\in\mathbb{N}\). Such characterizations allow to find new extensions of the Favard theorem in this setting.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
11E39 Bilinear and Hermitian forms
15A63 Quadratic and bilinear forms, inner products
30E05 Moment problems and interpolation problems in the complex plane
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