Abstract
Let \(\left\{ {{\alpha _n}:n = 1,2,\ldots } \right\}\) be a sequence of (not necessarily distinct) points on the unit circle. Set \({\omega _0}(z) \equiv 1,{\omega _n}(z) = (z - {\alpha _1})(z - {\alpha _2}) \ldots (z - {\alpha_n})for n = 1,2,\ldots,\)and let G denote the linear space spanned by the functions\(\left\{ {1/{\omega _n}:n = 0,1,2,\ldots,} \right\}\).LetMbe a quasi-definite linear functional on\(\mathcal{L} \cdot \mathcal{L}\), and define the inner product\(, by f,g = M(f{g_ * })\), where \(g * (z) = \overline {g(1\left){\vphantom{1z}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{z}})}\). (In particular M may be a positive definite functional given by\(M(f) = \int_{ - \pi }^\pi {f({e^{it}})d\mu (t)},\),where µ is a measure such that all functions in \(\mathcal{L} \cdot \mathcal{L}\) are µ-integrable).Let \(\left\{ {{\varphi _n}} \right\}\)be an orthogonal system obtained from the basis \(\left\{ {1/{\omega _n}} \right\}\) by the Gram-Sdunidt method and define the associated
functions\(\left\{ {{\varphi _n}} \right\}\) by \({\varphi _n}(z) = M(\frac{{t + z}}{{t - z}}\left[ {{\varphi _n}(t) - {\varphi _n}(z)} \right])\).Interpolation of \({\psi _n}/{\varphi _n}\)to \(M(\frac{{t + z}}{{t - z}})\) (or to the formal series
\({\mu _0} + 2\sum\nolimits_{m = 1}^\infty {{\mu _m}z{\omega _{m - 1}}} (z),{\mu _m} = M(1/{\omega _m})\) at interpolation points \({\alpha _k}\)is
discussed.
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Bultheel, A., Gonzalez-Vera, P., Hendriksen, E., Njåstad, O. (1994). Orthogonality and Boundary Interpolation. In: Cuyt, A. (eds) Nonlinear Numerical Methods and Rational Approximation II. Mathematics and Its Applications, vol 296. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0970-3_4
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