Equilibrium conditions for the Malmquist-Takenaka systems

A Family of Orthogonal Rational Functions and Other Orthogonal Systems with a skew-Hermitian Differentiation Matrix

“…The nature of the questions they have asked was different -essentially, they proved that the above system is a basis (which need not be orthogonal) of H 2 , the Hardy space of complex analytic functions in the open unit disc. In our case the θ k ≡ 2i are all the same and outside the unit circle, yet it seems fair (and consistent with, say, (Pap & Schipp 2015)) to call (3) a Malmquist-Takenaka system. Fig.…”

“…The MT system has been discovered independently by Malmquist (1926) and Takenaka (1926) and investigated by many mathematicians, in different contexts: approximation theory (Bultheel & Carrette 2003, Bultheel, González-Vera, Hendriksen & Njåstad 1999, Higgins 1977, Weideman 1994, harmonic analysis (Eisner & Pap 2014, Pap & Schipp 2015, signal processing (Wiener 1949) and spectral methods (Christov 1982). Some of these references are aware of the original work of Malmquist and Takenaka, while others reinvent the construct.…”

Minimal energy point systems on the unit circle and the real line

“…In connection with potential theory, it was studied whether the discretization nodes satisfy certain equilibrium conditions, namely, they arise from critical points of a logarithmic potential energy. The question whether the critical points are minima was proposed by Pap and Schipp [10,11]. In this paper we follow this line of research.…”

“…Moreover, it is standard (see, e.g., [10], pp. 186-187) that one can define the functions δ → τ j (δ) such that they are continuously differentiable, strictly increasing functions, and τ 1 (δ) < .…”

“…Similar distinction also appears in [14], see the definitions of Γ-normal electrostatic equilibrium and total electrostatic equilibrium on p. 2255. This total electrostatic equilibrium appears in Theorem 2, [10] which will be used later. Proposition 4.…”