Padé Interpolation by F-Polynomials and Transfinite Diameter

We define F-polynomials as linear combinations of dilations by some frequencies of an entire function F. In this paper, we use Padé interpolation of holomorphic functions in the unit disk by F-polynomials to obtain explicitly approximating F-polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set K⊂ℂ, then optimal choices for the frequencies of interpolating F-polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfinite diameter of K. In case of the Laplace transforms of measures on K, we show that the coefficients of interpolating F-polynomials stay bounded provided that the frequencies are Fekete points. Finally, we give a sufficient condition for measures on the unit circle that ensures that the sums of the absolute values of the coefficients of interpolating F-polynomials stay bounded.