On a characterization of finite Blaschke products

Boundary Interpolation by Finite Blaschke Products

Given 2n distinct points z1, z′ 1, z2, z ′ 2, . . . , zn, z ′ n (in this order) on the unit circle, and n points w1, . . . , wn on the unit circle, we show how to construct a Blaschke product B of degree n such that B(zj) = wj for all j and, in addition, B(z′ j) = B(z ′ k) for all j and k. Modifying this example yields a Blaschke product of degree n− 1 that interpolates the zj ’s to the wj ’s. ...

The Location of Critical Points of Finite Blaschke Products

A theorem of Bôcher and Grace states that the critical points of a cubic polynomial are the foci of an ellipse tangent to the sides of the triangle joining the zeros. A more general result of Siebert and others states that the critical points of a polynomial of degree N are the algebraic foci of a curve of class N − 1 which is tangent to the lines joining pairs of zeroes. We prove the analogous...

A Characterization of Blaschke Addition

A characterization of Blaschke addition as a map between origin-symmetric convex bodies is established. This results from a new characterization of Minkowski addition as a map between origin-symmetric zonoids, combined with the use of Lévy-Prokhorov metrics. A full set of examples is provided that show the results are in a sense the best possible.

Computable Analysis and Blaschke Products

We show that if a Blaschke product defines a computable function, then it has a computable sequence of zeros in which the number of times each zero is repeated is its multiplicity. We then show that the converse is not true. We finally show that every computable, radial, interpolating sequence yields a computable Blaschke product.

AN Lp DEFINITION OF INTERPOLATING BLASCHKE PRODUCTS