Analogues of finite Blaschke products as inner functions

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...

Regularized Inner Products of Modular Functions

The numerator over c is the Ramanujan sum ∑ (a,c)=1 cos( 2πma c ). This identity clearly displays the oscillations of σ(m) around its mean value π m 6 . Also, (1.1) makes sense as a limit when m = 0 and gives the extension σ(0) = − 1 24 . There is a nice generalization of Ramanujan’s formula, that goes back to Petersson and Rademacher, that connects it with the theory of modular forms (see Knop...

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