Vol. 63, No. 1, 1976

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Interpolating sequences for functions satisfying a Lipschitz condition

Eric P. Kronstadt

Vol. 63 (1976), No. 1, 169–177

Let D be the unit disk in C, Lip (D) the space of functions, f, holomorphic in D, continuous on D and satisfying a Lipschitz condition:

|f (z)− f(w )| ≦ M |z − w | ∀z,w ∈ D

If S = {ai}x=1D is a discrete sequence with no accumulation points in D, let Lip (S) be the set of functions, g, defined on S satisfying

|g(ai)− g(aj)| ≦ M |ai − aj| ∀i,j = 1,2,⋅⋅⋅

We say S is a Lip (D) interpolating (LI) sequence if the restriction mapping from Lip (D) to Lip (S) given by f f|S is surjective. Our aim is to describe some of the properties of such sequences and to give some examples. Specifically we show that an LI sequence must either be a uniformly separated sequence, or the union of two such sequences which approach one another as they tend to ∂D.

Mathematical Subject Classification
Primary: 30A80, 30A80
Received: 4 November 1974
Published: 1 March 1976
Eric P. Kronstadt