2009/05/05 by Rachid Zarouf, Zarouf, Rachid
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.0905.0572
openalex publication_date 2009/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a finite set σ of the unit disc \mathbbD and a holomorphic function f in \mathbbD which belongs to a class X we are looking for a function g in another class Y which minimizes the norm |g|Y among all functions g such that g|σ=f|σ. Generally speaking, the interpolation constant considered is c(σ, X, Y)=sup_f∈ X, ∥ f∥X≤1inf\|g|Y: g|σ=f|σ\ . When Y=H∞, our interpolation problem includes those of Nevanlinna-Pick (1916), Caratheodory-Schur (1908). Moreover, Carleson's free interpolation (1958) has also an interpretation in terms of our constant c(σ, X, H∞). If X is a Hilbert space belonging to the scale of Hardy and Bergman weighted spaces, we show that c(σ, X, H∞)≤ aϕX(1-(1-r)/(n)) where n=#σ, r=maxλ∈σ|λ| and where ϕX(t) stands for the norm of the evaluation functional f↦ f(t) on the space X. The upper bound is sharp over sets σ with given n and r. If X is a general Hardy-Sobolev space or a general weighted Bergman space (not necessarily of Hilbert type), we also found upper and lower bounds for c(σ, X, H∞) (sometimes for special sets σ) but with some gaps between these bounds. This constrained interpolation is motivated by some applications in matrix analysis and in operator theory.