Effective H ∞ interpolation

Abstract. Given a finite subset σ of the unit disc D and a holomorphic function f in D belonging to a class X, we are looking for a function g in another class Y which satisfies g |σ = f |σ and is of minimal norm in Y. Then, we wish to compare ‖g‖Y with ‖f‖X. More precisely, we consider the interpolation constant c (σ, X, Y) = sup f∈X, ‖f‖X ≤1infg |σ =f |σ ‖g ‖ Y. When Y = H ∞ , our interpolation problem includes those of Nevanlinna-Pick and Caratheodory-Schur. Moreover, Carleson’s free interpolation problem can be interpreted in terms of the constant c (σ, X, H ∞). For Y = H ∞, X = H p (the Hardy space) or X = L 2 a (the Bergman space), we obtain an upper bound for the constant c (σ, X, H ∞ ) in terms of n = card σ and r = maxλ∈σ |λ|. Our upper estimates are shown to be sharp with respect to n and r. 1

TOEPLITZ OPERATORS WITH RADIAL SYMBOLS ON BERGMAN SPACE AND SCHATTEN-VON NEUMANN CLASSES

In the present paper, we study spectral properties of Toeplitz operators with (quasi-) radial symbols on Bergman space. More precisely, the problem we are interested in is to understand when a given Toeplitz operator belongs to a Schatten-von Neumann class. The methods of the approximation theory (i.e., Legendre polynomials) are used to advance in this direction

TOEPLITZ OPERATORS WITH RADIAL SYMBOLS ON BERGMAN SPACE AND SCHATTEN-VON NEUMANN CLASSES

International audienceIn the present paper, we study spectral properties of Toeplitz operators with (quasi-) radial symbols on Bergman space. More precisely, the problem we are interested in is to understand when a given Toeplitz operator belongs to a Schatten-von Neumann class. The methods of the approximation theory (i.e., Legendre polynomials) are used to advance in this direction