We study a tangential interpolation problem with an arbitrary set of interpolating points on the distinguished boundary of the unit polydisk for Schur–Agler class. The description of all solutions is parametrized in terms of a linear fractional transformation.  2002 Elsevier Science (USA). All rights reserved.

We find necessary density conditions for Marcinkiewicz-Zygmund inequalities and interpolation for spaces of spherical harmonics in S with respect to the L norm. Moreover, we prove that there are no complete interpolation families for p 6= 2.

We show that, in complex interpolation, an operator function that is compact on one side of the interpolation scale will be compact for all proper interpolating spaces if the right hand side (Y , Y ) is reduced to a single space. A corresponding result, in restricted generality, is shown if the left hand side (X, X) is reduced to a single space. These results are derived from the fact that a ho...

Characterization of Schur functions in terms of their Taylor coefficients is due to C. Carathéodory and I. Schur. We discuss the boundary analogue of this problem.

We consider the problem of growth of the sequence of Lebesgue constants corresponding to the Newton interpolation and estimate the growth of this sequence in the case of a nested family of Chebyshev's points.

Given > 0, we compute a function taking prescribed values at N given points in R2, whose C2-norm is within a factor (1 + ) of least possible. The computation takes C( )N logN computer operations.

Suppose that (A0, A1) and (B0, B1) are Banach couples, and that T is a linear operator which maps A0 compactly into B0 and A1 boundedly (or even compactly) into B1. Does this imply that T : [A0, A1]θ → [B0, B1]θ compactly? (Here, as usual, [A0, A1]θ denotes the complex interpolation space of Alberto Calderón.) This question has been open for 44 years. Affirmative answers are known for it in man...

We discuss the aysmptotics of the points that maximize the determinant of the interpolation matrix for interpolants of the form I1(x) = n ∑ i=1 aie αxti and I2(x) = n ∑

Several recent papers have treated the latent space of deep generative models, e.g., GANs or VAEs, as Riemannian manifolds. The argument is that operations such as interpolation are better done along geodesics that minimize path length not in the latent space but in the output space of the generator. However, this implicitly assumes that some simple metric such as L2 is meaningful in the output...