Free interpolation in some banach spaces of analytic functions

We state a series of results regarding the interpolation in the spaces of analytic functions being the Taylor coefficient of the expansion at O, and. One asserts that a sequence with one condensation point, having a structure similar to a geometric progression, is an interpolation sequence for these spaces, i.e., the restriction operator on these sets maps these spaces onto the corresponding collection of sequences. In this case the restriction operator has a continuous right inverse which is explicitly constructed. This note is a continuation of the author's paper. Ref. Zh. Mat. 1973, 4B164.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 186–187, 1976.     

Free interpolation of germs of analytic functions in Hardy spaces

One proves theorems on the interpolation of germs of analytic functions, defined in the neighborhoods of the interpolation nodes, in the Hardy spaces H^P(0 < p +), generalizing the corresponding results of N. K. Nikol'skii and V. I. Vasyunin for the classes H and H². One obtains estimates of the norms of the interpolating functions in terms of the parameter of the set on which the interpolation is performed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 36–45, 1982.     

Free interpolation in Bergman spaces

A series of conditions is given, imposed on a subset Λ of the unit disk D, sufficient that the collection of all restrictions to the set Λ of functions from the Bergman space be naturally isomorphic with the space ?^p(Λ).     

Nonclassical interpolation in spaces of analytic functions smooth up to the boundary

One considers the following problem. Let X be the space of smooth functions on the circumference and let Y be the space of functions analytic in the disk and smooth up to the boundary. One has to find necessary and sufficient conditions on the closed subset E of the circumference that ensure the inclusion^X¦_E^y¦_E. The problem is solved in the case when the space X is a Carleman class and Y is either an analytic Carleman class having weaker smoothness properties than X, or a Hölder class A^s with arbitrary exponent s.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 7–26, 1982.     

Free holomorphic functions and interpolation

In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna–Pick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function , where is an arbitrary subset of the open unit ball , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball such that

Unconditionality in spaces of smooth functions

Our main result shows that subspaces of L¹([0, 1]) on which the blow-up operators act compactly are isometric to dual spaces, and their natural predual belongs to the Banach-Mazur closure of quotient spaces of . Related general results are shown for subspaces X of or of reflexive K?the function spaces, which imply that when X consists of smooth functions it embeds into a Banach space with an unconditional basis. Received: 25 September 2008     

Conjugate interpolation problems in spaces of entire functions

A generalization of the Taylor expansions of entire functions with variable center is used to illustrate the notion of conjugate interpolation problem. The radii of existence, uniqueness, and convergence for such problems are found. A relationship between the interpolation functional systems of two mutually conjugate problems is revealed. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 616–628, April, 2000.     

A remark on interpolation in spaces of vector functions

Let B(H) be the space of bounded operators in a Hubert space H, let B _p ^s (_p) be the Besov class of functions, analytic in the unit circle and taking values in the Schatten-von Neumann class _p (H), and let . The fundamental result is that (B _p ^1/p (r_p),X)_Q,q=B _q ^1/g (r_q), 1 p , 0<Q<1, q=p/(1–).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 162–164, 1985.     

Multivariate convexity preserving interpolation by smooth functions

A set of multivariate data is called strictly convex if there exists a strictly convex interpolant to these data. In this paper we characterize strict convexity of Lagrange and Hermite multivariate data by a simple property and show that for strict convex data and given smoothness requirements there exists a smooth strictly convex interpolant. We also show how to construct a multivariate convex smooth interpolant to scattered data. Partially supported by DGICYT PS93-0310 and by the EC project CHRX-CT94-0522.     

Multivariate convexity preserving interpolation by smooth functions

A set of multivariate data is called strictly convex if there exists a strictly convex interpolant to these data. In this paper we characterize strict convexity of Lagrange and Hermite multivariate data by a simple property and show that for strict convex data and given smoothness requirements there exists a smooth strictly convex interpolant. We also show how to construct a multivariate convex smooth interpolant to scattered data. Communicated by T.N.T. Goodman     

Lagrange interpolation for continuous piecewise smooth functions

This note is devoted to Lagrange interpolation for continuous piecewise smooth functions. A new family of interpolatory functions with explicit approximation error bounds is obtained. We apply the theory to the classical Lagrange interpolation.     

Multiple interpolation and extremal functions in the Bergman spaces

Multiple interpolation sequences for the Bergman space are characterized. In addition, relationships between interpolation sequences and extremal functions are explored.     

Geometric conditions for interpolation in weighted spaces of entire functions

We use L² estimates for the equation to find geometric conditions on discrete interpolating varieties for weighted spaces A_p(ℂ) of entire functions such that |f(z)|≤Ae^Bp(z) for some A, B>0. In particular, we give a characterization when p(z)=e^|z| and more generally, when In p(e^r) is convex andIn p(r) is concave. Acknowledgements and Notes. The author wishes to thank X. Massaneda for useful talks and remarks.     

Lipschitz and darbo conditions for the superposition operator in some non-ideal spaces of smooth functions

Let X be a Banach space of differentiable functions and A: XX be a superposition operator. We consider for A the conditions