Minimal projections on hyperplanes in sequence spaces

Summary The projection constants of hyperplanes in the classical sequence spaces (c ₀ ) and (l ₁ ) are determined, together with the projections of minimum norm. Entrata in Redazione I'll dicembre 1972.     

Minimal projections in spaces of functions of N variables

We will construct a minimal and co-minimal projection from L^p([0,1]ⁿ) onto L^p([0,1]^n₁)++L^p([0,1]^n_k), where n=n₁++n_k (see Theorem 2.9). This is a generalization of a result of Cheney, Halton and Light from (Approximation Theory in Tensor Product Spaces, Lecture Notes in Mathematics, Springer, Berlin, 1985; Math. Proc. Cambridge Philos. Soc. 97 (1985) 127; Math. Z. 191 (1986) 633) where they proved the minimality in the case n=2. We provide also some further generalizations (see Theorems 2.10 and 2.11 (Orlicz spaces) and Theorem 2.8). Also a discrete case (Theorem 2.2) is considered. Our approach differs from methods used in [8,13,20].     

Minimal linear spaces

A decent linear space DLS(k) is a linear space with minimal line size at least three and with maximal line size exactly k. Denote by v_k (resp. b_k) the minimum number of points (resp. lines) in a DLS(k). We determine the numbers v_k and b_k for all k and prove that each DLS(k) with b_k lines has v_k points. Thus the DLS(k)'s with b_k lines are the minimal linear spaces.     

Oblique projections in Hilbert spaces

In this study we define some classes of projections in a Hilbert space and extend results obtained in a former paper, [2]. The theory allows better insight into examples described in [2], and it is demonstrated how a certain family of projections can be expressed by means of familiar operators.     

s-numbers of projections in Banach spaces

Given ans-number sequences te {h, x, y, c, d, a, Γ}, we find a characterization of the following property of a Banach spaceX:(P _s). There is a constantC>0 such that, for anyn-dimensional subspaceE ofX, we can find a projectionP fromX ontoE with sup k ks _k(P)≦Cn. As an application, we prove thatX has weak type 2 if and only ifX is finite dimensionally norming, thus answering a question of Casazza and Shura. Weak Hilbert spaces are also characterized in a new way, the main tool in the proof being a characterization of weak cotype 2 by means of projections. The latter is applied to the study of U.A.P., too.     

Minimal widths of metric spaces

The minimal width of an arbitrary metric space is defined as the greatest lower bound of its Kolmogorov widths under all isometric embeddings in all possible Banach spaces and is computed or estimated in a number of examples. Supported by RFBR grant No. 96-15-96249. Human Moscow State Technical University. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 33, No. 4, 49, October–December, 1999. Translated by R. S. Ismagilov     

Bicircular projections on some Banach spaces

In this paper we show that the bicircular projections are precisely the Hermitian projections and prove some immediate consequences of this result.     

Superapproximation for projections on spline spaces

Summary. In this paper general conditions are given for the superapproximation of projections on non-uniform mesh multiple knot splines in L_p-spaces. Various known results are contained as special cases.Mathematics Subject Classification (2000): 41A15, 41A25, 41A28, 65R26AcknowledgmentThe comments of the referees, among them attracting the authors attention to Theorem 2.7 in [7], are gratefully acknowledged.     

BERGMAN TYPE PROJECTIONS ON L^p SPACES

The paper is a survey on the action of Bergman type projections on various Lp on three types of holomorphic function spaces： weighted Bergman spaces, the Bloch spaces. The focus is space, and diagonal Besov spaces.     

Minimal kernels of weakly complete spaces

Let X be a weakly complete space i.e. X a complex space endowed with a C^k-smooth, k?0, plurisubharmonic exhaustion function. We give the notion of minimal kernelΣ¹=Σ¹(X) of X by the following property: x∈Σ¹ if no continuous plurisubharmonic exhaustion function is strictly plurisubharmonic near x. The study of the geometric properties of the minimal kernels is the aim of present paper. After stating that the minimal kernel Σ¹ of a weakly complete space can be defined by a single plurisubharmonic exhaustion function ?, called minimal, using the characterization in terms of Bremermann envelopes, we prove the following, crucial, result: if X is a weakly complete manifold and ? a minimal function for X, the nonempty level sets Σ_c¹=Σ¹∩{?=c} have the local maximum property. In the last section we discuss the special case of weakly complete surfaces. We prove that if dim_cX=2 and c is a regular value of a minimal function ? then the nonempty level sets Σ_c¹=Σ¹∩{?=c} are compact spaces foliated by holomorphic curves.     

Smoothing of Radon projections type of data by bivariate polynomials

Given information about a function in two variables, consisting of a finite number of Radon projections, we study the problem of smoothing this data by a bivariate polynomial. It turns out that the smoothing problem is closely connected with the interpolation problem. We propose several schemes consisting of sets of parallel chords in the unit disk which ensure uniqueness of the bivariate polynomial having prescribed Radon projections along these chords. Regular schemes play an important role in both interpolation and smoothing of such kind of data. We prove that the existence and uniqueness of the best smoothing polynomial relies on a regularity property of the scheme of chords. Results of some numerical experiments are presented too.