A constructive approach to minimal projections in Banach spaces

Let X be a Banach space and Y a finite-dimensional subspace of X. Let P be a minimal projection of X onto Y. It is shown (Theorem 1.1) that under certain conditions there exist sequences of finite-dimensional “approximating subspaces” X_m and Y_m of X with corresponding minimal projections P_m: X_m → Y_m, such that lim_m→∞ P_m = P. Moreover, a certain related sequence of projections i_m○P_m○π_m: X → Y has cluster points in the strong operator topology, each of which is a minimal projection of X onto Y. When X = C[a, b] the result reduces to a theorem of [7.]. It is shown (Corollary 1.11) that the hypothesis of Theorem 1.1 holds in many important Banach spaces, including C[a, b], L^P[a, b] and l^P for 1 p < ∞, and c₀, the space of sequences converging to zero in the sup norm.     

Codimension-one minimal projections onto Haar subspaces

Let H_n be an n-dimensional Haar subspace of and let H_n−1 be a Haar subspace of H_n of dimension n−1. In this note we show (Theorem 6) that if the norm of a minimal projection from H_n onto H_n−1 is greater than 1, then this projection is an interpolating projection. This is a surprising result in comparison with Cheney and Morris (J. Reine Angew. Math. 270 (1974) 61 (see also (Lecture Notes in Mathematics, Vol. 1449, Springer, Berlin, Heilderberg, New York, 1990, Corollary III.2.12, p. 104) which shows that there is no interpolating minimal projection from C[a,b] onto the space of polynomials of degree n, (n2). Moreover, this minimal projection is unique (Theorem 9). In particular, Theorem 6 holds for polynomial spaces, generalizing a result of Prophet [(J. Approx. Theory 85 (1996) 27), Theorem 2.1].     

Extrapolation spaces and minimal regularity for evolution equations

A new approach to extrapolation spaces for unbounded linear operators is applied to evolution equations in a Banach space in order to derive existence and properties of its solutions under minimal assumptions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.     

A characterization of maximal and minimal Fermat curves

In this article we provide a characterization of maximal and minimal Fermat curves using the classification of supersingular Fermat curves.     

A characterization of minimal valid inequalities for mixed integer programs

A simple condition on the underlying subadditive function is shown to characterize minimal valid inequalities. This result is proved in a very general master problem framework and completes the characterization there. We explain the condition also in the context of value functions and finally give some related, unresolved questions.     

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.     

A characterization of minimal locally finite varieties

In this paper we describe a one-variable Mal'cev-like condition satisfied by any locally finite minimal variety. We prove that a locally finite variety is minimal if and only if it satisfies this Mal'cev-like condition and it is generated by a strictly simple algebra which is nonabelian or has a trivial subalgebra. Our arguments show that the strictly simple generator of a minimal locally finite variety is unique, it is projective and it embeds into every member of the variety. We give a new proof of the structure theorem for strictly simple abelian algebras that generate minimal varieties.

     

Center projections for smooth difference equations of mixed type

We study a class of mixed type difference equations that enjoy a special smoothening property, in the sense that solutions automatically satisfy an associated functional differential equation of mixed type. Using this connection, a finite dimensional center manifold is constructed that captures all solutions that remain sufficiently close to an equilibrium. The results enable a rigorous analysis of a recently developed model in economic theory, that exhibits periodic oscillations in the interest rates of a simple economy of overlapping generations.     

A characterization of multivariate normality through univariate projections

This paper introduces a new characterization of multivariate normality of a random vector based on univariate normality of linear combinations of its components.     

Generalized bi-circular projections on minimal ideals of operators

We characterize generalized bi-circular projections on a minimal norm ideal of operators in where is a separable infinite dimensional Hilbert space.

     

Almost locally minimal projections in finite dimensional Banach spaces

A projectionP on a Banach spaceX is called “almost locally minimal” if, for every α>0 small enough, the ballB(P,α) in the spaceL(X) of all operators onX contains no projectionQ with whereD is a constant. A necessary and sufficient condition forP to be almost locally minimal is proved in the case of finite dimensional spaces. This criterion is used to describe almost locally minimal projections on ℓ ₁ ⁿ . Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas, 1997. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Approximation and shape-preserving properties of q-Stancu operator

We introduce the definition of q-Stancu operator and investigate its approx- imation and shape-preserving property. With the help of the sign changes of f(x) and Ln = f ( f,q;x) the shape-preserving property of q-Stancu operator is obtained.