Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite dimensional Haar subspace of C(X,R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.     

Optimal approximation of elliptic problems by linear and nonlinear mappings III: Frames

We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space and is a bounded Lipschitz domain; the error is always measured in the H^s-norm. We study the order of convergence of the corresponding nonlinear frame widths and compare it with several other approximation schemes. Our main result is that the approximation order is the same as for the nonlinear widths associated with Riesz bases, the Gelfand widths, and the manifold widths. This order is better than the order of the linear widths iff p<2. The main advantage of frames compared to Riesz bases, which were studied in our earlier papers, is the fact that we can now handle arbitrary bounded Lipschitz domains—also for the upper bounds.     

Simultaneous approximation and interpolation of increasing functions by increasing entire functions

We prove that, under suitable assumptions, an isomorphism g of dense subsets A,B of the real line can be taken to approximate a given increasing Cⁿ surjection f with the derivatives of g agreeing with those of f on a closed discrete set. For example, we have the following theorem. Let be a nondecreasing Cⁿ surjection. Let be a positive continuous function. Let be a closed discrete set on which f is strictly increasing. Let each of {A_i}, {B_i} be a sequence of pairwise disjoint countable dense subsets of such that for each and xE we have xA_i if and only if f(x)B_i. Then there is an entire function such that and the following properties hold.

(a) For all , Dg(x)>0.

(b) For k=0,…,n and all , |D^kf(x)−D^kg(x)|<ε(x).

(c) For k=0,…,n and all xE, D^kf(x)=D^kg(x).

(d) For each , g[A_i]=B_i.

This provides a version for increasing functions of a theorem of Hoischen. In earlier work, we proved that it is consistent that a similar theorem, omitting clause (c), holds when the sets A_i,B_i are of cardinality ₁ and have second category intersection with every interval. (See the introduction for the exact statement.) In this paper, we show how to incorporate clause (c) into the statement of the earlier theorem.

Interpolation and approximation in

Assume a standard Brownian motion W=(W_t)_t[0,1], a Borel function such that f(W₁)L₂, and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space , obtained via the real interpolation method, by the behavior of , where is a deterministic time net and the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers a_X(f(X₁);τ) can be used to describe the L₂-error in discrete time simulations of the martingale generated by f(W₁) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.     

Linear interpolation and Sobolev orthogonality

There is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints , defined by

where f belongs to a certain Sobolev space, a_ij() are piecewise continuous functions over [a,b], b_ijk are real numbers, and the points t_k belong to [a,b] (the nonnegative integer m depends on each concrete interpolation scheme). For each f in this Sobolev space and for each integer l greater than or equal to the number of constraints considered, we compute the unique best approximation of f in , denoted by p_f, which fulfills the interpolatory data , and also the condition that best approximates f⁽ⁿ⁾ in (with respect to the norm induced by the continuous part of the original discrete–continuous bilinear form considered).     

Smooth approximation of Lipschitz functions on Riemannian manifolds

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous , and for every positive number r>0, there exists a C^∞ smooth Lipschitz function such that |f(p)−g(p)|?ε(p) for every p∈M and Lip(g)?Lip(f)+r. Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville-Godefroy-Zizler's smooth variational principle.     

On relations between weak approximation properties and their inheritances to subspaces

It is shown that for the separable dual X^∗ of a Banach space X, if X^∗ has the weak approximation property, then X^∗ has the metric weak approximation property. We introduce the properties W^∗D and MW^∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M^⊥ is complemented in the dual space X^∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W^∗D (respectively, metric weak approximation property and MW^∗D), then M has the weak approximation property (respectively, bounded weak approximation property).     

On holomorphic domination, I

Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C¹-smooth -closed (0,1)-form on the space X=L₁[0,1] of summable functions, then there is a C¹-smooth function u on X with on X.     

C-Weierstrass for compact sets in Hilbert space

The C¹-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space . The same theorem is also proved for Whitney 1-jets on X when X satisfies the following further condition: There exist finite dimensional linear subspaces such that ?_n?1H_n is dense in and π_n(X)=X∩H_n for each n?1. Here, is the orthogonal projection. It is also shown that when X is compact convex with and satisfies the above condition, then C¹(X) is complete if and only if the C¹-Whitney extension theorem holds for X. Finally, for compact subsets of , an extension of the C¹-Weierstrass approximation theorem is proved for C¹ maps with compact derivatives.     

Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay

In this paper, we obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space BC((-∞,0];R^d) which denotes the family of bounded continuous R^d-value functions defined on (-∞,0] with norm under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition. The solution is constructed by the successive approximation.     

An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function (x;α,h)≡exp(-[α²/h²]x²) where α is a user-choosable constant. Unfortunately, computing the coefficients of (x-jh;α,h) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C_j(x;α,h) is defined by the set of linear combinations of the Gaussians such that C_j(kh)=1 when k=j and C_j(kh)=0 for all integers . We show that the cardinal functions for the uniform grid are C_j(x;h,α)=C(x/h-j;α) where C(X;α)≈(α²/π)sin(πX)/sinh(α²X). The relative error is only about 4exp(-2π²/α²) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h→0, but only to an “error saturation” proportional to exp(-π²/α²). Because the error in our approximation to the master cardinal function C(X;α) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f(x). The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.     

Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay

In this paper, we prove the invariance of Stepanov-like pseudo-almost periodic functions under bounded linear operators. Furthermore, we obtain existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations u^′(t)=A(t)u(t)+h(t) and on , assuming that A(t) satisfy “Acquistapace–Terreni” conditions, that the evolution family generated by A(t) has exponential dichotomy, that R(λ₀,A()) is almost periodic, that B,C(t,s)_t≥s are bounded linear operators, that f is Lipschitz with respect to the second argument uniformly in the first argument and that h, f, F are Stepanov-like pseudo-almost periodic for p>1 and continuous. To illustrate our abstract result, a concrete example is given.     

Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

Let M be an m-dimensional, C^k manifold in , for any , and for any τ>0 let

     

Fourier transforms and generalized Lipschitz classes in uniform metric

For functions f∈L¹(R)∩C(R) with Fourier transforms in L¹(R) we give necessary and sufficient conditions for f to belong to the generalized Lipschitz classes Hω,m and hω,m in terms of behavior of .     

Lipschitz regularity for scalar minimizers of autonomous simple integrals

We prove Lipschitz regularity for a minimizer of the integral , defined on the class of the AC functions having x(a)=A and x(b)=B. The Lagrangian may have L(s,) nonconvex (except at ξ=0), while may be non-lsc, measurability sufficing for ξ≠0 provided, e.g., L^**() is lsc at (s,0) s. The essential hypothesis (to yield Lipschitz minimizers) turns out to be local boundedness of the quotient φ/ρ() (and not of L^**() itself, as usual), where φ(s)+ρ(s)h(ξ) approximates the bipolar L^**(s,ξ) in an adequate sense. Moreover, an example of infinite Lavrentiev gap with a scalar 1-dim autonomous (but locally unbounded) lsc Lagrangian is presented.     

Vector hyperinterpolation on the sphere

We present new results on hyperinterpolation for spherical vector fields. Especially we consider the operator , which may be described as an approximation to the L² orthogonal projection . In detail, we prove that is the projection with the least uniform norm and that has the optimal value for its norm in the C→L² setting. These results are already known for the scalar case. In the continuous space setting, we could prove only a sub-optimal bound for the Lebesgue constant of the vector hyperinterpolation operator.     

Coarse version of the Banach-Stone theorem

We show that if there exists a Lipschitz homeomorphism T between the nets in the Banach spaces C(X) and C(Y) of continuous real valued functions on compact spaces X and Y, then the spaces X and Y are homeomorphic provided . By l(T) and l(T−1) we denote the Lipschitz constants of the maps T and T−1. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is . We also estimate the distance of the map T from the isometry of the spaces C(X) and C(Y).     

Continuous extension operators and convexity

Given a nonempty closed subset A of a Hilbert space X, we denote by L(A) the space of all bounded Lipschitz mappings from A into X, equipped with the supremum norm. We show that there is a continuous mapping F_c:L(A)?L(X) such that for each g∈L(A), F_c(g)|_A=g, , and . We also prove that the corresponding set-valued extension operator is lower semicontinuous.     

A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix

A non-linear structure preserving matrix method for the computation of a structured low rank approximation of the Sylvester resultant matrix S(f,g) of two inexact polynomials f=f(y) and g=g(y) is considered in this paper. It is shown that considerably improved results are obtained when f(y) and g(y) are processed prior to the computation of , and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of , which leads to a linear structure preserving matrix method, or they can be incremented during the computation of , which leads to a non-linear structure preserving matrix method. It is shown that the non-linear method yields a better structured low rank approximation of S(f,g) and that the assignment of f(y) and g(y) is important because may be a good structured low rank approximation of S(f,g), but may be a poor structured low rank approximation of S(g,f) because its numerical rank is not defined. Examples that illustrate the differences between the linear and non-linear structure preserving matrix methods, and the importance of the assignment of f(y) and g(y), are shown.