Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.     

Contractive Korovkin approximations

Our results are related to $\text{L}$¹-shadows in L_p-spaces. For p = 1 we will complete the characterization of $\text{L}$¹-shadows and $\text{L}$^1,1-shadows. For 1 < p < ∞ S. J. Bernau has shown that the $\text{L}$¹-shadow of a set in L_p is the range of a contractive projection. We will show that the corresponding theorem is not true for all reflexive spaces, but is true for locally uniformly convex reflexive spaces.     

Infinite-interval nonlinear approximations

Uniform nonlinear approximation on an infinite interval is studied. This study was motivated by a specific nonlinear approximation problem arising in the construction of a stiff ordinary differential equation system solver. General results include characterization, uniqueness, and discretization theorems, as well as a computational algorithm for this problem.     

Adaptive greedy approximations

The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function’s structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model. We derive a notion of the coherence of a signal with respect to a dictionary from our characterization of the approximation errors of a pursuit. The dictionary elements slected during the initial iterations of a pursuit correspond to a function’s coherent structures. The tail of the expansion, on the other hand, corresponds to a noise which is characterized by the invariant measure of the pursuit map. When using a suitable dictionary, the expansion of a function into its coherent structures yields a compact approximation. We demonstrate a denoising algorithm based on coherent function expansions.     

Galerkin eigenvector approximations

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace--and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a special emphasis on nonselfadjoint problems, thus extending earlier studies by Chatelin, Babuska and Osborn, and Knyazev. Consequences for the numerical treatment of elliptic PDEs discretized either with finite element methods or with spectral methods are discussed. New lower bounds to the of a pair of operators are developed as well.

     

Sequences of Diophantine approximations

We sharpen a technique of Gelfond to show that, in a sense, the only possible gap-free sequences of “good” Diophantine approximations to a fixed α ∈ C are trivial ones. For example, suppose that a > 1 and that (δ_n)_n=1^∞ and (σ_n)_n=1^∞ are two positive, strictly increasing unbounded sequences satisfying δ_n+1 ≤ aδ_n and σ_n+1 ≤ aσ_n. If there is a sequence of nonzero polynomials P_n ∈ Z[x] with deg P_n ≤ δ_n, deg P_n + log height P_n ≤ σ_n, and ∣P_n(α)∣ ≤ e^{?(2a+1)δ_nσ_n}, then each P_n(α) = 0.     

On calculating residuated approximations

For an order-preserving map f : L → Q between two complete lattices L and Q, there exists a largest residuated map ρ _f under f, which is called the residuated approximation of f. Andreka, Greechie, and Strecker introduced the notion of the shadow σ _f of f Iterations of the shadow are called the umbral mappings. The umbral mappings form a decreasing net that converges to the residuated approximation ρ _f of f. The umbral number u _f of f is the smallest ordinal number α such that the equation \({\sigma^{(\alpha)}_{f} = \rho_{f}}\) holds. In order to speed up the computation of the umbral number u _f of f and find some relation between the structure of L and u _f , we present the concept of the order skeleton of a lattice \({L, \tilde{L} = L/\sim}\), determined by a certain congruence relation ~ on L where each equivalence class [x] is the maximal autonomous chain containing x. If [x] is finite for each \({x \in L}\), then \({L_{o} := \{ \Lambda [x]\,|\, x \in L \}}\) is a join-subcomplete sub-semilattice of L isomorphic to the order skeleton \({\tilde{L}}\) of L; for every order-preserving mapping f : L → Q from such a lattice L to a complete lattice Q, we define f _o : L _o → Q by \({f_{o} := f|_{{L}_{o}}}\) and prove that \({u_{f} = u_{{f}_{o}}}\). For a lattice L with no infinite chains, the order skeleton \({\tilde{L}}\) of L is distributive if and only if the shadow σ _f of f is residuated for every complete lattice Q and every mapping f : L → Q. Related topics are discussed.     

Best simultaneous Lp approximations

In this paper we study simultaneous approximation of n real-valued functions in L _p[a, b] and give a generalization of some related results.     

On invariant random approximations

A result regarding invariant random approximation is proved. Supported by Kuwait University Research Grant No. SM-119.     

On signed normal-Poisson approximations

For lattice distributions a convolution of two signed Poisson measures proves to be an approximation comparable with the normal law. It enables to get rid of cumbersome summands in asymptotic expansions and to obtain estimates for all Borel sets. Asymptotics can be constructed two-ways: by adding summands to the leading term or by adding summands in its exponent. The choice of approximations is confirmed by the Ibragimov-type necessary and sufficient conditions. Received: 20 November 1996 / Revised version: 5 December 1997     

Diophantine approximations and foliations

In this paper we indicate the proof of an effective version of the Green-Griffiths conjecture for surfaces of general type and positive second Segre class (i.e.c ₁ ² >c ₂). Naturally this effective version is stronger than the Green-Griffiths conjecture itself.     

Topological approximations of multisets

Rough set theory is a powerful mathematical tool for dealing with inexact, uncertain or vague information. The core concept of rough set theory are information systems and approximation operators of approximation spaces. In this paper, we define and investigate three types of lower and upper multiset approximations of any multiset. These types based on the multiset base of multiset topology induced by a multiset relation. Moreover, the relationships between generalized rough msets and mset topologies are given. In addition, an illustrative example is given to illustrate the relationships between different types of generalized definitions of rough multiset approximations.     

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q ₁ ^rot , EQ ₁ ^rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.