Hermite Function Description of Feichtinger’s Space S0

For let be the set of all for which , where is the orthonormal base of Hermite functions. We show that if and only if and that if and only if . Here S₀ is Feichtinger’s space.     

Dirac Concentrations in a Chemostat Model of Adaptive Evolution(In honor of the immense scientific influence of Ham Brezis)

This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection. In this paper, the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies.Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.     

Announcement of New Editorial Policy

In view of the rapid diversification of the field of approximation theory and its applications, the editorial committee has decided to expand the scope of the journal. Until recently, the central themes have included 1. approximation theory 2. numerical approximation 3. harmonic analysis 4. fractals and wavelets     

Announcement of New Editorial Policy

In view of the rapid diversification of the field of approximation theory and its applications, the editorial committee has decided to expand the scope of the journal.     

Announcement of New Editorial Policy

Inviewoftherapiddiversificationofthefieldofapproximationtheoryanditsapplica tions,theeditorialcommitteehasdecidedtoexpandthescopeofthejournal.　　Untilrecently ,thecentralthemeshaveincluded1 .approximationtheory2 .numericalapproximation3 .harmonicanalysis4.fractalsandwavelets　　SinceVolume 1 9,Issue 1 ,thejournalwillfocuson( 1 )approximationtheory( 2 )computationalanalysis( 3 )dynamicsystems( 4 )fractals( 5)functionspaces( 6) geometricanalysis( 7)harmonicanalysis( 8)numericalapproximation( 9)…     

Announcement of New Editorial Policy

In view of the rapid diversification of the field of approximation theory and its applica-tions, the editorial committee has decided to expand the scope of the journal.     

The Implementation of an In‐Core Compiler

The present need in teaching institutions to convey to hundreds of students at least an elementary level of programming skills has led to the development of a number of fast compilers, usually designed to remain in core throughout the compilation and execution of a batch of student programs. Such compilers have usually been developed by institutions with considerable resources in terms of hardware and man‐power.

It is the author's belief that the writing of a special purpose compiler is not necessarily a large job, measurable in man‐years of effort and hence beyond the reach of a small installation. In this paper, methods of design and programming that have made such a project feasible with a limited amount of programmer effort are described, together with the reasons for implementing the compiler in its present form. Some suggested extensions are listed, as are some statistics covering two years’ use of the system.     

In memory of Tibor Gallai Paul Erdős

In memory of Tibor Gallai Paul Erds     

Optimal Order of Convergence and (In)Tractability of Multivariate Approximation of Smooth Functions

We study the approximation problem for C ^∞ functions f:[0,1] ^d →? with respect to a W _p ^m -norm. Here, m=[m,m,…,m], d times, with the norm of the target space defined in terms of up to m partial derivatives with respect to all d variables. The optimal order of convergence is infinite, hence excellent, but the problem is still intractable and suffers from the curse of dimensionality if m≥1. This means that the order of convergence supplies incomplete information concerning the computational difficulty of a problem. For m=0 and p=2, we prove that the problem is not polynomially tractable, but that it is weakly tractable.     

Number of quasimodes of “bouncing-ball” type

A new two-scale expansion is proposed for eigenfunctions of bouncing-ball type and the corresponding eigenvalues of the Laplace operator with the Dirichlet condition in a domain of the plane. The eigenfunctions are concentrated in a neighborhood of a stable diameter of the domain and are numbered with two indices (p, q) where p is the number of longitudinal nodes and q the number of nodes in a direction orthogonal to the diameter. The validity of the asymptotic expansions is guaranteed for 0 q const p^–1 and >0 as p +.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 172–182, 1981.