Une propriété de composition dans l'espace Hs

Let us assume that $1<p?q?+\infty$, $1/p<s?[s]<1$, and $[s]?1$. If f and g are functions in the Besov space $B_p^{s\text{,}q}(\mathbb{R})$, such that g is real valued and such that $f(0)=0$, then the composed function $f○g$ belongs to $B_p^{s\text{,}q}(\mathbb{R})$. To cite this article: G. Bourdaud, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, $L=F(\sqrt{d})$ a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, $M_f$, $M_g$ their matrices. We say that the pair $(f,g)$ is a k-pair if there exist $S\in G{L}_n(L)$ such that all the entries of the $k\times k$ upper-left corner of the matrices $S{M}_f{S}^t$ and $S{M}_g{S}^t$ are in F. We give certain criteria to determine whether a given pair $(f,g)$ is a k-pair. We consider the transfer ${\mathrm{cor}}_{L(t)/F(t)}$ determined by the $F(t)$-linear map $s:L(t)\to F(t)$ with $s(1)=0$, $s(\sqrt{d})=1$, and prove that if $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a $[\frac{k+1}{2}]$-pair. If, additionally, the form $f+tg$ does not have a totally isotropic subspace of dimension $p+1$ over $L(t)$, we show that $(f,g)$ is a $(k?2p)$-pair. In particular, if the form $f+tg$ is anisotropic, and $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a k-pair.     

A failing eigenfunction expansion associated with an indefinite Sturm–Liouville problem

We consider the indefinite Sturm–Liouville problem $?f^{″}=\lambda rf$, $f^{\prime }(?1)=f^{\prime }(1)=0$ where $r\in L^1[?1,1]$ satisfies $xr(x)>0$. Conditions are presented such that the (normed) eigenfunctions $f_n$ form a Riesz basis of the Hilbert space $L_{|r|}^2[?1,1]$ (using known results for a modified problem). The main focus is on the non-Riesz basis case: We construct a function $f\in L_{|r|}^2[?1,1]$ having no eigenfunction expansion $f=\sum {\beta }_n{f}_n$. Furthermore, a sequence $({\alpha }_n)\in l^2$ is constructed such that the “Fourier series” $\sum {\alpha }_n{f}_n$ does not converge in $L_{|r|}^2[?1,1]$. These problems are closely related to the regularity property of the closed non-semibounded symmetric sesquilinear form $\mathfrak{t}[u,v]=\int u^{\prime }{\overline{v}}^{\prime }pdx$ with Dirichlet boundary conditions in $L^2[?1,1]$ where $p=1/r$. For the associated operator $T_{\mathfrak{t}}$ we construct elements in the difference between $\mathrm{dom}\mathfrak{t}$ and the domain of the associated regular closed form, i.e. $\mathrm{dom}{|T_{\mathfrak{t}}|}^{1/2}$.     

The D+E[Γ⁎] construction from Prüfer domains and GCD-domains

Let $D?E$ denote an extension of integral domains, Γ be a nonzero torsion-free grading monoid with $\Gamma \cap ?\Gamma =\{0\}$, ${\Gamma }^{?}=\Gamma ?\{0\}$ and $D+E[{\Gamma }^{?}]=\{f\in E[\Gamma ]|f(0)\in D\}$. In this paper, we give a necessary and sufficient criteria for $D+E[{\Gamma }^{?}]$ to be a Prüfer domain or a GCD-domain.     

Approximations in Sobolev spaces by prolate spheroidal wave functions

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ${\psi }_{n,c},c>0$. This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space $H^s([?1,1])$. The quality of the spectral approximation and the choice of the parameter c when approximating a function in $H^s([?1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([?1,1])$ as the restriction to $[?1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.     

On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) $K_1$ and $K_2$ over a field k, a k-isomorphism of $K_1[t]$ ($K_1(t)$) and $K_2[t]$ ($K_2(t)$) implies a k-isomorphism of $K_1$ and $K_2$. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:1. Let K be an affine field over an algebraically closed field k of any characteristic. Suppose $K(t)?k(t_1,t_2,t_3)$, then $K?k(t_1,t_2)$.2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. Let $A=K[x,y,z,w]/M$ be the coordinate ring of M. Suppose $A[t]?k[x_1,x_2,x_3,x_4]$, then $\mathrm{frac}(A)?k(x_1,x_2,x_3)$, where $\mathrm{frac}(A)$ is the field of fractions of A.In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171].