We consider a diffusion process $X$ which is observed at times $i/n$ for $i=0,1,...,n$, each observation being subject to a measurement error. All errors are independent and centered gaussian with known variance ${\rho }_n$. There is an unknown parameter within the diffusion coefficient, to be estimated. In this first paper the case when $X$ is indeed a gaussian martingale is examined: we can prove that the LAN property holds under quite weak smoothness assumptions, with an explicit limiting Fisher information. What...

In this paper we will prove a Mittag-Leffler type theorem for $\overline{\partial }$-closed $(0,n-1)$-forms in ${ℂ}^n$ by addressing the question of constructing such differential forms with prescribed periods in certain domains.

Let $G_{n,k}$ (${\tilde{G}}_{n,k}$) denote the Grassmann manifold of linear $k$-spaces (resp. oriented $k$-spaces) in ${ℝ}^n$, $d_{n,k}=k(n-k)=\text{dim}{G}_{n,k}$ and suppose $n\ge 2k$. As an easy consequence of the Steenrod obstruction theory, one sees that $(d_{n,k}+1)$-fold Whitney sum $(d_{n,k}+1){\xi }_{n,k}$ of the nontrivial line bundle ${\xi }_{n,k}$ over $G_{n,k}$ always has a nowhere vanishing section. The author deals with the following question: What is the least $s$ ($=s_{n,k}$) such that the vector bundle $s{\xi }_{n,k}$ admits a nowhere vanishing section ? Obviously, $s_{n,k}\le d_{n,k}+1$, and for the special case in which $k=1$, it is known that $s_{n,1}=d_{n,1}+1$....

In this paper we extend the Duman-King idea of approximation of functions by positive linear operators preserving $e_k\left(x\right)=x^k$, $k=0,2$. Using a modification of certain operators $L_n$ preserving $e_0$ and $e_1$, we introduce operators $L_n^{*}$ which preserve $e_0$ and $e_2$ and next we define operators $L_{n;r}^{*}$ for $r$-times differentiable functions. We show that $L_n^{*}$ and $L_{n;r}^{*}$ have better approximation properties than $L_n$ and $L_{n;r}$.