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 l The basic tool is the decomposition ofN pairs of free charged bosons with respect toglN and the commuting withglN Lie algebra of infinite matrices?l. Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K]. It was then observed independentely by Lusztig and Ginzburg-Vasserot (see [L1], [GV]) that this construction admits an affine analogue in terms of periodic flags of lattices. As a consequence we prove a recent conjecture of Lusztig (see [L1]). The aim of this paper is to begin a study of the cohomology modules Hi(X(w),Lλ) We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. For instance, we find that |f(u)| ≤ L(u) + O(ε2/3), where L(u) is the Boas-Kac-Lukosz bound, and show by means of an example that this version is the sharpest possible with respect to its behaviour as a function of ε as ε ↓ 0. These spaces include, for example, the $L^p$ spaces, One outcome is a simple proof that for $g_{m \alpha , n \beta}$ to span $L^2,$ the lattice $(m \alpha , n \beta )$ must have at least unit density. We investigate the $L_p$-error of approximation to a function $f\in L_p({\Bbb T}^d)$ by a linear combination $\sum_{k}c_ke_k$ of $n$ Autocorrelation Functions as Translation Invariants in L1 and L2 13 (1962), 425-428] states that the collection of nth-order autocorrelation functions ${\cal M} = \{M^n(\cdot): n=1,2,\dots\}$ is a complete set of translation invariants for real-valued L1 functions on a locally compact abelian group. Indeed, if the coefficients of L are in $W^{1,2}\cap L^{\infty},$ then L can be rewritten in divergence form for which the notion of a "weak" solution can be applied. Higher dimensional analogues of these results, which apply to functions $f\in L^p[-R,R]^d$ and $C^{m-2-k}[-R,R]^d,$ are proven. The sampling results are sharp in the sense that if any condition is omitted, there exist nonzero $f\in L^p[-R,R]^d$ and $C^{m-2-k}[-R,R]^d$ satisfying the rest. It is shown that the one-dimensional sampling sets correspond to Bessel sequences of complex exponentials that are not Riesz bases for $L^2[-R,R].$ A signal processing application in which such sampling sets arise naturally is described in detail. The investigation of $L^q$ norm Bernstein inequalities $(0>amp;lt;q\leq \infty ),$ for polynomials with unimodular coefficients, leads to unexpected results. It is shown that these transformations are bounded in the space $L^p,\ 1>amp;lt;p>amp;lt;\infty,$ with respect to the measure that makes LB selfadjoint. Hence, we are able to approximate (in the L2-norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Our work builds on a uniqueness result for reconstructing an L2 signal from irregular sampling of its wavelet transform of Grochenig and the related work of Benedetto, Heller, Mallat, and Zhong.

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