

  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 GinzburgVasserot (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 BoasKacLukosz 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), 425428] states that the collection of nthorder autocorrelation functions ${\cal M} = \{M^n(\cdot): n=1,2,\dots\}$ is a complete set of translation invariants for realvalued 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 $C^{m2k}[R,R]^d,$ 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 $C^{m2k}[R,R]^d$ satisfying the rest.


It is shown that the onedimensional 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 L2norm) 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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