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 condition We also prove the shifted cocycle condition for the twistors, thereby completing Fr?nsdal's findings. It is known [M4] that K?-orbits S and G?-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact. We give a simple necessary and sufficient condition for a Schubert It is also shown that on the nilmanifold $\Gamma\backslash (H^3\times H^3)$ the balanced condition is not stable under small deformations. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In the spirit of work of Kerman and Sawyer, a condition is given that is necessary and sufficient for the Fourier transform norm inequality 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. A sufficient condition for the sequence of sampling functions to constitute a frame is derived. Here we show that U actually is invertible under a much weaker condition. We prove a trigonometric inequality of Ingham's type for nonharmonic Fourier series when the gap condition between frequencies does not hold any more. It is shown that this is possible when the window g ε L2(?) generating the Weyl-Heisenberg frame satisfies an appropriate regularity condition at the integers. When, in addition, the Tolimieri-Orr condition A is satisfied, the minimum energy dual windowoγ ε L2(?) can be sampled as well, and the two sampled windows continue to be related by duality and minimality. A necessary condition for Calderón-Zygmund singular integral operators For any closed subspace V0/L2 (R), we present a necessary and sufficient condition under which there is a sampling expansion for everyf ε V0-Several examples are given. A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported We present a necessary and sufficient condition on a compactly supported function g(t) generating a Weyl-Heisenberg frame for L2 (?) for its minimal dual (Wexler-Razdual) γ0 (t) to be compactly supported. We furthermore provide a necessary and sufficient condition for a band-limited function g(t) generating a Weyl-Heisenberg frame for L2 (?) to have a band-limited minimal dual γ0 (t). This somewhat technical result does provide a method for simple constructions of low pass filters whose only smoothness assumption is a Holder condition at the origin. A necessary and sufficient condition is presented for a set to be a Pompeiu subset of any compact homogeneous space with a finite invariant measure. The condition, which is expressed in terms of the intertwining operators of each primary summand of the quasi-regular representation, is then interpreted in the case of the compact Heisenberg manifolds.

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