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Let be a totally positive function of finite type, that is, for , , and , and let . Then the set is a frame for if and only if . This result is a first positive contribution to a conjecture of Daubechies from 1990. Until now, the complete characterization of lattice parameters , that generate a frame has been known for only six window functions . Our main result now yields an uncountable class of window functions. As a by-product of the proof method, we also derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.
This article studies a distinguished collection of so-called generalized Heegner cycles in the product of a Kuga–Sato variety with a power of a CM elliptic curve. Its main result is a -adic analogue of the Gross–Zagier formula which relates the images of generalized Heegner cycles under the -adic Abel–Jacobi map to the special values of certain -adic Rankin -series at critical points that lie outside their range of classical interpolation.
In this article, we classify -dimensional () complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any -dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat and hence a finite quotient of the Gaussian shrinking soliton or the round cylinder . More generally, for , a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton or the product , where is Einstein.
We study degenerations of complex surfaces with no vanishing cycles first described by J. Wahl. Given such a degeneration, we construct an exceptional vector bundle on the general fiber in the case . For , we show that our construction establishes a bijective correspondence between the possible singular surfaces and the set of exceptional bundles on modulo a natural equivalence relation.