Carleson’s Theorem: Proof, Complements, Variations

Carleson’s Theorem: Proof, Complements, Variations

2 00 3 Carleson ’ s Theorem : Proof , Complements , Variations

L. Carleson’s celebrated theorem of 1965 [14] asserts the pointwise convergence of the partial Fourier sums of square integrable functions. We give a proof of this fact, in particular the proof of Lacey and Thiele [47], as it can be presented in brief self contained manner, and a number of related results can be seen by variants of the same argument. We survy some of these variants, complements...

Another proof of Banaschewski's surjection theorem

We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...

KNOTS DETERMINED BY THEIR COMPLEMENTS 3 Proof

The surgery theory of Browder, Lashof and Shaneson reduces the study of high-dimensional smooth knots n , ! S n+2 with 1 = Zto homotopy theory. We apply Williams's Poincar e embedding theorem to the unstable normal invariant : S n+2 ? ! (M=@M) of a Seifert surface M n+1 , ! S n+2. Then a knot is determined by its complement if the Z-cover of the complement is (n + 2)=3]-connected; we improve Fa...

Proof of Theorem A

Since the full proof of this theorem 10] is simple yet quite tedious, we connne ourselves to the guidelines of the proof. Observing the angles between the three sample points s i ; s j ; s k forming the minimal angle and the corresponding VD vertex v, we take advantage of the fact that d(v; s i) < d(s i ; s k) (Theorem 4.2(ii)) and, by trigonometric arguments, show that sin sin(120) R M