Foundation of paralogical nonstandard analysis and its application to
some famous problems of trigonometrical and orthogonal series.PartII.
Jaykov Foukzon Israel Institute of Technology, Haifa, Israel Tel. 03-
517-26-90 Telaviv st.Rambam 7a/2 I. Introduction. L. Carleson s
celebrated theorem of 1965 [1] asserts the pointwise convergence of
the partial Fourier sums of square integrable functions. The Fourier
transform has a formulation on each of the Euclidean groups ,
and .Carleson s original proof worked on . Fefferman s proof
translates very easily to . M at e [2] extended Carleson s proof to .
Each of the statements of the theorem can be stated in terms of a
maximal Fourier multiplier theorem [5]. Inequalities for such
operators can be transferred between these three Euclidean groups,
and was done P. Auscher and M.J. Carro [3]. But L. Carleson s
original proof and another proofs very long and very complicated. We
give a very short and very simple proof of this fact. Our proof uses
PNSA technique only, developed in part I, and does not uses
complicated technical formations unavoidable by the using of purely
standard approach to the present problems. In contradiction to
Carleson s method, which is based on profound properties of
trigonometric series, the proposed approach is quite general and
allows to research a wide class of analogous problems for the general
orthogonal series.
http://www.mathpreprints.com/math/Preprint/Jaykov/20040330/1/