General Mathematics

1311 Submissions

[2] viXra:1311.0160 [pdf] submitted on 2013-11-23 14:01:45

Foundation of Paralogical Nonstandard Analysis and Its Application to Somefamous Problems of Trigonometrical and Orthogonal Series. Part I.

Authors: Jakov Foukzon
Comments: 22 Pages.

This article about foundation of paralogical nonstandard analysis and its applications to the continuous function without a derivative presented by absolutely convergent rigonometrical series and another famous problems of trigonometrical and orthogonal series.
Category: General Mathematics

[1] viXra:1311.0159 [pdf] submitted on 2013-11-23 14:07:45

Foundation of Paralogical Nonstandard Analysis and Its Application to Somefamous Problems of Trigonometrical and Orthogonal Series. Part II.

Authors: Jaykov Foukzon
Comments: 14 Pages.

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 R , Z andΤ .Carleson’s original proof worked on Τ.Fefferman’s proof translates very easily to R. M´at´e [2] extended Carleson’s proof to Z.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.
Category: General Mathematics