Some Problems in the Theory of Simple and Multiple Trigonometric and Orthogonal Series

Abstract

The structure of a hydromagnetic shock front wherein an electrostatic turbulence continuously develops from a two-stream instability, is investigated to aim at orders of magnitude for the width of the front, the heating of electrons and ions, and the magnitude of excitation of fluctuations. Use is made of a stationary macroscopic model that includes two phenomenological parameters. The meaning of these is examined on the basis of a microscopic theory of weak turbulence; an argument is put forward which shows that an appreciable heating of electrons can be consistent with such a regime. Conditions entailing a turbulent shock are given.

Generalizations of the wave equation

“…But B is compact, so finitely many such D cover B . By extending the edges of these D until they reach the edge of B we may produce a decomposition of B into a finite collection of nonoverlapping boxes {Bj} such that each B¡ is contained in a D and consequently in an S satisfying (14). Apply (11) again to see that T satisfies (14).…”

“…We now show that T satisfies (14). Let B be a box contained in T. Each point in B is the center of a box D contained in an S satisfying (14). But B is compact, so finitely many such D cover B .…”

“…Obviously T itself satisfies (12) and (13). We now show that T satisfies (14). Let B be a box contained in T. Each point in B is the center of a box D contained in an S satisfying (14).…”

“…If the dimensions of B are both < j¡, then ASB > 0 by definition of A" ; while if B is too large, simply write it as a finite union of small (dimensions < j¡) nonoverlapping boxes and combine the definition of An with property (11) to conclude that <S satisfies (14). Now let T be the union of all sets S satisfying conditions (12), (13), and (14). Obviously T itself satisfies (12) and (13).…”

On a conjecture of F. Móricz

“…Meanwhile we can easily demonstrate that the convergence of S µν in L 2 (Q) does not depend on the manner in which µ and ν tend to infinity. For a discussion of different kinds of multiple limits we refer the reader to [4].…”