Some theorems on Fourier coefficients

Some Stability Theorems for Nonharmonic Fourier Series

The theory of nonharmonic Fourier series in L2(-ir,tr) is concerned with the completeness and expansion properties of sets of complex exponentials {e'x"'}. It is well known, for example, that the completeness of the set {e'x"'} ensures that of {e'^"'} whenever 2 lA„ ~~ M»l < oo. In this note we establish two results which guarantees that if {elX"'} is a Schauder basis for l}(—n, it), then [e'^"...

Some Theorems on Coverings

Some Theorems on Continua

PROOF. Let {Gn} and {Hn} denote monotonie decreasing sequences of open sets which close down upon .Pi and F2 respectively. We may suppose that d and Hi are so chosen that G i # i = 0. Now, by Mullikin's theorem, there is a connected subset of C— C(Gw+27n) which has a limit point in Gn and a limit point in Hn. The closure of such a connected set is a subcontinuum of C which "extends" from Gn to ...

Some Theorems on Subseries

1. Absolutely convergent series. A simple calculation reveals that the arithmetic mean value of all subsums (including the void sum) of a given finite sum sn=ui+U2 + • • • +un is equal to sn/2. In this section we shall show (see Theorem 1 below) that an integral mean value can be found, consistent with the preceding, for the sums of all infinite subseries of a given absolutely convergent series...

Texture simulation based on tensorial Fourier coefficients

The main result of the paper is the derivation of the evolution equation of the tensorial texture coefficients of the crystallite orientation distribution function (codf). The evolution equation of each coefficient depends on the complete codf and the lattice spin, which is a constitutive quantity. For the solution of the differential equation based on a finite number of coefficients, the codf ...