Some theorems on Fourier series

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'^"...

Notes on Fourier Series

A function or a real variable f is said to be periodic with period P if f(x+ P ) = f(x) holds for all x. Hence, if we know the values of f on an interval of length P , we know its values everywhere. If f is a function defined on an interval [a, b), we can extend f to a function defined for all x which is periodic of period b− a. We simply define f(x) to be f(x+ n(b− a)), where n is the integer ...

On Walsh-fourier Series^)

Every function f(x) which is of period 1 and Lebesgue integrable on [0, 1 ] may be expanded in a Walsh-Fourier series(3), f(x)~ ?.?=n ak\pk(x), where ak=fof(x)ypk(x)dx, k=0, 1, 2, • • • . Fine exhibited some of the basic similarities and differences between the trigonometric orthonormal system and the Walsh system. He identified the Walsh functions with the full set of characters of the dyadic ...

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 ...