A complex-variable proof of the Wiener tauberian theorem

The Wiener-Pitt tauberian theorem

A Complex Tauberian Theorem and Mean Ergodie Semigroups

for short) if C lira f ( t ) := lira 1 i t t -oo t--.oo-[ Jo f ( s )ds = foo. If ,-~lim f ( t ) = foo, then C lim f ( t ) = foo; and if C lira f ( t ) = foo, then A lim f ( t ) = f ~ . The converse implicat ions are false, in general. Addi t ional condit ions which allow the inverse implicat ion are called Tauber ian conditions, and the corresponding s ta tements Tauber ian theorems. Here we ar...

The Wiener–ikehara Theorem by Complex Analysis

The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman’s contour integration method can be adapted to establish the Wiener–Ikehara theorem. A simple special case suffices for the PNT. But what about the twin-prime problem?

A Tauberian Theorem for Stretchings

R. C. Buck fl] has shown that if a regular matrix sums every subsequence of a sequence x, then x is convergent. I. J. Maddox [4] improved Buck's theorem by showing that if a non-Schur matrix sums every subsequence of a sequence x, then x is convergent. Actually Maddox proved a stronger result: If x is divergent and A sums every subsequence of x, then A is a Schur matrix, i.e., It should be rema...

Wiener Tauberian Theorems for Vector - Valued Functions

Different versions of Wiener’s Tauberian theorem are discussed for the generalized group algebra LI(G,A) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A) using A-valued Fourier transforms. A weak form of Wiener’s Tauberian property is introduced and it is proved that LI(G,A) is weakly Tauberian if and only if A is....