Vol. 40, No. 1, 1972

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Summability and Fourier analysis

George Ulrich Brauer

Vol. 40 (1972), No. 1, 33–43

An integration on βN, the Stone-Cech compactification of the natural numbers N, is defined such that if s is a bounded sequence and ϕ is a summation method evaluating s to σ, sdϕ = σ. The Fourier transform ϕ of a summation method ϕ is defined as a linear functional on a space of test functions analytic in the unit disc: if

      ∑∞  ˆ    n                  ∫  ˆ
f(z) =    f(n )Z  ,|z| < 1, then ϕ(f) = f(n)dϕ.

A functional which agrees with the Fourier transform of a regular summation method must annihilate the Hardy space H1. Our space of test functions is often the space M,, of functions f = f(n)zn, analytic in the unit disc, such that

                     ∫ 2π    t′p′i𝜃 p      t∕p
∥f ∥Mp = lim sup[(1− r) 0 |f(r e  )| d𝜃∕2π]

is finite for some p > 1. A functional L which is well defined on a space Mp for some p 2 such that L(1(1 z)) = 1 agrees with the Fourier transform of a summation method which is slightly stronger than convergence.

Mathematical Subject Classification 2000
Primary: 42A68
Secondary: 40H05, 30A78
Received: 6 August 1970
Published: 1 January 1972
George Ulrich Brauer