ISSN 1786-0091 SOME PROPERTIES FOR FUNCTIONS OF VMO(2ω)

Dedicated to Professor W.R. Wade on his sixtieth birthday Abstract. A function of bounded mean oscillation (BMO) is said to have vanishing mean oscillation or belong to VMO space if its mean oscillation is locally small in a uniform sense. Though there is an extensive literature on the BMO, very few mention is made on the properties for functions of VMO. In this note, we discuss the connection between modulus of continuity and the approximation of functions by Walsh polynomials in VMO space on the dyadic group 2ω, VMO(2ω), the analogy between VMO(2ω) and C(2ω), the estimate for certain type of convolution operators on VMO(2ω), the decom-position theorem for functions in VMO(2ω) and the characterization of Walsh series which happen to be the Walsh-Fourier series of a function in VMO(2ω). 1. Notation Our results are stated in the situation that the dyadic group 2ω is the additive subgroup of the ring of integers in the 2-series field K of formal Laurent series in one variable over the finite field GF(2). We need to set some basic notation. It is taken from Taibleson’s book [9] where the fundamentals are detailed. For the additive subgroup K+ of the 2-series field K, we may choose a Haar measure dx. Let d(αx) = |α|dx, α 6 = 0 and call |α | the valuation of α. Let P 0 = {x ∈ K: |x | ≤ 1} and P 1 = {x ∈ K: |x | < 1}. K is totally disconnected, hence the value is discrete valued. Thus there is an element ℘ of P 1 of maximum value. Then an element x ∈ K is represented as (1) x = k=j ak℘ k, ak ∈ GF(2), which can contain a finite number of terms with negative powers of ℘. The ring of integers P 0 = x = k=0ak