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 $2^{\omega}$, VMO($2^{\omega}$), the analogy between VMO($2^{\omega}$) and C($2^{\omega}$), the estimate for certain type of convolution operators on VMO($2^{\omega}$), the decomposition theorem for functions in VMO($2^{\omega}$) and the characterization of Walsh series which happen to be the Walsh-Fourier series of a function in VMO($2^{\omega}$).
Keywords: Dyadic group, BMO space, VMO space.
Classification (MSC2000): 42C10
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