DOUBLING CONDITION AND LINEARITY OF THE SEQUENCE SPACE Λp(ƒ)

For 1 ≤ p < +∞,every ƒ (≠ 0) ∈Lp(R,dx) defines a sequence space Λp(ƒ) (Honda et al. Proc. Japan Acad. Ser. A 84 (2008), 39-41) which is an additive group but not necessarily a linear space. The main purpose of this paper is to discuss the linearity of Λp(ƒ). First we show that if ƒ is a piecewise monotone function, then Λp(ƒ) is a linear space. Next, specializing the case to p = 2, we characterize Λ2(ƒ) as a set, and discuss the linearity of it. With this aim, we extend the definition of the doubling condition and define the doubling dimension H(φ) of a non-negative function on [0, +∞).Let ƒ be the Fourier transform of ƒ and define a function φƒ associated with ƒ. Then we show that H(|ƒ|)< ∞ implies the linearity of Λ2(ƒ). In addition, we show that if H(φƒ)< 2, then Λ2(ƒ) is linear and give several examples.