For 1 ≤ p < +∞,every ƒ (≠ 0) ∈
Lp(
R,
dx) defines a sequence space Λ
p(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 Λ
p(ƒ). First we show that if ƒ is a piecewise monotone function, then Λ
p(ƒ) is a linear space. Next, specializing the case to
p = 2, we characterize Λ
2(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 φ
ƒ associated with ƒ. Then we show that
H(|ƒ|)< ∞ implies the linearity of Λ
2(ƒ). In addition, we show that if
H(φ
ƒ)< 2, then Λ
2(ƒ) is linear and give several examples.
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