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For 1 ≤ p < +∞, every f(≠ 0) ∈ Lp(R, dx) defines a sequence space Λ_p(f) (Honda et al. Proc. Japan Acad. Ser. A 84 (2008), 39–41) which is an additive group but not necessarily a linear space. The mai...n purpose of this paper is to discuss the linearity of Λ_p(f). First we show that if f is a piecewise monotone function, then Λ_p(f) is a linear space. Next, specializing the case to p = 2, we characterize Λ_2(f) 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 ^^^f be the Fourier transform of f and define a function φ_f associated with ˆ^^f. Then we show that H(^^^f|) < ∞ implies the linearity of Λ_2(f). In addition, we show that if H(φ_f) < 2, then Λ_2(f) is linear and give several examples.続きを見る
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