A Note on Completeness of Real-Valued Functions {φnp: p=1, 2, …}

Abstract

imply that f(t)=0, a.e. on [α,β] (cf. [1]). Here μ denotes the Lebesgue measure on R. Throughout the remainder {np:p=1,2,...} will denote a sequence of positive numbers with limp→ ∞np=+∞ and φ will denote a real-valued function on R such that φ(αφ)≧0 and φ is strictly increasing on some interval [αφ,αφ+δ φ], where αφ is a real number and δφ is a positive number. In [3], the first author has showen that if φ is an absolutely continuous function on [αφ,αφ+δ φ] with φ'(t)≠0, a.e. on [αφ,αφ+δ φ], and if Σ ∞p=11/np=+∞, then {φnp:p=1,2,...} is complete on [αφ,αφ+δ φ] (see [3, Theorem 1 part (i)]). The following theorem shows that the above result holds under a strictly weaker condition on φ.