Dependence of α in Peak Norms and Best Peak Norms Approximation Original Research Article

Let C[0, 1] be the space of all continuous functions defined on [0, 1] and U be an n dimensional subspace of C[0, 1]. A peak norm, or α-norm for 0<α⩽1, α-norm is defined by ‖f‖α=1α sup{∫A |f| dμ ∣ μ(A)=α, A⊂[0, 1]}, where μ denotes the Lebesgue measure. We say p∈U is a best α-norm approximant to f from U if Dα(f)=‖f−p‖α=inf{‖f−u‖α ∣ u∈U}. In this paper we shall study ‖f‖α, Dα(f) and Pα(f)={p∈U ∣ ‖f−p‖α=Dα(f)} as functions of α for fixed f. We shall show their continuous dependence on α and differentiability with respect to α.