The uniqueness of the element of best mean approximation to a continuous function using splines with fixed nodes

Abstract

Suppose that on the Interval [a, b] the nodes

$$a = x_0< x_1< \ldots< x_m< x_{m + 1} = b$$

are given and the functions u₀(t)=ω₀(t)

$$u_i (t) = \omega _0 (t)\smallint _0^t \omega _1 (\varepsilon _1 )d\varepsilon _1 \ldots \smallint _a^{\varepsilon _{\iota - 1} } \omega _1 (\varepsilon _1 )d\varepsilon _\iota ,\varepsilon _0 = t(i = 1,2, \ldots ,n)$$

where the functions ω_i(t)> 0 have continuous (n−i)-th derivatives (i=0, 1, ..., n). S_n,m will designate the subspace of functions that have continuous (n−1)-st derivatives on [a, b] and coincide on each of the intervals [x_j, x_j+1] (j=0, 1, ..., m) with some polynomial from the system {u_i(t)} ⁿ_i=0 .THEOREM. For every continuous function on [a, b] there exists in S_n,m a unique element of best mean approximation.