Approximation Theory and Methods

Let A be an (n +1)-dimensional linear space in C [a, b]. In Section 7.3 A is defined to satisfy the Haar condition if the following property is obtained.

Condition (1). If ϕ is any element of A that is not identically zero, then the number of roots of the equation {ϕ(x) = 0; a ≤ x ≤ b} is less than (n +1).

The purpose of this appendix is to prove that the following three conditions are implied by Condition (1), and also that Condition (3) and Condition (4) are each equivalent to Condition (1).

Condition (2). If k is any integer in [1, n], and if {ζi;j = 1, 2, …, k) is any set of distinct points from the open interval (a, b), then there exists an element of A that changes sign at these points, and that has no other zeros. Moreover, there is a function in A that has no zeros in [a, b].

Condition (3). If ϕ is any element of A that is not identically zero, if the number of roots of the equation {ϕ(x) = 0; a ≤ x ≤ b} is equal to j, and if k of these roots are interior points of [a, b] at which ϕ does not change sign, then (j + k) is less than (n + 1).

Condition (4). If{ϕi i = 0, 1, … n) is any basis of A, and if {ξi j = 0, 1, …, n} is any set of (n +1) distinct points in [a, b], then the (n +1) × (n +1) matrix whose elements have the values {ϕi(ξi) = 0, 1,…, n; j = 0, 1,…, n} is non-singular.