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In this paper we consider the problem of characterizing those situations under which the best uniform linear approximation to an arbitrary continuous function is unique. The problem has been solved by Haar where the set of approximants is a finite dimensional subspace, but in this paper we generalize this by allowing the set of approximants to be any subset of a finite dimensional space. Some previous work has been done on this problem by Rice [2, p. 87 ff.] for a number of partial results.

This paper is concerned with nonlinear optimization problems in normed linear spaces. Necessary and sufficient conditions for optimal points are given and the range of applicability of these conditions is studied. The results are applied to nonlinear approximation problems.

This paper is concerned with Chebyshev approximation by spline functions with free knots. If a zero of a Chebyshev spline function occurs at a knot, the multiplicity of the zero is suitably extended. Theorems on uniqueness on the whole approximation interval and on subintervals are stated in terms of alternation properties.

Birkhoff and de Boor first posed the question of the existence of a convergent bicubic spline interpolation scheme for non-rectangular domains. In this paper that query is answered affirmatively for L-shaped domains. Specifically, it is shown that ∥sf − f∥= O(hr) where sf is the bicubic spline interpolant associated with a smooth function f, h is the maximum mesh spacing, r − 4 for uniform partitions, and r = 3 for nonuniform partitions.

Let R be a region in n -space and Q a linear quadrature formula for R of the form (f)= ∑ r=1 k r f(x r ) . It is known that if Q(ƒ) = ∝ R ƒ whenever ƒ is a polynomial of degree 3 or lower, then k ⩾ n + 1. It is known that the minimum possible value of k depends on the region R , being 2 n for the n -cube and n + 2 for the n -simplex ( n > 1). In 1956 Hammer and Stroud conjectured that k ⩾ n + 2 for every R , when n > 1. In this paper we construct an R , and a Q with the required property, with k = n + 1.

In this paper a generalized inner product 〈 x | y 〉 is defined as a binary function with complex values which satisfies the following: (i) for any nonzero vector y and any complex number ζ there exists a vector x such that 〈 x | y | = ζ , ( ii ) 〈 x 1 + x 2 | y 1 + y 2 〉 = 〈 x 1 | y 1 〉 + 〈 x 2 | y 1 〉 + 〈 x 1 | y 2 〉 + 〈 x 2 | y 2 〉, ( iii ) 〈 y | x 〉 = f [〈 x | y 〉], where f is a continuous function and, (iv) 〈 x | μy 〉 = g [ μ , 〈 x | y 〉], where g is a continuous function. These conditions induce several functional equations which are then solved. By making a linear combination of 〈 x | y 〉 with its complex conjugate a new function ( x | y ) is obtained which is either symmetric, antisymmetric, or Hermitian. The functions 〈 x | y 〉 and ( x | y ) have the same orthogonal vectors.

It is shown in this paper (Theorem 1) that if α and β are real numbers such that α gb is irrational and algebraic, then there exist two (necessarily distinct) distributions S and T on R, both with compact supports, such that δ′ = ΔαS + ΔβT. Here ΔαS means S − Sα and Sα denotes the translate of S by α. It is also shown that δ′ has no such representation if α gb has rational or certain transcendental values, and that S and T can be chosen to have order two and no lower order. If ϑ belongs to any of the spaces D, E, S or their duals then ϑ ∗ S and ϑ ∗ T belong to the same space as ϑ, and ϑ′ = Δ α (ϑ ∗ S) + Δ β (ϑ ∗ T) . The formula δ′ = ΔαS + ΔβT is generalized to Rn by means of the tensor product of distributions, and it follows from this formula that there is no discontinuous translation-invariant linear form on any of the spaces D (Rn), E (Rn), S (Rn) or their duals. The same thing is also proved for E (Tn) and its dual where Tn denotes the n-dimensional torus group.

In this paper, we show the existence of nonnegative interpolation formulas for functions which are solutions of second-order uniformly elliptic equations over bounded domains.