Let S N 2 S_N^2 denote the nonlinear manifold of second order splines defined on [0, 1] having at most N N interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function f f by elements of S N 2 S_N^2 . Approximation relative to the L 2 [ 0 , 1 ] {L_2}[0,1] norm is treated first, with the results then extended to the best L 1 {L_1} and best one-sided L 1 {L_1} approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function f f satisfying f > 0 f > 0 has a unique best approximant from S N 2 S_N^2 provided either log ⁡ f \log f is concave, or N N is sufficiently large, N ⩾ N 0 ( f ) N \geqslant {N_0}(f) ; for any N N , there is a smooth function f f , with f > 0 f > 0 , having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.

Abstract.Let SN" be the set of &th-order splines on [0,1] having at most N -1 interior knots, counting multiplicities. We prove the following sharp asymptotic behavior of the error for the best L2

Let C = { ( x ( s ) , t ( s ) ) : a ⩽ s ⩽ b } C = \{ (x(s),t(s)):a \leqslant s \leqslant b\} be a Jordan arc in the x-t plane satisfying ( x ( a ) , t ( a ) ) = ( a , t ∗ ) , ( x ( b ) , t ( b ) ) = ( b , t ∗ ) (x(a),t(a)) = (a,{t_ \ast }),(x(b),t(b)) = (b,{t_\ast }) , and t ( s ) > t ∗ t(s) > {t_\ast } when a > s > b a > s > b . Let a > x ∗ > b a > {x_\ast } > b . We prove the existence of Gauss interpolation formulas for C and the point ( x ∗ , t ∗ ) ({x_\ast },{t_\ast }) , for solutions u of the one-dimensional heat equation, u t = u x x {u_t} = {u_{xx}} . Such formulas approximate u ( x ∗ , t ∗ ) u({x_\ast },{t_\ast }) in terms of a linear combination of its values on C. The formulas are characterized by the requirement that they are exact for as many basis functions (the heat polynomials) as possible.

Abstract. Let fip . -. , Hk be odd positive integers and »i = Z¡=.x(p¡ + 1). Let {«(.}|=j be an extended Tchebycheff system on [a, b]. Let L be a positive linear functional on U = span( {u,}). We prove that L has a unique representation in the

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