Let X be a compact metric space, C(X) be the space of continuous real-valued functions on X and $A_{1},A_{2}$ be two closed subalgebras of C(X) containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_{1}+A_{2}$. We prove a Chebyshev-type alternation theorem for a function $u_{0} \in A_{1}+A_{2}$ to be a best approximation to f.

We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set:

$$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$

$\mathcal {C}[[\mathcal {V}]]$

$L^2([-1,1])$

We find the potential function whose gradient best approximates an observed square integrable function on a bounded open set subject to prescribed weight factors. With an appropriate choice of topology, we show that the gradient operator is a bounded linear operator and that the desired potential function is obtained by solving a second-order, self-adjoint, linear, elliptic partial differential equation. The main result makes a precise analogy with a standard procedure for the best approximate solution of a system of linear algebraic equations. The use of bounded operators means that the definitive equation is expressed in terms of well-defined functions and that the error in a numerical solution can be calculated by direct substitution into this equation. The proposed method is illustrated with a hypothetical example.

The notion of adequate (resp. strongly adequate) function has been recently introduced tocharacterize the essentially strictly convex (resp. essentially firmly subdifferentiable)functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In thispaper we provide various necessary and sufficient conditions in order that the lowersemicontinuous hull of an extended real-valued function on a reflexive Banach space isessentially strictly convex. Some new results on nearest (farthest) points are derivedfrom this approach.

We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold $\text{Sym(}p\text{,}\,\mathbb{R}{{\text{)}}^{++}}\,\times \,\text{Sym(}q\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ block diagonally embedded in $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\,\le \,2$ or $q\,\le \,2$.

The convergence behavior of best uniform rational approximations with numerator degree m and denominator degree n to the function |x|α, α > 0, on [-1, 1] is investigated. It is assumed that the indices (m, n) progress along a ray sequence in the lower triangle of the Walsh table, i.e. the sequence of indices {(m, n)} satisfies

In addition to the convergence behavior, the asymptotic distribution of poles and zeros of the approximants and the distribution of the extreme points of the error function on [-1, 1] will be studied. The results will be compared with those for paradiagonal sequences (m = n + 2[α/2]) and for sequences of best polynomial approximants.