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.

The theory of reproducing kernel Hilbert spaces is applied to a minimization problem with prescribed nodes. We re-prove and generalize some results previously obtained by Gunawan et al. [2,3], and also discuss the Hölder continuity of the solution to the problem.

Let $f\in \,{{L}_{2\pi }}$ be a real-valued even function with its Fourier series $\frac{{{a}_{0}}}{2}\,+\,\sum _{n=1}^{\infty }\,{{a}_{n}}\,\cos \,nx$ , and let ${{S}_{n}}\left( f,x \right)$ , $n\,\,\ge \,\,1$ , be the $n$ -th partial sum of the Fourier series. It is well known that if the nonnegative sequence $\{{{a}_{n}}\}$ is decreasing and ${{\lim }_{n\to \infty }}\,{{a}_{n}}\,=\,0$ , then

$$\underset{n\to \infty }{\mathop{\lim }}\,{{\left\| f-{{S}_{n}}\left( f \right) \right\|}_{L}}=0\text{ifanyonlyif}\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\log n=0.$$

We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the ${{L}^{1}}$ -convergence of a function $f\in {{L}_{2\pi }}$ in complex space. We also give results on ${{L}^{1}}$ -approximation of a function $f\in {{L}_{2\pi }}$ under the MVBV condition.

It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szegö polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szegö polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients etc. are derived.

Bivariate polynomials with a fixed leading term ${{x}^{m}}{{y}^{n}}$ , which deviate least from zero in the uniform or ${{L}^{2}}$ -norm on the unit disk $D$ (resp. a triangle) are given explicitly. A similar problem in ${{L}^{p}}$ , $1\,\le \,p\,\le \,\infty $ is studied on $D$ in the set of products of linear polynomials.

We consider some variations of Muntz's classical theorem on when Span{xλi } is dense in C[0,1]. We prove, for example, that if then the collection of spaces is dense in C[0,1] if and only if lim sup(log n)/λn = ∞. Another variation concerns the denseness of the union of spaces of the form The derivations of these results require an examination of the location of the zeros of the associated Chebyshev polynomials.

Suppose X and Y are closed subspaces of (ΣXn)p and (ΣYn)q (1 < p ≦ q < ∞, dim Xn < ∞, dimYn < ∞), respectively. If K(X, Y), the space of the compact linear operators from X to Y, is dense in L(X, Y), the space of the bounded linear operators from X to Y, in the strong operator topology, then K(X, Y) is an M-ideal in L(X, Y).

A theorem concerning M-summands in dual spaces is used to prove that certain known M-ideals are not M-summands. In some cases where this information was already known, our procedure greatly simplifies the earlier proofs. Finally, we give a condition to determine which M-ideals in dual spaces are M-summands and which are not.

Let B be a Douglas algebra which admits best approximation. It will be shown that the following are equivalent: (1) The unit ball of (L∞/B) has no extreme points; (2) For any Blaschke product b with , there exists h ∈ B such that = 1 and h|E≢0, where E is the essential set of B.

It will also be proven that if B⊇H∞+C and its essential set E contains a closed Gδ set, then the unit ball of (L∞/B) has no extreme points. Many known results concerning this subject will follow from these results.

It is reasonable to expect that, under suitable conditions, Padé approximants should provide nearly optimal rational approximations to analytic functions in the unit disc. This is shown to be the case for e z in the sense that main diagonal Padé approximants are shown to converge as expeditiously as best uniform approximants. Some more general but less precise related results are discussed.

Best biased and one-sided Chebyshev approximation with respect to a varisolvent approximating function on an interval are considered. The uniform limit of best biased approximations is the (unique) best one-sided approximation if the best one-sided approximation is of maximum degree. Examples are given where the best one-sided approximation is not of maximum degree and failure of uniform convergence and of existence occurs.