Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${\left({\lambda }_j\right)}_{j=1}^{\infty }$ is a sequence of distinct positive numbers. Then $span1,x^{\lambda ₁},x^{\lambda ₂},...$ is dense in C[0,1] if and only if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)=\infty$. Moreover, if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)<\infty$, then every function from the C[0,1] closure of $span1,x^{\lambda ₁},x^{\lambda ₂},...$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an earlier result...

We present a solution to the (strict) Bernstein-Nachbin approximation problem in the general complex case. As a corollary, we get proofs of the analytic, the quasi-analytic, and the bounded criteria for localizability in the general complex case. This generalizes the known results of the real or self-adjoint complex cases, in the same way that Bishop’s Theorem generalizes the Weierstrass-Stone Theorem. However, even in the real or self-adjoint complex cases, the results that we obtain are stronger...

The aim of this paper is to obtain a generalization of W. A. Woyczyński and B. Ram results concerning integrability of power series in terms of their coefficients for the class GM of general monotonic sequences.

We prove an a priori error estimate for the hp-version of the boundary element method with hypersingular operators on piecewise plane open or closed surfaces. The underlying meshes are supposed to be quasi-uniform. The solutions of problems on polyhedral or piecewise plane open surfaces exhibit typical singularities which limit the convergence rate of the boundary element method. On closed surfaces, and for sufficiently smooth given data, the solution is H1-regular whereas, on open surfaces, edge...

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_j$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n,\nu }$ be the space of piecewise linear continuous functions on the torus with knots $t_j:0\le j\le N-1$. Finally, let $P_{n,\nu }$ be the orthogonal projection operator from L²([0,1)) onto $V_{n,\nu }$. The main result is $li{m}_{n\to \infty ,\nu =1}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=su{p}_{n\in \mathbb{N},0\le \nu \le n}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=2+(33-18\surd 3)/13$. This shows in particular that the Lebesgue constant of the classical Franklin...

To each set of knots $t_i=i/2n$ for i = 0,...,2ν and $t_i=(i-\nu )/n$ for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space ${}_{\nu ,n}$ of all piecewise linear and continuous functions on I = [0,1] with knots $t_i$ and the orthogonal projection $P_{\nu ,n}$ of L²(I) onto ${}_{\nu ,n}$. The main result is $li{m}_{(n-\nu )\wedge \nu \to \infty }\left|\right|P_{\nu ,n}\left|\right|₁=su{p}_{\nu ,n:1\le \nu \le n}\left|\right|P_{\nu ,n}\left|\right|₁=2+(2-\surd 3)²$. This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².