Search Results

The rate of polynomial approximation $$E_{n}(f,\mathcal{K}):=\inf\{\max_{z\in \mathcal{K}}|f(z)-P_{n}(z)|,~\text{deg}~P_{n}\leq n\}$$ to an entire function $f$ on a compact set $\mathcal{K}$ of the c …

First, let us ignore the nonnegativity constraint on the approximating polynomial. By linear transformation of the variable, one may suppose that $I_{0}=[-A,-1]$ and $I_{1}=[1,B]$ and by a linear tr …

T. Ganelius has shown inequalities of the type $$ Z_k(E,F)\leq const. e^{-k/C(E,F)},\qquad k=1,2,\ldots,\qquad (*) $$ under additional assumptions on $E$ and $F$, see [1] T. Ganelius, Some extremal f …

Bounds in the univariate case, see e.g. here, were established by V.A. Markov in 1892. S.N. Bernstein has given an extension of the result to the multivariate case in S.N. Bernstein, On certai …

Mean square approximation of a continuous function by rational functions on an interval is studied in the monograph Walsh, J. L. Interpolation and approximation by rational functions in the comp …

Denote the minimal approximation error to the function $f(x)=|x|$ in the uniform norm on $[-1,1]$ by $$ E_{mn}(f,[-1,1])=\inf_{r\in\mathcal{R}_{mn}}\|f-r\|_{\infty,[-1,1]}, $$ where $\mathcal{R}_{mn} …

If the main interest lies in numerical methods designed to compute specifically with functions defined on a disk, the following recent manuscript may be useful: https://arxiv.org/pdf/1604.03061.pdf …

Let $T_{n}(x)=\sum_{\nu=0}^{n}t_{n,\nu}x^{\nu}$ denote the Chebyshev polynomial (of the first kind) of degree $n$ and let $x_{n,\nu}=\cos\nu\pi/n$ for $0\leq\nu\leq n$. The answer follows from the f …

Marcinkiewicz and Grûnwald (1936) have independently shown that there exists a function continuous on $[-1,1]$ such that its Lagrange interpolants on the Chebyshev nodes diverge at all points of $[-1, …

Uniform approximation by polynomials on a finite set of points is studied in Section 1.3 of T.J. Rivlin, An introduction to the approximation of functions, Dover, 2003. An explicit solution to …

A standart assumption for the density of $C^{\infty}(\overline\Omega)$ in $W^{k,p}(\Omega)$ is that the domain $\Omega$ satisfies the segment condition, namely, for every point $x\in\partial\Omega$, t …

1) No, in general, convergence does not hold because of the presence of spurious poles. A typical result, see [1], is Theorem. Let $(n_{\nu})_{1}^{\infty}$ be a sequence of positive integers satisfyi …

This is the main result in a paper by J.L. Walsh from 1927 (in german): J.L. Walsh, Über die Entwicklung einer Funktion einer komplexen Veränderlichen nach Polynomen. Math. Ann. 96 (1927), no. 1, 437– …

Yes, indeed there is. Explicit expressions for multipoint Padé approximants to the exponential (and power) function at points $z=0,\ldots,m+n$, were given in A. Zhedanov, Explicit multipoint rati …

When $[a,b]=[-1,1]$, the $\inf$ on the right-hand side is attained (uniquely) by the monic Chebyshev polynomial $T_{n+1}$. It is well known that its roots belong to $[-1,1]$ and are simple. For a ge …