Exact value of widths of a certain class of solutions of linear differential equations (Q1103136)

The distance of a subset A of a normed vector space R from a subspace V, i.e. the expression \(E(A,x):=\sup_{x\in A}\inf_{y\in V}\| x- y\|\) is called the width of A with respect to V. This concept, introduced by A. N. Kolmogorov in 1963, is of increasing interest. - In the above paper the author uses the vector space \(C_{2\pi}\) of all real \(2\pi\)-periodic functions. The subset \(A=K(P)\) is defined by all functions of \(C_{2\pi}\) satisfying a certain differential inequality: Let \(P=P_ r(z)\) be a monic polynomial of degree \(r\in {\mathbb{N}}\) with real coefficients. Then K(P) consists of al \(2\pi\)-periodic functions x(t) which are absolutely continuous with their derivatives up to the order r-1 and satisfy \[ \| P(\frac{d}{dt})x\|_{\infty}= \sup_{t}| P(\frac{d}{dt})x(t)| \leq 1. \] The subspace is the common space \(T_{2n-1}\) of all real trigonometric sums of order at most n-1. The author gives under a suitable restriction of n for the corresponding width \[ \epsilon_ n:=\sup_{x\in K(P)}\inf_{y\in T_{2n-1}}\| x-y\| \] in the uniform norm an exact expression which is based on a special periodic function \(B_ r\), defined by a Fourier series in which the values of P at the points \(i\nu\), \(\nu\in {\mathbb{Z}}\) occur.