Coefficient problems for real bounded-spectrum functions
Let \(R\) denote the real line, let \(\mathbb{C}\) denote the complex plane, and let \(B^\infty_R= \{f:R\to R,\;f\in L^\infty(R)\), \(f\) extends to an entire function in \(\mathbb{C}\}\). Let \(W\) be a finite set of points in the closed upper half plane, let \(m\) be a positive integer, and let \(c:W\to C^{m+1}\), where \(c(w)=(c_0(w), c_1(w),\dots, c_m(w))\) and \(c(w)\in R^{m+1}\) if \(w\in R\). Let \(B_{c,W}= \{f\in B^\infty_R,\;f^{(j)} (w)=c_j(w)\), \(0\leq j\leq m\), \(w\in W\}\), and let \(d(W,c)= \inf\{\| f\|_\infty: f\in B_{c,W}\}\). A function \(f\in B_{c,W}\) is called an extremal function if \(\| f\|_\infty=d(W,c)\). The author proves that there exist extremal functions of a special form \(f(x)=d(W,c) \sin\varphi (x)\), where \(\varphi\) is determined by two polynomials that satisfy a number of specific conditions. Further, it is proved that if either \(W\cap R=\emptyset\) or \(\max\{| c_0(w)|: w\in W\cap R\}<d(W,c)\), then there is a unique extremal function.
scientific article; zbMATH DE number 1262944