A new Chebyshev family with applications to Abel equations
Let \(g:[a,b]\rightarrow\mathbb{R}\) be a continuous and non-identically vanishing function, and let \(J\) be the open interval given by the connected component of the set \(\{y\in\mathbb{R}:1-yg(t)>0\), \(t\in[ a,b]\}\) which contains the origin. For \(k\in\mathbb{N}\), \(\alpha\in\mathbb{R}\), consider the family of analytic functions on \(J,\) \[ I_{k,\alpha}(y):=\int_{a}^{b}\frac {g^{k}(t)}{(1-yg(t))^{\alpha}}dt,\;k=0,1,2,\dots,n. \] Then, for any \(n\in\mathbb{N}\) and \(\alpha\in\mathbb{R\smallsetminus Z}^{-}\), the set \(\{I_{0,\alpha},\dots,I_{n,\alpha}\}\) is an extended complete Chebyshev (ECT) system on \(J\) [\textit{S. Karlin, W. J. Studden}, Tchebycheff systems: With applications in analysis and statistics. Pure and Applied Mathematics. Vol. 15. New York etc.: Interscience Publishers (1966; Zbl 0153.38902)]. When \(\alpha\in {\mathbb{Z}}^{-}\) the family of functions considered above is an ECT-system on \(J\) if and only if \(n\leq-\alpha\). In particular, if \(\{I_{0,\alpha },I_{1,\alpha},\dots,I_{n,\alpha}\}\) is an ECT-system, any non-trivial function of the form \(\phi_{\alpha}(y):=\sum\limits_{k=0}^{n}a_{k}I_{k,\alpha}(y),\) \(a_{k}\in\mathbb{R}\), has at most \(n\) zeros in \(J\), counting multiplicities. This fact is used to determine upper bounds for the number of isolated \(2\pi\)-periodic solutions which appear when one performs first order analysis in \(\varepsilon\) of the generalized Abel equations: \[ \frac{dx}{dt} = \frac{\cos(t)} {q-1}x^{q}+\varepsilon P_{n}(\cos(t),\sin(t))x^{p}, \] where \(q,p\in \mathbb{N}\smallsetminus\{0,1\}\), \(q\neq p\) and \(P_{n}\) is a polynomial of degree \(n\). One obtains that the maximum number of \(2\pi\)-periodic solutions of the generalized Abel equation, obtained by a first order analysis is \(n\) when \(q\) is even, and \(2n\) when \(q\) is odd. These upper bounds are sharp.
number of zeroes of real analytic functions
scientific article; zbMATH DE number 5988443