Lothar Berg, Manfred Krüppel
Linear combinations of shifted eigenfunctions of two-scale difference equations
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock
39Axx Difference equations,
Abstract: In this paper, we continue our considerations in \cite{bk1} on eigenfunctions of two-scale difference equations. Using the results in \cite{bk1} concerning equivalent eigenfunctions and equivalent characteristic
polynomials, we derive sum relations for shifts of an
eigenfunction in the interval $(-1,1)$. In particular, we deal with the case that the characteristic polynomial contains a cyclic factor. We give necessary and sufficient conditions for the linear independence of shifts of an eigenfunction, and we determine a basis for the coefficient vector in the case of linear dependence. Here the representation of the characteristic polynomial by means
of the corresponding minimal polynomial is basically used. Our main emphasis is laid on linear combinations of such shifts yielding polynomials, where both the possible coefficients and the possible polynomials are characterized. The results are specialized to the constant polynomial equal to 1, i.e. to partitions of unity. But also linear combinations of shifts of an eigenfunction, yielding certain distributions,are investigated.
Keywords: Two-scale difference equations, distributional solutions, eigenfunctions, Appell polynomials, sums of shifted eigenfunctions, cyclic zeros and cyclic polynomials, linear independence, partitions of unity.