Theorem of Budan-Fourier and Birkhoff interpolation
G. G. Lorentz · Birkhäuser Basel eBooks · 1983
In this note we would like to point out the close relation which exists between the theorem of Budan-Fourier and lacunary or Birkhoff interpolation — a subject to which P. Turán himself made important contributions. In this way we obtain a generalization of this theorem, which in its simplest form asserts that the number Z0 of zeros of a polynomial P of degree n in (a, b) satisfies 1 $$Z_0 \leqq S^ - \{ P^{(k)} (a)\} - S^ + \leqq P^{(k)} (b)\} .$$