Markov interlacing property for exponential polynomials

作者：Milev Lozko^*; Naidenov Nikola

来源：Journal of Mathematical Analysis and Applications, 2010, 367(2): 669-676.

DOI：10.1016/j.jmaa.2010.02.033

• Exponential polynomials
• Muntz polynomials
• Markov interlacing property
• Tchebycheff systems

Let U be an extended Tchebycheff system on the real line. Given a point (x) over bar = (x(1),...,x(n)), where x(1) < ... < x(n), we denote by f((x) over bar; t) the polynomial from U(n), which has zeros x(1),...,x(n). (It is uniquely determined up to multiplication by a constant.) The system U(n) has the Markov interlacing property (M) if the assumption that (x) over bar and (y) over bar interlace implies that the zeros of f'((x) over bar; t) and f'((y) over bar; t) interlace strictly, unless (x) over bar = (y) over bar. We formulate a general condition which ensures the validity of the property (M) for polynomials from U(n). We also prove that the condition is satisfied for some known systems, including exponential polynomials and Sigma(n)(i=0)b(i)e(alpha ix) and Sigma(n)(i=0)b(i)e(-(x-beta i)2). As a corollary we obtain that property (M) holds true for Muntz polynomials Sigma(n)(i=0)b(i)x(gamma i), too.

• 出版日期2010-7-15