Markov interlacing property for exponential polynomials

Let U n be an extended Tchebycheff system on the real line. Given a point x ¯ = ( x 1 , … , x n ) , where x 1 < ⋯ < x n , we denote by f ( x ¯ ; 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 ¯ and y ¯ interlace implies that the zeros of f ′ ( x ¯ ; t ) and f ′ ( y ¯ ; t ) interlace strictly, unless x ¯ = y ¯ . 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 ∑ i = 0 n b i e α i x and ∑ i = 0 n b i e − ( x − β i ) 2 . As a corollary we obtain that property (M) holds true for Müntz polynomials ∑ i = 0 n b i x γ i , too.