On Biorthogonalization of a Dirichlet System Over a Finite Interval

• Main Authors: Mher Martirosyan, Davit Martirosyan
• Format: Article
• Language: English
• Published: Republic of Armenia National Academy of Sciences 2019-04-01
• Series: Armenian Journal of Mathematics
• Subjects: Dirichlet Polynomials; Biorthogonal Systems; Blaschke Product; Gram Matrix; Bernstein-Type Inequality
• Online Access: http://armjmath.sci.am/index.php/ajm/article/view/268

Ultimately aiming to estimate Dirichlet polynomials, a representation problem for special biorthogonal systems of exponentials is explored in $L^2(0,a)$. If $a=+\infty$, a method of construction of such systems through suitable Blaschke products is known, but the method ceases to operate when $a$ is finite. It turns out that the Blaschke product cannot be even adjusted to maintain the old method for the new situation. The biorthogonal system is then represented by a single determinant of a modified Gram matrix of the original system. Bernstein-type inequalities for Dirichlet polynomials and their higher order derivatives are established. The best constants and extremal polynomials are obtained in terms of the Gram matrix.