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On Biorthogonalization of a Dirichlet System Over a Finite Interval
DOI:
https://doi.org/10.52737/18291163-2019.11.4-1-9Keywords:
Dirichlet Polynomials, Biorthogonal Systems, Blaschke Product, Gram Matrix, Bernstein-Type InequalityAbstract
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.
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2019-04-17
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On Biorthogonalization of a Dirichlet System Over a Finite Interval. (2019). Armenian Journal of Mathematics, 11(4), 1-9. https://armjmath.sci.am/index.php/ajm/article/view/268