Vol. 11 No. 4 (2019): On Biorthogonalization of a Dirichlet System Over a Finite Interval
Published:
2019-04-19
Articles
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Articles
On Biorthogonalization of a Dirichlet System Over a Finite Interval
AbstractUltimately 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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