On Biorthogonalization of a Dirichlet SystemOver a Finite Interval

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Submitted: 2024-12-10; Published: 2024-12-10

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Abstract

Ultimately aiming to estimate Dirichlet polynomi-als, a representation problem for special biorthogonal systems ofexponentials is explored inL2(0,a). Ifa= +∞, a method ofconstruction of such systems through suitable Blaschke productsis known, but the method ceases to operate whenais finite.It turns out that the Blaschke product cannot be even ad-justed to maintain the old method for the new situation. Thebiorthogonal system is then represented by a single determinantof a modified Gram matrix of the original system. Bernstein-type inequalities for Dirichlet polynomials and their higher or-der derivatives are established. The best constants and extremalpolynomials are obtained in terms of the Gram matrix.