Asymptotics of the roots of Bernstein polynomials used in the construction of modified Daubechies wavelets.
Bernstein polynomials are \[ B_L(x)=\sum_{\ell=0}^{L-1}{4L\choose \ell} x^{\ell}(1-x)^{4L-1}+ \sum_{\ell=L}^{3L-1}\left({3\over 2}-{\ell\over 2L}\right){4L\choose \ell} x^{\ell}(1-x)^{4L-1}, \quad L=1,2,\dots\,. \] The paper studies the distribution of the zeros of these polynomials when \(L\) is large. The zeros play a role in the construction of modified compactly supported wavelets. It is proved that the limiting curve for the zeros is the boundary of the domain of convergence of the Bernstein polynomials in the complex plane.
scientific article; zbMATH DE number 1928164