On the norms and roots of orthogonal polynomials in the plane and L p -optimal polynomials with respect to varying weights

AbstractFor a measure on a subset of the complex plane we consider L p -optimal weighted polynomials, namely,monic polynomials of degree nwith a varying weight of the form w n = e nV which minimize the L p -norms, 1 p1. It is shown that eventually all but a uniformly bounded number of the roots of theL p -optimal polynomials lie within a small neighborhood of the support of a certain equilibrium measure;asymptotics for the nth roots of the L p norms are also provided. The case p= 1is well known andcorresponds to weighted Chebyshev polynomials; the case p= 2 corresponding to orthogonal polynomialsas well as any other 1 p<1is our contribution. 1 Introduction, background and results In approximation theory an important role is played by the so-called Chebyshev polynomials associated toa compact set KC, namely monic polynomials of degree nthat minimize the supremum norm over K.As a natural generalization, one can consider weighted Chebyshev polynomials with respect to a varyingweight of the form w