Widom factors for generalized Jacobi measures

• We study optimal lower and upper bounds for Widom factors W-infinity,W-n(K, w) associated with Chebyshev polynomials for the weights w(x) = N/1 + x and w(x) = N/1 - x on compact subsets of [-1,1]. We show which sets saturate these bounds. We consider Widom factors W-2,W-n(mu) for L-2(mu) extremal polynomials for measures of the form d mu(x) = (1 - x)(alpha)(1 + x)(beta)d mu(K)(x) where alpha + beta >= 1, alpha, beta is an element of N boolean OR {0} and mu K is the equilibrium measure of a compact regular set K in [-1, 1] with +/- 1 is an element of K. We show that for such measures the improved lower bound (which was first studied in [4]) [W2,n(mu)](2) >= 2S(mu) holds. For the special cases d mu(x) = (1 - x(2))d mu K(x), d mu(x) = (1 - x)d mu K(x), d mu(x) = (1 + x)d mu K(x) we determine which sets saturate this lower bound and discuss how saturated lower bounds for [W2,n(mu)](2) and W-infinity,W-n(K,w) are related.

Ämnesord

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Nyckelord

Widom factors

Chebyshev polynomials

Orthogonal polynomials

Jacobi polynomials

Extremal polynomials

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