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We construct an explicit even jump distribution      d α  ( x ) d\alpha (x)       on      [ −  1 , 1 ] [-1,1]       for which      γ  n ( d α  ) {\gamma _n}(d\alpha )      , the leading coefficient of the      n n      th orthonormal polynomial for      d α  ( x ) d\alpha (x)      , has any given asymptotic behavior. For example, if      { ξ  n } 1 ∞  \{ {\xi _n}\} _1^\infty       is a strictly increasing sequence with limit      ∞  \infty      , and      ξ  3 n + 1 / ξ  3 n →  1 {\xi _{{3^{n + 1}}}}/{\xi _{{3^n}}} \to 1       as      n →  ∞  n \to \infty      , we can ensure that      lim n →  ∞  γ  n ( d α  ) 2 −  n / ξ  n = 1 {\lim _{n \to \infty }}{\gamma _n}(d\alpha ){2^{ - n}}/{\xi _n} = 1      . As a consequence, we have      lim n →  ∞  γ  n −  1 ( d α  ) / γ  n ( d α  ) = 1 / 2 {\lim _{n \to \infty }}{\gamma _{n - 1}}(d\alpha )/{\gamma _n}(d\alpha ) = 1/2      ; that is,      d α  d\alpha       belongs to Nevai’s class      M \mathcal {M}      . This positively and explicitly answers a question of Al. Magnus.

Jump distributions on [-1,1] whose orthogonal polynomials have leading coefficients with given asymptotic behavior