leja

Leja points, or Leja sequences, refer to a method for choosing interpolation nodes in a compact subset of the complex plane. They are used in polynomial interpolation, numerical quadrature, and related approximation tasks. The construction is greedy: select the first point x1 in the target set; for n ≥ 2, choose xn in the set to maximize the product ∏_{k=1}^{n-1} |xn − xk|. In practice, this means each new point tends to be as far as possible, in a multiplicative sense, from the previously chosen points. The resulting sequence provides a stable foundation for interpolating polynomials of increasing degree and often yields favorable conditioning of the associated Vandermonde matrices.

Leja sequences have variants on specific domains, such as the unit disk or the unit circle, and they can be extended to higher dimensions by forming product sets. They are studied in approximation theory, numerical analysis, and potential theory, where their asymptotic distribution relates to equilibrium measures on the domain. They are related to, but distinct from, Fekete points, which maximize a global Vandermonde determinant over all choices of n points rather than via a greedy rule.

Applications include constructing interpolation grids for spectral methods, developing quadrature rules, and analyzing polynomial approximation behavior on complex domains. The topic is discussed in mathematical literature and computational guides on numerical interpolation.