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We study the stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients. Time is discretized by explicit multi-level or Runge-Kutta methods of order < 3 (forward Euler time-differencing is included), and we study spatial discretizations by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. We prove that these fully explicit spectral approximations are stable provided their time step, At,doi:10.1090/s0025-5718-1991-1066833-9 fatcat:tiku24lu55f7pcwgtbk7foc6ze