📝 SummarXiv

Abstract

We analyze decay of Chebyshev coefficients and local Chebyshev approximations for functions of finite regularity on finite intervals, focusing on the framework where the interval length tends to zero while the number of approximation nodes remains fixed. For all four families of Chebyshev polynomials, we derive error estimates that quantify the dependence on the interval length for (i) the decay of Chebyshev coefficients, (ii) the approximation error between continuous and discrete Chebyshev coefficients, and (iii) the convergence of Chebyshev-based quadrature rules. These results fill a gap in the existing theory and provide a unified and rigorous description of how approximation accuracy scales on shrinking intervals, offering new theoretical insight and practical guidance for high-order numerical methods on decomposed domains. Numerical experiments corroborate the theoretical results, confirming the decay rates and illustrating the error behavior in practice.