We present constructive a priori error estimates for H0 2-projection into a space of polynomials on a one-dimensional interval. Here, "constructive" indicates that we can obtain the error bounds in which all constants are explicitly given or are represented in a numerically computable form. Using the properties of Legendre polynomials, we consider a method by which to determine these constants to be as small as possible. Using the proposed technique, the optimal constant could be enclosed in a very narrow interval with result verification. Furthermore, constructive error estimates for finite element H0 2-projection in one dimension are presented. These types of estimates will play an important role in the numerical verification of solutions for nonlinear fourth-order elliptic problems as well as in the guaranteed a posteriori error analysis for the finite element method or the spectral method (e.g. Hashimoto et al. (2006) , Nakao et al. (2008) , Watanabe et al. (2009) ).
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