It is proved that the weighted Hölder spaces \(\Lambda_{\alpha, p}\) on the interval \([-1, 1]\) with the weight \(\varphi(x) = \sqrt {1-x^2}\) are linearly isomorphic to some sequence spaces. The isomorphism is given by the coefficients of functions with respect to a system of orthonormal splines with knots uniformly distributed according to the measure with density \(1/\varphi\). An application to the characterization of the order of polynomial approximation is given.