Using a traditional orthonormal wavelet basis which is obtained by the integer translates and dilates of a mother wavelet, in section 2 the authors propose a general way of constructing a wavelet-like (with a similar multiresolution analysis structure) orthonormal basis in a separable Hilbert space. Applying this general method to the Chebyshev polynomials and Meyer wavelets, the authors obtain in section 3 an interesting polynomial wavelet-like orthonormal basis on the interval. The applications of these polynomial wavelet-like bases to the numerical resolution of degenerate elliptic operators are also addressed in section 4.