In this paper, we briefly review the connection between subband coding, wavelet approximation and general compression problems. Wavelet or subband coding is successful in compression applications partly because of the good approximation properties of wavelets. First, we revisit some rate-distortion bounds for wavelet approximation of piecewise smooth functions. We contrast these results with rate-distortion bounds achievable using oracle based methods. We indicate that such bounds are achievable in practice using dynamic programming. Finally, we conclude with an outlook on open questions in the area of compression and representations.