Here we present and analyze an optical spectrum analyzer at the nanometer scale that is able to distribute different frequency contents of the radiation of an optical dipole source into different directions in the space. The spectrum analyzer is composed of arrays of optical Yagi-Uda nanoantennas, forming relatively narrow radiation patterns operating at different frequencies. The optical Yagi-Uda nanoantennas composed of plasmonic core-shell nanoparticles are used as an example of building blocks for this idea in our study. Narrow radiation beams in such antenna arrays are realized by tailoring the scattering phase of the nanoparticles. The sensitivity of such an antenna array to the operating wavelength and the angular distribution of the radiation pattern, which is essential for the operation of the spectrum analyzer proposed here, is studied theoretically. The chromatic dispersion and the angular variation of the radiation pattern of such an optical spectrum analyzer are discussed in detail.