Cosine modulated filter banks are filter banks whose impulse responses are obtained by modulating a window with cosines. Among their applications are video and audio compression and multitone modulation. Their continuous time counterpart is known as local cosine bases. While there is an extended literature on the discrete time case, both for single and multiple overlapping, the continuous time case has received less attention and only the single overlapping case has been solved. This work gives a solution to the problem of continuous time local cosine bases with multiple overlapping via a general theory that emphasizes the deep connection between discrete and continuous time. A sampling theorem for local cosine basis and an efficient algorithm to compute the expansion of a signal are also given.