Local Sine and Cosine Bases

19 and = 1 + 1 m ; m a positive integer. If we let w(x) 1 p 2 R 1 ?1 e ixx ()dd, then w is a \mother function" that generates a wavelet basis (giving us a Multi Resolution Analysis) m ; m a positive integer. x6. Concluding remarks. We repeat that the local bases we developed in x2. were introduced by Coifman and Meyer, and their use in obtaining the smooth wavelet bases were pointed out to us by these two authors. Some of the ideas involved in developing the properties of the local bases are also found in Malvar's paper 4]. The particular emphasis on the r^ ole played by the projections P I , however, does not appear in Malvar's paper 5] and is not promminent in the exposition 1] of Coifman and Meyer. As mentioned in the Abstract, one of our goals is to make this material accessible to the largest possible audience. One can develop a parallel treatment connected with the discrete sine and cosine transforms. We shall present this and its connection with the corresponding discrete version of the smooth wavelet bases in a subsequent paper.