The generalized Battle-Lemarie wavelets

Starting with an investigation of the Haar scaling function, we introduce the scaling coefficient function. Then the Battle-Lemarie bases are extended to the generalized Battle-Lemarie bases by introducing a time-shift factor into the Battle-Lemarie scaling coefficient functions. The rationality of the presented extended method is theoretically proved, and some basic properties of the generalized Battle-Lemarie bases, such as orthonormality, time frequency localization, exponential decay features, and so on, are discussed. Meanwhile a series of neoteric symmetric multiresolution analysis bases is generated. The generalized Battle-Lemarie wavelets are better than the corresponding original Battle-Lemarie wavelets in regularity.