Wavelet Frame On The Quaternion-valued Functions

Let H be the set of all quaternion and P the affine group of the real line R.We introduced the square integrable quaternion-valued function spaces L~2(R,H)via the quaternion valued inner product.By using the theory of square integrable group representations,we establish the theory of continuous wavelet transforms on L~2(R,H)associated with the affine group P .We obtain wavelet frame on quaternion-valued function spaces L~2(R,H)via the theory of wavelet frame onL~2(R).We give the definition of quaternion wavelet frame,and list some wavelet frame criteria.Furthermore,we give a related example to illustrate the existence of the frame.