The Cauchy wavelet transform for ultradistributions

The Cauchy wavelet is defined in 1-dimension, and the corresponding Cauchy wavelet transform (CWT) is constructed for functions. This wavelet and its corresponding kernel function for the transform are extended to n-dimension first as a product whose components concern the n components of the variable in $R^n$. The structure of the CWT and its kernel function in this n-dimensional setting is then noted with the kernel function containing derivatives of the classical Cauchy kernel associated with tubes of the form $R^n+i{C}_{\nu }\subset C^n$ where $C_{\nu }$ is any of the $2^n$ n-rants in $R^n$. With this view of the form of the CWT we then extend the kernel function to have the form of derivatives of the Cauchy kernel defined with respect to complex variables in tubes $T^C=R^n+iC\subset C^n$ where C is a cone in $R^n$. We then apply a defined ultradistribution to this extended kernel function to obtain the CWT that we study. Properties of this CWT for ultradistributions which we obtain concern the analyticity of the transform, pointwise growth, norm growth, and boundary limit properties of the transform.