Calderon reproducing formula associated with the Gegenbauer operator on the half-line

In this paper, we introduce the generalized shift operator generated by the Gegenbauer differential operator , and define a generalized convolution ⊗ on the half-line corresponding to the Gegenbauer differential operator. We investigate the Calderon reproducing formula associated with the convolution ⊗ involving finite Borel measures, leading to results on the Lp-norm and pointwise approximation for functions on the half-line.