Cauchy Integral and Boundary Value for Vector-Valued Tempered Distributions

Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued functions in tubes TC=Rn+iC obtain values. In a certain case, study structure of this value, is shown to be Fourier transform distributional derivative continuous function polynomial growth. A set used value one–one and onto relationship with distributions, generalize Schwartz space DL2?(Rn); distribution defines between these two sets. By combining previously stated results, Cauchy integral representation terms value.