Reconstruction of the Cauchy–Riemann Operator by Complex Integration Operators along Circles

Abstract

One well-known integral condition for a function to be holomorphic is the following classical Morera theorem: if a function $f:\mathcal{O} \to \mathbb{C}$ is continuous in a domain $\mathcal{O} \subset \mathbb{C}$ and has zero integrals over all rectifiable contours in $\mathcal{O}$, then $f$ is holomorphic in $\mathcal{O}$. This fact allows for far-reaching generalizations in various directions. In particular, if a continuous function $f:\mathbb{C} \to \mathbb{C}$ has zero integrals over all circles of fixed radii ${{r}_{1}}$ and ${{r}_{2}}$ in $\mathbb{C}$ and ${{r}_{1}}{\text{/}}{{r}_{2}}$ is not the ratio of two zeros of the Bessel function ${{J}_{1}}$, then $f$ is holomorphic on the whole complex plane (entire). One example of the function $\frac{\partial }{{\partial z}}\left( {{{J}_{0}}\left( {\lambda \left| z \right|} \right)} \right)$ with a suitable parameter $\lambda$ shows that this condition on ${{r}_{1}}{\text{/}}{{r}_{2}}$ cannot be omitted. In this article, we study the problem of recovering the derivative $\frac{{\partial f}}{{\partial \bar {z}}}$ from given contour integrals of $f$. Our main result is Theorem 4, which gives a new formula for finding $\frac{{\partial f}}{{\partial \bar {z}}}$ in terms of integrals of $f$ over circles with the above condition. The key step in the proof of Theorem 4 is the expansion of the Dirac delta function in terms of a system of radial distributions supported in ${{\bar {B}}_{r}}$ biorthogonal to some system of spherical functions. A similar approach can be used to invert a number of convolution operators with radial distributions in $\mathcal{E}{\kern 1pt} '{\kern 1pt} ({{\mathbb{C}}^{n}})$.