Reconstruction of the Cauchy--Riemann operator by complex integration operators along circles

N. P. Volchkova ; Vit. V. Volchkov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2023), pp. 20-33

Résumé

One of the well-known integral conditions for a function to be holomorphic is the following classical G. 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 $r_1/r_2$ is not the ratio of two zeros of the Bessel function $J_{1}$, then $f$ is holomorphic on the whole complex plane (entire). An example of the function $\frac{\partial}{\partial {z}}\big(J_0(\lambda |z|)\big)$ with a suitable parameter $\lambda$ shows that this condition on $r_1/r_2$ cannot be omitted. In this article, we study the problem of recovering the derivative $\frac{\partial f}{\partial \overline{z}}$ from given contour integrals of $f$. Our main result is Theorem 4, which gives a new formula for finding $\frac{\partial f}{\partial \overline{z}}$ 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 $\overline{B}_r$ biorthogonal to some system of spherical functions. A similar approach can be used to invert a number of convolution operators with radial distributions in $\mathcal{E}'(\mathbb{R}^n)$.

Détail

Keywords: Cauchy–Riemann operator, Bessel functions
Mots-clés : Fourier transform, convolution.