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.

Bibliographie

[1] Volchkov V.V., Integral geometry and convolution equations, Kluwer, Dordrecht, 2003 | MR | Zbl

[2] Volchkov V.V., Volchkov Vit.V., Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group, Springer, London, 2009 | MR | Zbl

[3] Volchkov V.V., Volchkov Vit.V., Offbeat integral geometry on symmetric spaces, Birkhäuser, Basel, 2013 | MR | Zbl

[4] Zalcman L., “Analyticity and the Pompeiu problem”, Arch. Rat. Anal. Mech., 47:3 (1972), 237–254 | DOI | MR | Zbl

[5] Smith J.D., “Harmonic analysis of scalar and vector fields in $\mathbb R^n$”, Proc. Cambridge Philos. Soc., 72:3 (1972), 403–416 | DOI | MR | Zbl

[6] Delsarte J., “Note sur une propriété nouvelle des fonctions harmoniques”, C. R. Acad. Sci. Paris Sér. A–B, 246 (1958), 1358–1360 | MR | Zbl

[7] Berenstein K.A., Struppa D., “Kompleksnyi analiz i uravneniya v svertkakh”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem. Fundament. napravleniya, 54, VINITI, 1989, 5–111

[8] Zalcman L., “Offbeat integral geometry”, Amer. Math. Monthly, 87:3 (1980), 161–175 | DOI | MR | Zbl

[9] Radon J., “Über die Bestimmung von funktionen durch ihre Integralwerte iängs gewisser mannigfaltigkeiten”, Ber. Verh. Sächs. Akad. Wiss., 69 (1917), 262–277

[10] Casey S.D., Walnut D.F., “Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms”, SIAM Review, 36:4 (1994), 537–577 | DOI | MR | Zbl

[11] Berenstein C.A., Taylor B.A., Yger A., “Sur quelques formules explicites de déconvolution”, J. Opt., 14:2 (1983), 75–82 | DOI | MR

[12] Berenstein C.A., Yger A., “Le problème de la déconvolution”, J. Funct. Anal., 54:2 (1983), 113–160 | DOI | MR | Zbl

[13] Berenstein C.A., Yger A., “Analytic Bezout identities”, Adv. Appl. Math., 10:1 (1989), 51–74 | DOI | MR | Zbl

[14] Volchkov V.V., “Okonchatelnyi variant lokalnoi teoremy o dvukh radiusakh”, Matem. sb., 186:6 (1995), 15–34 | MR | Zbl

[15] Berenstein C.A., Gay R., Yger A., “Inversion of the local Pompeiu transform”, J. Anal. Math., 54:1 (1990), 259–287 | DOI | MR | Zbl

[16] Volchkova N.P., Volchkov Vit. V., “Problema dekonvolyutsii dlya indikatorov otrezkov”, Matem. zametki SVFU, 26:3 (2019), 1–14

[17] Hörmander L., The analysis of linear partial differential operators, v. I, Springer-Verlag, New York, 1990 | MR

[18] Khelgason S., Gruppy i geometricheskii analiz, Mir, M., 1987 | MR

[19] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 2, Nauka, M., 1974

[20] Hielscher R., Quellmalz M., “Reconstructing an function on the sphere from its means along vertical slices”, Inverse Probl. Imaging, 10:3 (2016), 711–739 | DOI | MR | Zbl

[21] Salman Y., “Recovering functions defined on the unit sphere by integration on a special family of sub-spheres”, Anal. Math. Phys., 7:2 (2017), 165–185 | DOI | MR | Zbl

[22] Rubin B., “Reconstruction of functions on the sphere from their integrals over hyperplane sections”, Anal. Math. Phys., 9:4 (2019), 1627–1664 | DOI | MR | Zbl