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 Mat. Sb., 1997, Volume 188, Number 9, Pages 13–30 (Mi msb255)

Solution of the support problem for several function classes

V. V. Volchkov

Donetsk National University

Abstract: Several classes of functions with zero integrals over all balls of a fixed radius are studied. A description of functions in such classes as series in special functions is obtained and a theorem on a removable singularity is proved. These results make it possible to solve completely the support problem for several classes of functions with zero ball averages. As a consequence, a far-reaching generalization of the well-known Zalcman's two-radii theorem is obtained.

DOI: https://doi.org/10.4213/sm255

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English version:
Sbornik: Mathematics, 1997, 188:9, 1279–1294

Bibliographic databases:

UDC: 517.5
MSC: 53C65, 44A35

Citation: V. V. Volchkov, “Solution of the support problem for several function classes”, Mat. Sb., 188:9 (1997), 13–30; Sb. Math., 188:9 (1997), 1279–1294

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Volchkov, “Injectivity sets of the Pompeiu transform”, Sb. Math., 190:11 (1999), 1607–1622
2. Volchkov, VV, “Functions with zero integrals over parallelepipeds”, Doklady Akademii Nauk, 369:4 (1999), 444
3. V. V. Volchkov, “Extremal problems on Pompeiu sets. II”, Sb. Math., 191:5 (2000), 619–632
4. Vit. V. Volchkov, “On functions with zero spherical means of complex hyperbolic spaces”, Math. Notes, 68:4 (2000), 436–443
5. V. V. Volchkov, “A definitive version of the local two-radii theorem on hyperbolic spaces”, Izv. Math., 65:2 (2001), 207–229
6. V. V. Volchkov, “Theorems on ball mean values in symmetric spaces”, Sb. Math., 192:9 (2001), 1275–1296
7. Volchkov, VV, “A local two-radius theorem on symmetric spaces”, Doklady Mathematics, 64:3 (2001), 398
8. Ochakovskaya, OA, “On functions with zero integrals over balls of fixed radius on the half-space”, Doklady Mathematics, 64:3 (2001), 413
9. Vit. V. Volchkov, “Functions with zero ball means on the quaternionic hyperbolic space”, Izv. Math., 66:5 (2002), 875–903
10. Volchkov, VV, “Final version of the local two-radius theorem on the quaternion hyperbolic space”, Doklady Mathematics, 65:3 (2002), 389
11. Siberian Math. J., 43:2 (2002), 215–222
12. Vit. V. Volchkov, “Uniqueness Theorems for Periodic (in Mean) Functions on Quaternion Hyperbolic Space”, Math. Notes, 74:1 (2003), 30–37
13. V. V. Volchkov, “A local two-radii theorem for quasianalytic classes of functions”, Math. Notes, 80:4 (2006), 468–477
14. V. V. Volchkov, “Uniqueness theorems for solutions of the convolution equation on symmetric spaces”, Izv. Math., 70:6 (2006), 1077–1092
15. V. V. Volchkov, “Local two-radii theorem in symmetric spaces”, Sb. Math., 198:11 (2007), 1553–1577
16. V. V. Volchkov, “On the Injectivity of the Local Pompeiu Transform on the Sphere”, Math. Notes, 81:1 (2007), 51–60
17. Ochakovskaya, OA, “Liouville-type theorems for functions with zero integrals over balls of fixed radius”, Doklady Mathematics, 76:1 (2007), 530
18. O. A. Ochakovskaya, “Precise characterizations of admissible rate of decrease of a non-trivial function with zero ball means”, Sb. Math., 199:1 (2008), 45–65
19. Ochakovskaya, OA, “MAJORANTS OF FUNCTIONS WITH VANISHING INTEGRALS OVER BALLS”, Ukrainian Mathematical Journal, 60:6 (2008), 1003
20. Ochakovskaya, OA, “Majorants of functions with zero integrals over balls of fixed radius”, Doklady Mathematics, 77:3 (2008), 446
21. Ochakovskaya O.A., “On the Injectivity of the Pompeiu Transform for Integral Ball Means”, Ukrainian Math J, 63:3 (2011), 416–424
22. V. V. Volchkov, Vit. V. Volchkov, “Spherical means on two-point homogeneous spaces and applications”, Izv. Math., 77:2 (2013), 223–252
23. Ochakovskaya O.A., “Injectivity Classes of the Pompeiu Transformation”, Ukr. Math. J., 64:12 (2013), 1893–1902
24. O. A. Ochakovskaya, “Radial majorants of functions with zero integrals over balls of a fixed radius”, Izv. Math., 78:3 (2014), 580–595
25. Volchkov V.V., Savost'yanova I.M., “Smoothing of the Singularities of Functions Whose Integrals Over the Balls on a Sphere Are Zero”, Ukr. Math. J., 67:2 (2015), 314–322
26. Vit. V. Volchkov, N. P. Volchkova, “The extension problem for functions with zero weighted spherical means”, Russian Math. (Iz. VUZ), 61:8 (2017), 13–21
27. Vit. V. Volchkov, N. P. Volchkova, “The removability problem for functions with zero spherical means”, Siberian Math. J., 58:3 (2017), 419–426
28. Vit. V. Volchkov, N. P. Volchkova, “Vektornye polya s nulevym potokom cherez sfery fiksirovannogo radiusa”, Vladikavk. matem. zhurn., 20:4 (2018), 20–34
29. N. P. Volchkova, Vit. V. Volchkov, “A one-radius theorem on a sphere with pricked point”, Ufa Math. J., 11:4 (2019), 3–12
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