Generalizations of the wave equation
HTML articles powered by AMS MathViewer
- by J. Marshall Ash, Jonathan Cohen, C. Freiling and Dan Rinne PDF
- Trans. Amer. Math. Soc. 338 (1993), 57-75 Request permission
Abstract:
The main result of this paper is a generalization of the property that, for smooth $u$, ${u_{xy}} = 0$ implies $(\ast )$ \[ u(x,y) = a(x) + b(y).\] Any function having generalized unsymmetric mixed partial derivative identically zero is of the form $(\ast )$. There is a function with generalized symmetric mixed partial derivative identically zero not of the form $(\ast )$, but $(\ast )$ does follow here with the additional assumption of continuity. These results connect to the theory of uniqueness for multiple trigonometric series. For example, a double trigonometric series is the ${L^2}$ generalized symmetric mixed partial derivative of its formal $(x,y)$-integral.References
- J. Marshall Ash, Uniqueness of representation by trigonometric series, Amer. Math. Monthly 96 (1989), no. 10, 873–885. MR 1033355, DOI 10.2307/2324582
- J. Marshall Ash, A new proof of uniqueness for multiple trigonometric series, Proc. Amer. Math. Soc. 107 (1989), no. 2, 409–410. MR 984780, DOI 10.1090/S0002-9939-1989-0984780-8
- J. Marshall Ash, A. Eduardo Gatto, and Stephen Vági, A multidimensional Taylor’s theorem with minimal hypothesis, Colloq. Math. 60/61 (1990), no. 1, 245–252. MR 1096374, DOI 10.4064/cm-60-61-1-245-252
- J. Marshall Ash and Grant V. Welland, Convergence, uniqueness, and summability of multiple trigonometric series, Trans. Amer. Math. Soc. 163 (1972), 401–436. MR 300009, DOI 10.1090/S0002-9947-1972-0300009-X
- Karl Bögel, Über die mehrdimensionale Differentiation, Jber. Deutsch. Math.-Verein. 65 (1962/63), no. Abt. 1, 45–71 (German). MR 146311
- Z. Buczolich, A general Riemann complete integral in the plane, Acta Math. Hungar. 57 (1991), no. 3-4, 315–323. MR 1139326, DOI 10.1007/BF01903683
- Roger Cooke, A Cantor-Lebesgue theorem in two dimensions, Proc. Amer. Math. Soc. 30 (1971), 547–550. MR 282134, DOI 10.1090/S0002-9939-1971-0282134-X
- C. Freiling and D. Rinne, A symmetric density property: monotonicity and the approximate symmetric derivative, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1098–1102. MR 936773, DOI 10.1090/S0002-9939-1988-0936773-3
- C. Freiling and D. Rinne, A symmetric density property for measurable sets, Real Anal. Exchange 14 (1988/89), no. 1, 203–209. MR 988365 A. Khintchine, Recherches sur la structure des fonctions mesurables, Fund. Math. 9 (1927), 212-279. A Rajchman and A. Zygmund, Sur la relation du procédé de sommation de Cesàro et celui de Riemann, Bull. Internat. Acad. Polon. Sér. A (1925), 69-80. (Also in Selected Papers of Antoni Sygmund, Vol. 3 (A. Hulanicki et al., eds., Kluwer, Dordrecht, 1989, pp. 81-92.)
- Donald Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 377518, DOI 10.1090/S0002-9947-1975-0377518-3
- Victor L. Shapiro, Uniqueness of multiple trigonometric series, Ann. of Math. (2) 66 (1957), 467–480. MR 90700, DOI 10.2307/1969904 L. V. Zhizhiashvili, Some problems in the theory of simple and multiple trigonometric series, Russian Math. Surveys 28 (1975), 65-127.
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 338 (1993), 57-75
- MSC: Primary 35L05; Secondary 26B40, 42B99
- DOI: https://doi.org/10.1090/S0002-9947-1993-1088475-X
- MathSciNet review: 1088475