Duke Mathematical Journal

Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions

Hermann Render

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In this article, a positive answer is given to the following question posed by Hayman [35, page 326]: if a polyharmonic entire function of order k vanishes on k distinct ellipsoids in the Euclidean space Rn, then it vanishes everywhere. Moreover, a characterization of ellipsoids is given in terms of an extension property of solutions of entire data functions for the Dirichlet problem, answering a question of Khavinson and Shapiro [39, page 460]. These results are consequences from a more general result in the context of direct sum decompositions (Fischer decompositions) of polynomials or functions in the algebra A(BR) of all real-analytic functions defined on the ball BR of radius R and center zero whose Taylor series of homogeneous polynomials converges compactly in BR. The main result states that for a given elliptic polynomial P of degree 2k and for sufficiently large radius R>0, the following decomposition holds: for each function fA(BR), there exist unique q,rA(BR) such that f=Pq+r and Δkr=0. Another application of this result is the existence of polynomial solutions of the polyharmonic equation Δku=0 for polynomial data on certain classes of algebraic hypersurfaces

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Duke Math. J., Volume 142, Number 2 (2008), 313-352.

First available in Project Euclid: 27 March 2008

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Zentralblatt MATH identifier

Primary: 31B30: Biharmonic and polyharmonic equations and functions
Secondary: 35A20: Analytic methods, singularities 14P99: None of the above, but in this section 12Y05: Computational aspects of field theory and polynomials


Render, Hermann. Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions. Duke Math. J. 142 (2008), no. 2, 313--352. doi:10.1215/00127094-2008-008. https://projecteuclid.org/euclid.dmj/1206642157

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  • M. L. Agranovsky and Y. Krasnov, Quadratic divisors of harmonic polynomials in $\mathbbR^n$, J. Anal. Math. 82 (2000), 379--395.
  • E. Almansi, Sull'integrazione dell'equazione differenziale $\Delta ^2nu=0$, Ann. Mat. 2 (1899), 1--51.
  • D. H. Armitage, Cones on which entire harmonic functions can vanish, Proc. Roy. Irish Acad. Sect. A 92 (1992), 107--110.
  • N. Aronszajn, T. M. Creese, and L. J. Lipkin, Polyharmonic Functions, Oxford Math. Monogr., Oxford Sci. Publ., Oxford Univ. Press, New York, 1983.
  • S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory, Grad. Texts in Math. 137, Springer, New York, 1992.
  • S. Axler, P. Gorkin, and K. Voss, The Dirichlet problem on quadratic surfaces, Math. Comp. 73 (2004), 637--651.
  • S. Axler and W. Ramey, Harmonic polynomials and Dirichlet-type problems, Proc. Amer. Math. Soc. 123 (1995), 3765--3773.
  • B. Bacchelli, M. Bozzini, C. Rabut, and M.-L. Varas, Decomposition and reconstruction of multidimensional signals using polyharmonic pre-wavelets, Appl. Comput. Harmon. Anal. 18 (2005), 282--299.
  • M. B. Balk and M. Y. Mazalov, ``On the Hayman uniqueness problem for polyharmonic functions'' in Clifford Algebras and Their Application in Mathematical Physics (Aachen, Germany, 1996), Fund. Theories Phys. 94, Kluwer, Dordrecht, Germany, 1998, 11--16.
  • —, ``On uniqueness conditions for entire polyharmonic functions'' in Partial Differential and Integral Equations (Newark, Del., 1997), Int. Soc. Anal. Appl. Comput. 2, Kluwer, Dordrecht, Germany, 1999, 219--232.
  • V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187--214.
  • B. Beauzamy, Extremal products in Bombieri's norm, Rend. Istit. Mat. Univ. Trieste 28 (1996), suppl., 73--89.
  • J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, rev. of the French original, Ergeb. Math. Grenzgeb. (3) 36, Springer, Berlin, 1998.
  • M. Brelot and G. Choquet, ``Polynômes harmoniques et polyharmoniques'' in Second colloque sur les équations aux dérivées partielles (Brussels, 1954), 45--66.
  • M. Chamberland and D. Siegel, Polynomial solutions to Dirichlet problems, Proc. Amer. Math. Soc. 129 (2001), 211--217.
  • C. De Boor and A. Ron, The least solution for the polynomial interpolation problem, Math. Z. 210 (1992), 347--378.
  • W. F. Donoghue Jr., Distributions and Fourier Transforms, Pure Appl. Math. 32, Academic Press, New York, 1969.
  • P. Ebenfelt, Singularities encountered by the analytic continuation of solutions to Dirichlet's problem, Complex Variables Theory Appl. 20 (1992), 75--91.
  • P. Ebenfelt, D. Khavinson, and H. S. Shapiro, ``Algebraic aspects of the Dirichlet problem'' in Quadrature Domains and Their Applications (Santa Barbara, Calif., 2003), Oper. Theory Adv. Appl. 156, Birkhäuser, Basel, 2005, 151--172.
  • P. Ebenfelt and H. Render, On the mixed Cauchy problem with data on singular conics, to appear in J. London Math. Soc. (2).
  • —, The Goursat problem for a generalized Helmholtz operator in the plane, to appear in J. Anal. Math.
  • P. Ebenfelt and H. S. Shapiro, The Cauchy-Kowalevskaya theorem and generalizations, Comm. Partial Diffential Equations 20 (1995), 939--960.; Erratum, Comm. Partial Differential Equations 20 (1995), 2221--2222. $\!$;
  • —, The mixed Cauchy problem for holomorphic partial differential operators, J. Anal. Math. 65 (1995), 237--295.
  • —, A quasi-maximum principle for holomorphic solutions of partial differential equations in $\mathbbC^n$, J. Funct. Anal. 146 (1997), 27--61.
  • P. Ebenfelt and M. Viscardi, On the solution of the Dirichlet problem with rational holomorphic boundary data, Comput. Methods Funct. Theory 5 (2005), 445--457.
  • J. Edenhofer, Integraldarstellung einer m-polyharmonischen Funktion, deren Funktionswerte und erste $m-1$ Normalableitungen auf einer Hypersphäre gegeben sind, Math. Nachr. 68 (1975), 105--113.
  • E. Fischer, Über die Differentiationsprozesse der Algebra, J. für Math. 148 (1917), 1--78.
  • L. Flatto, D. J. Newman, and H. S. Shapiro, The level curves of harmonic functions, Trans. Amer. Math. Soc. 123 (1966), 425--436.
  • T. Futamura, K. Kishi, and Y. Mizuta, A generalization of Bôcher's theorem for polyharmonic functions, Hiroshima Math. J. 31 (2001), 59--70.
  • —, Removability of sets for sub-polyharmonic functions, Hiroshima Math. J. 33 (2003), 31--42.
  • E. A. GonzáLez-Velasco, Fourier Analysis and Boundary Value Problems, Academic Press, San Diego, Calif., 1996.
  • P. Griffiths and J. Harris, Principles of Algebraic Geometry, reprint of the 1978 original, Wiley Classics Lib., Wiley, New York, 1994.
  • L. J. Hansen and H. S. Shapiro, Graphs and functional equations, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), 125--146.
  • —, Functional equations and harmonic extensions, Complex Variables Theory Appl. 24 (1994), 121--129.
  • W. K. Hayman, ``A uniqueness problem for polyharmonic functions'' in Linear and Complex Analysis: Problem Book 3, Part II, Lecture Notes in Math. 1574, Springer, Berlin, 1994, 326--327.
  • W. K. Hayman and B. Korenblum, Representation and uniqueness theorems for polyharmonic functions, J. Anal. Math. 60 (1993), 113--133.
  • L. I. Hedberg, Approximation in the mean by solutions of elliptic equations, Duke Math. J. 40 (1973), 9--16.
  • L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Amer. Math. Soc., Providence, 1963.
  • D. Khavinson and H. S. Shapiro, Dirichlet's problem when the data is an entire function, Bull. London Math. Soc. 24 (1992), 456--468.
  • O. I. Kounchev, ``Extremal problems for the distributed moment problem'' in Potential Theory (Prague, 1987), Plenum, New York, 1988, 187--195.
  • —, Sharp estimate of the Laplacian of a polyharmonic function and applications, Trans. Amer. Math. Soc. 332, no. 1 (1992), 121--133.
  • —, Minimizing the Laplacian of a function squared with prescribed values on interior boundaries --.-Theory of polysplines, Trans. Amer. Math. Soc. 350, no. 5 (1998), 2105--2128.
  • —, Multivariate Polysplines: Applications to Numerical and Wavelet Analysis, Academic Press, San Diego, Calif., 2001.
  • O. I. Kounchev and H. Render, Cardinal interpolation with polysplines on annuli, J. Approx. Theory 137 (2005), 89--107.
  • —, Polyharmonic splines on grids $\mathbbZ\times a\mathbbZ^n$ and their limits, Math. Comp. 74 (2005), 1831--1841.
  • —, Polyharmonicity and algebraic support of measures, Hiroshima Math. J. 37 (2007), 25--44.
  • E. Ligocka, On duality and interpolation for spaces of polyharmonic functions, Studia Math. 88 (1988), 139--163.
  • W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines, J. Approx. Theory 60 (1990), 141--156.
  • A. Meril and D. C. Struppa, Equivalence of Cauchy problems for entire and exponential type functions, Bull. London Math. Soc. 17 (1985), 469--473.
  • D. J. Newman and H. S. Shapiro, Certain Hilbert spaces of entire functions, Bull. Amer. Math. Soc. 72 (1966), 971--977.
  • —, ``Fischer spaces of entire functions'' in Entire Functions and Related Parts of Analysis (La Jolla, Calif., 1966), Amer. Math. Soc., Providence, 1968, 360--369.
  • M. Nicolesco, Recherches sur les fonctions polyharmoniques, Ann. Sci. École Norm. Sup. (3) 52 (1935), 183--220.
  • P. S. Pedersen, A basis for polynomial solutions to systems of linear constant coefficient PDE's, Adv. Math. 117 (1996), 157--163.
  • R. Remmert, Funktionentheorie, I, Grundwissen Math. 5, Springer, Berlin, 1984.
  • T. Sauer, Gröbner bases, H-bases and interpolation, Trans. Amer. Math. Soc. 353, no. 6 (2001), 2293--2308.
  • B.-W. Schulze and G. Wildenhain, Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung, Lehrbücher Monogr. Gebiete Exakten Wiss. Math. Reihe 60, Birkhäuser, Basel, 1977.
  • H. S. Shapiro, An algebraic theorem of E. Fischer, and the holomorphic Goursat problem, Bull. London Math. Soc. 21 (1989), 513--537.
  • —, personal communication, Feb. 2006.
  • J. Siciak, ``Holomorphic continuation of harmonic functions'' in Collection of Articles Dedicated to the Memory of Tadeusz Wa\dzewski, Ann. Polon. Math. 29, Państwowe Wydawnictwo Nauk., Warsaw, 1974, 67--73.
  • S. L. Sobolev, Cubature Formulas and Modern Analysis: An Introduction, Gordon and Breach, Montreux, Switzerland, 1992.
  • E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, Princeton, 1971.
  • R. J. Walker, Algebraic Curves, reprint of the 1950 ed., Springer, New York, 1978.
  • D. Zeilberger, Chu's 1303 identity implies Bombieri's 1990 norm-inequality (Via an identity of Beauzamy and Dégot), Amer. Math. Monthly 101 (1994), 894--896.