Polyharmonic Hardy Spaces on the Complexified Annulus and Error Estimates of Cubature Formulas

详细信息    查看全文

• 作者：Ognyan Kounchev (1) (2)
  Hermann Render (3)
  
• 关键词：Primary 65D30 ; 32A35 ; Secondary 41A55 ; Hardy space ; numerical integration ; cubature formulas ; error estimate
• 刊名：Results in Mathematics
• 出版年：2012
• 出版时间：4 - December 2012
• 年：2012
• 卷：62
• 期：3
• 页码：377-403
• 全文大小：377KB
• 参考文献：1. Andrews, G.E., Askey, R. Roy, R.: Special functions. In: Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
  2. Armitage D.H., Goldstein M.: Quadrature and harmonic / L ¹-approximation in annuli. Trans. Am. Math. Soc. 312, 141鈥?54 (1989)
  3. Aronszajn N., Creese T.M., Lipkin L.J.: Polyharmonic Functions. Clarendon Press, Oxford (1983)
  4. Avanissian V.: Cellule d鈥橦armonicit茅 et Prolongement Analytique Complexe. Hermann, Paris (1985)
  5. Axler S., Bourdon P., Ramey W.: Harmonic Function Theory, 2nd edn. Springer, New York (2001)
  6. Bakhvalov N.S.: On the optimal speed of integrating analytic functions. U.S.S.R. Comput. Math. Math. Phys. 7, 63鈥?5 (1967) CrossRef
  7. Bergman S.: The Kernel Function and Conformal Mapping. American Mathematical Society, Providence (1970)
  8. Coifman R., Weiss G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569鈥?45 (1977) CrossRef
  9. Cools R.: An encyclopaedia of cubature formulas. J. Complexity 19, 445鈥?53 (2003) CrossRef
  10. Davis P.: Interpolation and Approximation. Dover Publications Inc., New York (1975)
  11. Davis, P., Rabinowitz, P.: Methods of numerical integration. 2nd edn. In: Computer Science and Applied Mathematics Academic Press, Inc. Orlando (1984)
  12. Dirac P.: Wave equations in conformal space. Ann. Math. 37(2), 429鈥?42 (1936) CrossRef
  13. Fujita K., Morimoto M.: On the double expansion of holomorphic functions. J. Math. Anal. Appl. 272, 335鈥?48 (2002) CrossRef
  14. Gautschi W., Varga R.S.: Error bounds for Gaussian quadrature of analytic functions. SIAM J. Numer. Anal. 20, 1170鈥?186 (1983) CrossRef
  15. Goetz M.: Optimal quadrature for analytic functions. J. Comput. Appl. Math. 137, 123鈥?33 (2001) CrossRef
  16. Goldstein M., Hau脽 mann W., Rogge L.: A harmonic quadrature formula characterizing bi-infinite cylinders. Michigan Math. J. 42, 175鈥?91 (1995) CrossRef
  17. Hoffman K.: Banach Spaces of Analytic functions. Prentice Hall, Englewood Cliffs (1962)
  18. Kerzman N., Stein E.M.: The Cauchy Kernel, the Szeg枚 Kernel, and the Riemann mapping function. Math. Ann. 236, 85鈥?3 (1978) CrossRef
  19. Kzaz M.: Convergence acceleration of some Gaussian quadrature formulas for analytic functions. J. Appl. Numer. Math. 10, 481鈥?96 (1992) CrossRef
  20. Kounchev O.: Multivariate Polyspnes. Applications to Numerical and Wavelet Analysis. Academic Press, San Diego (2001)
  21. Kounchev, O., Render H.: Reconsideration of the multivariate moment problem and a new method for approximating multivariate integrals. Electronic version at arXiv:math/0509380v1 [math.FA]
  22. Kounchev O., Render H.: Holomorphic continuation via Laplace-Fourier series. Contemp. Math. 455, 197鈥?05 (2008) CrossRef
  23. Kounchev O., Render H.: The moment problem for pseudo-positive definite functionals. Arkiv f艙r Matematik 48, 97鈥?20 (2010) CrossRef
  24. Kounchev, O., Render, H.: Multivariate moment problem, Hardy spaces, and orthogonality, a monograph, in preparation
  25. Kounchev, O., Render, H.: Polyharmonic functions of infinite order on annular regions. Tohoku Math. J, accepted
  26. Kounchev, O., Render, H.: Polyharmonic Hardy spaces on the Klein-Dirac quadric. Pliska Studia Mathematica Bulgarica, submitted
  27. Kowalski M.A., Werschulz A.G., Wozniakowski H.: Is Gauss quadrature optimal for analytic functions?. Numerische Mathematik 47(1), 89鈥?8 (1985) CrossRef
  28. Krantz S.: Explorations in Harmonic Analysis. Springer, Berlin (2009) CrossRef
  29. Krantz S.: Geometric Function Theory. Birkh盲user, Basel (2005)
  30. Krein, M., Nudelman, A.: The Markov moment problem and extremal problems. American Mathematical Society, Providence (1977)
  31. Krylov V.: Approximate calculation of integrals. Translated by Arthur H. Stroud. The Macmillan Co., New York-London (1962)
  32. McCullough S., Shen L.-C.: On the Szeg枚 kernel of an annulus. Proc. Am. Math. Soc. 121, 1111鈥?121 (1994)
  33. Milovanovic G., Spalevic M.M.: Error bounds for Gauss-Tur谩n quadrature formulae of analytic functions. Math. Comp. 72, 1855鈥?872 (2003) CrossRef
  34. Morimoto, M.: Analytic Functionals on the Sphere. Translation of Mathematical Monographs, vol. 178. American Mathematical Society, Providence (1998)
  35. Nikolov, N., Todorov, I.: Conformal Quantum Field Theory in Two and Four Dimensions. In: Dragovich, B., Sazdovi膰, B. (eds.) Belgrade, pp. 1鈥?9. Proceedings of the Summer School in Modern Mathematical Physics (2002). Online available at arxiv
  36. Nikolov, N.M., Todorov, I.T.: Conformal invariance and rationality in an even dimensional quantum field theory. Int. J. Mod. Phys. A19, 3605鈥?636 (2004). math-ph/0405005
  37. Nikolov, N., Todorov, I.: Lectures on Elliptic Functions and Modular Forms in Conformal Filed Theory. math-ph/ 0412039
  38. Penrose, R.: On the Origins of Twistor Theory. In: Gravitation and Geometry, a volume in honour of I. Robinson, Biblipolis, Naples (1987) online in https://users.ox.ac.uk/~tweb/00001/
  39. Range R.M.: Holomorphic Functions and Integral Representations in Several Complex Variables. Springer, Berlin (1986)
  40. Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1976)
  41. Rudin W.: Function Theory in Polydiscs. Benjamin, Inc., New York (1969)
  42. Rudin W.: Function Theory in the Unit Ball of / C ^{/ n} . Springer, NewYork (1980) CrossRef
  43. Sarason, D.: The / H ^{/ p} Spaces of an Annulus, vol. 56. Memoirs of AMS (1965)
  44. Sarason, D.: Holomorphic Spaces: A Brief and Selective Survey. In: Holomorphic Spaces, vol. 33. MSRI Publications, Cambridge University Press, Cambridge (1998)
  45. Seeley R.: Spherical harmonics. Am. Math. Monthly 73, 115鈥?21 (1966) CrossRef
  46. Shaimkulov B.A.: On holomorphic extendability of functions from part of the Lie sphere to the Lie ball. Siberian Math. J. 44, 1105鈥?110 (2003) CrossRef
  47. Sobolev S.L.: Cubature formulas and modern analysis. An introduction. Translated from the 1988 Russian edition. Gordon and Breach Science Publishers, Montreux (1992)
  48. Sobolev S., Vaskevich V.: The Theory of Cubature Formulas. Springer, Berlin (1997)
  49. Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
  50. Stein E.M., Weiss G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton University Press, Princeton (1971)
  51. Stroud A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)
  52. Stroud A.H.: Numerical Quadrature and Solution of Ordinary Differential Equations. Springer, New York (1974) CrossRef
  53. Tikhonov A.N., Samarskii A.A.: Equations of Mathematical Physics. Dover Publications, New York (1990)
  54. Vekua I.: New Methods for Solving Elliptic Equations. Wiley, New York (1967)
  
• 作者单位：Ognyan Kounchev (1) (2)
  Hermann Render (3)
  
  1. Acad. G. Bonchev, bl. 3, 1113, Sofia, Bulgaria
  2. IZKS, University of Bonn, Bonn, Germany
  3. School of Mathematical Sciences, Belfield, Dublin 4, Ireland
  
• ISSN：1420-9012

文摘

The present paper has a twofold contribution: first, we introduce a new concept of Hardy spaces on a multidimensional complexified annular domain which is closely related to the annulus of the Klein鈥揇irac quadric important in Conformal Quantum Field Theory. Secondly, for functions in these Hardy spaces, we provide error estimate for the polyharmonic Gau脽鈥揓acobi cubature formulas, which have been introduced in previous papers.