The Hardy Inequality for Hermite Expansions

详细信息    查看全文

• 作者：Zhongkai Li (1)
  Yufeng Yu (1)
  Yehao Shi (2)
  
  1. Department of Mathematics ; Capital Normal University ; Beijing ; 100048 ; China
  2. Elementary Education College ; Capital Normal University ; Beijing ; 100048 ; China
  
• 关键词：Hardy inequality ; Hermite expansion ; Hardy space ; 42B30 ; 42C10 ; 42B15
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：21
• 期：2
• 页码：267-280
• 全文大小：285 KB
• 参考文献：1. Askey, R, Wainger, S (1965) Mean convergence of expansions in Laguerre and Hermite series. Am. J. Math. 87: pp. 695-708 CrossRef
  2. Balasubramanian, R, Radha, R (2005) Hardy-type inequalities for Hermite expansions. J. Inequal. Pure Appl. Math. 6: pp. 1-4
  3. Chihara, TS (1978) An Introduction to Orthogonal Polynomials. Gordon and Breach, New York
  4. Janssen, AJEM (2005) Hermite function description of Feichtinger鈥檚 space $$S_0$$ S 0. J. Fourier Anal. Appl. 11: pp. 577-588 CrossRef
  5. Kanjin, Y (1997) Hardy鈥檚 inequalities for Hermite and Laguerre expansions. Bull. London Math. Soc. 29: pp. 331-337 CrossRef
  6. Kanjin, Y (2011) Hardy鈥檚 inequalities for Hermite and Laguerre expansions revisited. J. Math. Soc. Jpn. 63: pp. 753-767 CrossRef
  7. Li, Zh.-K., Shi, Y.-H.: Multipliers of Hardy spaces associated with generalized Hermite expansions. Constr. Approx. 39, 517鈥?40 (2014)
  8. Muckenhoupt, B (1970) Asymptotic forms for Laguerre polynomials. Proc. Am. Math. Soc. 24: pp. 288-292 CrossRef
  9. Muckenhoupt, B, Webb, DW (2000) Two-weight norm inequalities for Ces脿ro means of Laguerre expansions. Trans. Am. Math. Soc. 353: pp. 1119-1149 CrossRef
  10. Radha, R, Thangavelu, S (2004) Hardy鈥檚 inequalities for Hermite and Laguerre expansions. Proc. Am. Math. Soc. 132: pp. 3525-3536 CrossRef
  11. Rosenblum, M.: Generalized Hermite polynomials and the Bose-like oscillator calculus. In: Feintuch, A., Gohberg, I. (eds.) Operator Theory: Advances and Applications, vol. 73, pp. 369鈥?96. Birkh盲user, Basel (1994)
  12. R枚sler, M Bessel-type signed hypergroups on $$\mathbb{R}$$ R. In: Heyer, H, Mukherjea, A eds. (1995) Probability Measures on Groups and Related Structures XI. World Scientific, Singapore, pp. 292-304
  13. R枚sler, M (1998) Generalized Hermite polynomials and the heat equation for Dunkl operators. Commun. Math. Phys. 192: pp. 519-542 CrossRef
  14. Stein, EM (1993) Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton
  15. Szeg枚, G (1975) Orthogonal Polynomials. American Mathematical Society Colloquium Publisher, Providence
  16. Thangavelu, S (1993) Hermite and Laguerre Expansions. Princeton University Press, Princeton
  17. Thangavelu, S.: On regularity of twisted spherical means and special Hermite expansion. Proc. Indian Acad. Sci. Math. Sci. 103, 303鈥?20 (1993)
  18. Uchiyama, A (2001) Hardy Spaces on the Euclidean Space. Springer, Berlin CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Abstract Harmonic Analysis
  Approximations and Expansions
  Partial Differential Equations
  Applications of Mathematics
  Signal,Image and Speech Processing
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5851

文摘

The purpose of the paper is to prove a sharp form of Hardy-type inequality, conjectured by Kanjin, for Hermite expansions of functions in the Hardy space \(H^1({\mathbb {R}})\) , that is, \(\sum _{n=1}^{\infty }n^{-\frac{3}{4}}|a_n(f)|\le A\Vert f\Vert _{H^1({\mathbb {R}})}\) for all \(f\in H^1({\mathbb {R}})\) , where \(A\) is a constant independent of \(f\) .