### AIMS Mathematics

2016, Issue 1: 9-23. doi: 10.3934/Math.2016.1.9
Research article

# The Bedrosian Identity for Lp Function and the Hardy Space on Tube

• Received: 26 March 2016 Accepted: 11 April 2016 Published: 19 April 2016
• In this paper, we are devoted to establishing several necessary and sufficient conditions for $f\in L^{p}(\mathbb{R}^{n}),\ g\in L^{q}(\mathbb{R}^{n})$ with $\frac{1}{p}+\frac{1}{q}\leq 1$ to satisfy the Bedrosian identity $H(fg)=fHg$, where $H$ denotes the n-dimensional Hilbert transform. In addition, we also show that the distribution $f\in \mathcal{D}'_{L^{p}}(\mathbb{R}^{n})$ can be represented by functions in the Hardy space on tube.

Citation: Zhihong Wen, Guantie Deng. The Bedrosian Identity for Lp Function and the Hardy Space on Tube[J]. AIMS Mathematics, 2016, 1(1): 9-23. doi: 10.3934/Math.2016.1.9

### Related Papers:

• In this paper, we are devoted to establishing several necessary and sufficient conditions for $f\in L^{p}(\mathbb{R}^{n}),\ g\in L^{q}(\mathbb{R}^{n})$ with $\frac{1}{p}+\frac{1}{q}\leq 1$ to satisfy the Bedrosian identity $H(fg)=fHg$, where $H$ denotes the n-dimensional Hilbert transform. In addition, we also show that the distribution $f\in \mathcal{D}'_{L^{p}}(\mathbb{R}^{n})$ can be represented by functions in the Hardy space on tube.
 [1] Bedrosian E (1963) A product theorem for Hilbert transforms.Proc IEEE 51: 868-869. [2] Bochner S (1994) Group invariance of Cauchy formula in several variables.Annals Math 45: 686-707. [3] Brown J (1974) Analytic signals and product theorems for Hilbert transforms.IEEE Trans Circuits and Systems CAS-21: 790-792. [4] Butzer P, Nessel R (1971) Fourier analysis and approximation.Birkhauser Verlag, Bassel and New York:Academic Press. . [5] Cerejeiras P, Chen Q and K$\rm \ddot{a}$ehler U (2010) A necessary and sufficient condition for a Bedrosian identity.Math Methods Appl Sci 33: 493-500. [6] Cerejeiras P, Chen Q and K$\rm \ddot{a}$ehler U (2012) Bedrosian identity in Blaschke product case.Complex Anal Oper Theory 6: 275-300. [7] Chen Q, Huang N, Riemenschneider S and Xu Y (2006) A B-spline approach for empirical mode decompositions.Adv Comput Math 24: 171-195. [8] Chen Q and Micchelli C (2012) The Bedrosian identity for functions analytic in a neighborhood of the unit circle.Complex Anal Oper Theory 6: 781-798. [9] Cohen L (1995) Time-Frequency Analysis: Theory and Applications.Englewood Cliffs, NJ: Prentice-Hall. . [10] Deng G (2010) Complex Analysis.Beijing Norm University Press (In Chinese) . [11] Donoghue W (1969) Distributions and Fourier Transforms.New York: Academic Press . [12] Fei M (2014) Hp space on Tube.PHD thesis of Beijing Norm University (In press) . [13] Gabor D (1946) Theory of communication.J Inst Electr Eng 93: 429-457. [14] Gasquet C andWitomski P (1999) Fourier Analysis and Applications..New York: Springer-Verlag . [15] Huang N, Shen Z, Long S, Wu M, Shih E, Zheng Q, Tung C and Liu H (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis.Proc R Soc Lond Ser A Math Phys Eng Sci 454: 903-995. [16] Huang N and Attoh-Okine N (2005) The Hilbert-Huang Transform in Engineering.CRC/Taylor & Francis, Boca Raton. . [17] Huang N and Shen S (2005) Hilbert-Huang transform and its Applications..London: World Scientific . [18] Huang N, Shen Z and Long S (1999) A new view of nonlinear water waves: the Hilbert spectrum.Annu Rev Fluid Mech 31: 417-457. [19] Lin R and Zhang H (2014) Existence of the Bedrosian identity for singular integral operators.1407-0861. [20] Luszczki Z and Ziele$\acute{z}$ny Z (1961) Distributionen der R$\ddot{a}$ume D0 Lp als Randverteilungen analytischer Funktionen.Colloq Math 8: 125-131. [21] Nuttall A and Bedrosian E (1966) On the quadrature approximation for the Hilbert transform of modulated signals.Proceedings of the IEEE 54: 1458-1459. [22] Oppenheim A and Lim J (1981) The importance of phase in signal.Proc IEEE 69: 529-541. [23] Pandey J (1996) The Hibert transform of schwartz distributions and applications..New York: John Wiley & Sons, Inc. . [24] Picibono B (1997) On instantaneous amplitude and phase of signals.IEEE Trans Signal Process 45: 552-560. [25] Qian T, Chen Q and Li L (2005) Analytic unit quadrature signals with nonlinear phase.Phys D 203: 80-87. [26] Qian T, Xu Y, Yan D, Yan L and Yu B (2009) Fourier spectrum characterization of Hardy spaces and applications.Proc Am Math Soc 137: 971-980. [27] Schwartz L (1978) Theorie des distributions..Hermann, Paris. . [28] Stein E and Weiss G (1971) Introduction to Fourier analysis on Euclidean space..Princeton, New Jersey: Princeton University Press. . [29] Stein E, Weiss G and Weiss M (1964) $H^{p}$ class of holomophic functions in tube domains..Proc Nat Acad Sci USA 52: 1035-1039. [30] Tan L, Shen L and Yang L (2010) Rational orthogonal bases satisfying the Bedrosian identity..Adv Comput Math 33: 285-303. [31] Tan L, Yang L and Huang D (2009) Necessary and sufficient conditions for the Bedrosian identity..J Integral Equations Appl 21: 77-94. [32] Tan L, Yang L, Huang D (2010) Construction of periodic analytic signals satisfying the circular Bedrosian identity..IMA J. Appl. Math 75: 246-256. [33] Venouziou M and Zhang H (2008) Characterzing the Hilbert transform by the Bedrosian theorem..J Math anal Appl 338: 1477-1481. [34] Wang S (2009) Simple proofs of the Bedrosian equality for the Hilbert transform..Sci China Ser A 52: 507-510. [35] Xu Y and Yan D (2006) The Bedrosian identity for the Hilbert transform of product functions..Proc Am Math Soc 134: 2719-2728. [36] Yang L and Zhang H (2008) The Bedrosian identity for $H^p$ functions.J Math Anal Appl 345: 975-984. [37] Yu B and Zhang H (2008) The Bedrosian identity and homogeneous semi-convolution equations..J Integral Equations Appl 20: 527-568. [38] Zhang H (2014) Multidimensional analytic signals and the Bedrosian identity..Integral Equations Operator Theory 78: 301-321.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

0.882 0.9