May  2018, 25: 36-47. doi: 10.3934/era.2018.25.005

On the norm continuity of the hk-fourier transform

1. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana - Iztapalapa, Av. San Rafael Atlixco 186, CDMX, 09340, Mexico

2. 

Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 Sur S/N, Puebla, 72570, Mexico

Received  February 13, 2018 Published  May 2018

Fund Project: This work is partially supported by CONACyT-SNI and VIEP-BUAP (Puebla, Mexico)

In this work we study the Cosine Transform operator and the Sine Transform operator in the setting of Henstock-Kurzweil integration theory. We show that these related transformation operators have a very different behavior in the context of Henstock-Kurzweil functions. In fact, while one of them is a bounded operator, the other one is not. This is a generalization of a result of E. Liflyand in the setting of Lebesgue integration.

Citation: Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005
References:
[1]

R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, 32. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/032. Google Scholar

[2]

W. Beckner, Inequalities in Fourier analysis on $\mathbb{R}^n$, Proc. Nat. Acad. Sci., 72 (1975), 638-641. doi: 10.1073/pnas.72.2.638. Google Scholar

[3]

B. Bongiorno and T. V. Panchapagesan, On the Alexiewicz topology of the Denjoy space, Real Anal. Exchange, 21 (1995/96), 604–614. Google Scholar

[4]

H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press, San Diego, CA, 1972. Google Scholar

[5]

T. H. Hildebrandt, Introduction to the Theory of Integration, Publisher Academic Press, New York, 1963. Google Scholar

[6]

G. Jameson, Sine, cosine and exponential integrals, The Mathematical Gazette, 99 (2015), 276-289. doi: 10.1017/mag.2015.36. Google Scholar

[7]

R. Kannan and C. K. Krueger, Advanced Analysis on the Real Line, Springer-Verlag, Harrisburg, VA, 1996. Google Scholar

[8]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 1997. Google Scholar

[9]

E. Liflyand, Integrability spaces for the Fourier transform of a function of bounded variation, Journal of Mathematical Analysis and Applications, 436 (2016), 1082-1101. doi: 10.1016/j.jmaa.2015.12.042. Google Scholar

[10]

F.J. Mendoza-Torres, On pointwise inversion of the Fourier transform of BV0 functions, Ann. Funct. Anal., 1 (2010), 112-120. doi: 10.15352/afa/1399900593. Google Scholar

[11]

F. J. Mendoza-TorresM. G. Morales-MacíasJ. A. Escamilla-Reyna and J. H. ArredondoRuiz, Several aspects around the Riemann-Lebesgue lemma, Journal of Advance Research in Pure Mathematics, 5 (2013), 33-46. doi: 10.5373/jarpm.1458.052712. Google Scholar

[12]

M.G. Morales-Macías and J. H. Arredondo-Ruiz, Factorization in the space of Henstock-Kurzweil integrable functions, Azerbaijan Journal of Mathematics, 7 (2017), 116-131. Google Scholar

[13]

M. G. Morales-MacíasJ. H. Arredondo-Ruiz and F. J. Mendoza-Torres, An Extension of some properties for the Fourier transform operator on Lp($\mathbb{R}$) spaces, Revista de la Unión Matemática Argentina, 57 (2016), 85-94. Google Scholar

[14]

M. Reed and B. Simon, Methods of Modern Analysis, volume Ⅱ: Fourier Analysis, Self Adjointness, Academic Press, 1975. Google Scholar

[15]

M. Riesz and A. E. Livingston, A short proof of a classical theorem in the theory of Fourier integrals, Amer. Math. Montly, 62 (1955), 434-437. doi: 10.2307/2307003. Google Scholar

[16]

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. Google Scholar

[17]

E. Talvila, Henstock-Kurzweil Fourier transforms, Ilinois Journal of Mathematics, 46 (2002), 1207-1226. Google Scholar

[18]

M. Tvrdý, G. Antunes-Monteiro and A. Slavik, Kurzweil-Stieltjes Integral: Theory and Applications, Series in Real Analysis, World Scientific Publishing Co, Singapore, 2017.Google Scholar

show all references

References:
[1]

R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, 32. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/032. Google Scholar

[2]

W. Beckner, Inequalities in Fourier analysis on $\mathbb{R}^n$, Proc. Nat. Acad. Sci., 72 (1975), 638-641. doi: 10.1073/pnas.72.2.638. Google Scholar

[3]

B. Bongiorno and T. V. Panchapagesan, On the Alexiewicz topology of the Denjoy space, Real Anal. Exchange, 21 (1995/96), 604–614. Google Scholar

[4]

H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press, San Diego, CA, 1972. Google Scholar

[5]

T. H. Hildebrandt, Introduction to the Theory of Integration, Publisher Academic Press, New York, 1963. Google Scholar

[6]

G. Jameson, Sine, cosine and exponential integrals, The Mathematical Gazette, 99 (2015), 276-289. doi: 10.1017/mag.2015.36. Google Scholar

[7]

R. Kannan and C. K. Krueger, Advanced Analysis on the Real Line, Springer-Verlag, Harrisburg, VA, 1996. Google Scholar

[8]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 1997. Google Scholar

[9]

E. Liflyand, Integrability spaces for the Fourier transform of a function of bounded variation, Journal of Mathematical Analysis and Applications, 436 (2016), 1082-1101. doi: 10.1016/j.jmaa.2015.12.042. Google Scholar

[10]

F.J. Mendoza-Torres, On pointwise inversion of the Fourier transform of BV0 functions, Ann. Funct. Anal., 1 (2010), 112-120. doi: 10.15352/afa/1399900593. Google Scholar

[11]

F. J. Mendoza-TorresM. G. Morales-MacíasJ. A. Escamilla-Reyna and J. H. ArredondoRuiz, Several aspects around the Riemann-Lebesgue lemma, Journal of Advance Research in Pure Mathematics, 5 (2013), 33-46. doi: 10.5373/jarpm.1458.052712. Google Scholar

[12]

M.G. Morales-Macías and J. H. Arredondo-Ruiz, Factorization in the space of Henstock-Kurzweil integrable functions, Azerbaijan Journal of Mathematics, 7 (2017), 116-131. Google Scholar

[13]

M. G. Morales-MacíasJ. H. Arredondo-Ruiz and F. J. Mendoza-Torres, An Extension of some properties for the Fourier transform operator on Lp($\mathbb{R}$) spaces, Revista de la Unión Matemática Argentina, 57 (2016), 85-94. Google Scholar

[14]

M. Reed and B. Simon, Methods of Modern Analysis, volume Ⅱ: Fourier Analysis, Self Adjointness, Academic Press, 1975. Google Scholar

[15]

M. Riesz and A. E. Livingston, A short proof of a classical theorem in the theory of Fourier integrals, Amer. Math. Montly, 62 (1955), 434-437. doi: 10.2307/2307003. Google Scholar

[16]

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. Google Scholar

[17]

E. Talvila, Henstock-Kurzweil Fourier transforms, Ilinois Journal of Mathematics, 46 (2002), 1207-1226. Google Scholar

[18]

M. Tvrdý, G. Antunes-Monteiro and A. Slavik, Kurzweil-Stieltjes Integral: Theory and Applications, Series in Real Analysis, World Scientific Publishing Co, Singapore, 2017.Google Scholar

[1]

Sergiu Aizicovici, Yimin Ding, N. S. Papageorgiou. Time dependent Volterra integral inclusions in Banach spaces. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 53-63. doi: 10.3934/dcds.1996.2.53

[2]

Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems & Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43

[3]

Earl Berkson. Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces. Electronic Research Announcements, 2010, 17: 90-103. doi: 10.3934/era.2010.17.90

[4]

Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks & Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555

[5]

Pavel Krejčí, Harbir Lamba, Sergey Melnik, Dmitrii Rachinskii. Kurzweil integral representation of interacting Prandtl-Ishlinskii operators. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2949-2965. doi: 10.3934/dcdsb.2015.20.2949

[6]

X. Xiang, Y. Peng, W. Wei. A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces. Conference Publications, 2005, 2005 (Special) : 911-919. doi: 10.3934/proc.2005.2005.911

[7]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027

[8]

Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579

[9]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391

[10]

Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232

[11]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7

[12]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1

[13]

Goro Akagi, Mitsuharu Ôtani. Evolution equations and subdifferentials in Banach spaces. Conference Publications, 2003, 2003 (Special) : 11-20. doi: 10.3934/proc.2003.2003.11

[14]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203

[15]

Franco Obersnel, Pierpaolo Omari. Multiple bounded variation solutions of a capillarity problem. Conference Publications, 2011, 2011 (Special) : 1129-1137. doi: 10.3934/proc.2011.2011.1129

[16]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[17]

Michael Music. The nonlinear Fourier transform for two-dimensional subcritical potentials. Inverse Problems & Imaging, 2014, 8 (4) : 1151-1167. doi: 10.3934/ipi.2014.8.1151

[18]

Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339

[19]

Matti Viikinkoski, Mikko Kaasalainen. Shape reconstruction from images: Pixel fields and Fourier transform. Inverse Problems & Imaging, 2014, 8 (3) : 885-900. doi: 10.3934/ipi.2014.8.885

[20]

Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007

2018 Impact Factor: 0.263

Article outline

[Back to Top]