American Institute of Mathematical Sciences

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

On the norm continuity of the hk-fourier transform

Juan H. Arredondo 1, , Francisco J. Mendoza 2, and  Alfredo Reyes 1,

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.

Keywords: Fourier transform, Henstock-Kurzweil integral, integrability, bounded variation function, Banach spaces.
Mathematics Subject Classification: Primary: 26A39, 43A32; Secondary: 26A42, 26A45.
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-Torres, M. G. Morales-Macías, J. 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-Macías, J. 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-Torres, M. G. Morales-Macías, J. 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-Macías, J. 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

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