On the norm continuity of the hk-fourier transform

Juan H. Arredondo; Francisco J. Mendoza; Alfredo Reyes

doi:10.3934/era.2018.25.005

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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.
[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.
[3]	B. Bongiorno and T. V. Panchapagesan, On the Alexiewicz topology of the Denjoy space, Real Anal. Exchange, 21 (1995/96), 604–614.
[4]	H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press, San Diego, CA, 1972.
[5]	T. H. Hildebrandt, Introduction to the Theory of Integration, Publisher Academic Press, New York, 1963.
[6]	G. Jameson, Sine, cosine and exponential integrals, The Mathematical Gazette, 99 (2015), 276-289. doi: 10.1017/mag.2015.36.
[7]	R. Kannan and C. K. Krueger, Advanced Analysis on the Real Line, Springer-Verlag, Harrisburg, VA, 1996.
[8]	E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 1997.
[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.
[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.
[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.
[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.
[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 L^p($\mathbb{R}$) spaces, Revista de la Unión Matemática Argentina, 57 (2016), 85-94.
[14]	M. Reed and B. Simon, Methods of Modern Analysis, volume Ⅱ: Fourier Analysis, Self Adjointness, Academic Press, 1975.
[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.
[16]	W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.
[17]	E. Talvila, Henstock-Kurzweil Fourier transforms, Ilinois Journal of Mathematics, 46 (2002), 1207-1226.
[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.

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.
[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.
[3]	B. Bongiorno and T. V. Panchapagesan, On the Alexiewicz topology of the Denjoy space, Real Anal. Exchange, 21 (1995/96), 604–614.
[4]	H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press, San Diego, CA, 1972.
[5]	T. H. Hildebrandt, Introduction to the Theory of Integration, Publisher Academic Press, New York, 1963.
[6]	G. Jameson, Sine, cosine and exponential integrals, The Mathematical Gazette, 99 (2015), 276-289. doi: 10.1017/mag.2015.36.
[7]	R. Kannan and C. K. Krueger, Advanced Analysis on the Real Line, Springer-Verlag, Harrisburg, VA, 1996.
[8]	E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 1997.
[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.
[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.
[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.
[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.
[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 L^p($\mathbb{R}$) spaces, Revista de la Unión Matemática Argentina, 57 (2016), 85-94.
[14]	M. Reed and B. Simon, Methods of Modern Analysis, volume Ⅱ: Fourier Analysis, Self Adjointness, Academic Press, 1975.
[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.
[16]	W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.
[17]	E. Talvila, Henstock-Kurzweil Fourier transforms, Ilinois Journal of Mathematics, 46 (2002), 1207-1226.
[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.

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