-
Previous Article
Explicit geodesics in Gromov-Hausdorff space
- ERA-MS Home
- This Volume
-
Next Article
On the torsion in the center conjecture
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 |
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.
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 Lp($\mathbb{R}$) 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. 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. |
[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 Lp($\mathbb{R}$) 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. Google Scholar |
[1] |
Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020049 |
[2] |
Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014 |
[3] |
Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 |
[4] |
Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020432 |
[5] |
Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020109 |
[6] |
Bao Wang, Alex Lin, Penghang Yin, Wei Zhu, Andrea L. Bertozzi, Stanley J. Osher. Adversarial defense via the data-dependent activation, total variation minimization, and adversarial training. Inverse Problems & Imaging, 2021, 15 (1) : 129-145. doi: 10.3934/ipi.2020046 |
[7] |
Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280 |
[8] |
Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318 |
[9] |
Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 |
[10] |
Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285 |
[11] |
Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020443 |
[12] |
Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117 |
[13] |
Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 |
[14] |
Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300 |
[15] |
Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 |
[16] |
Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333 |
[17] |
Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365 |
[18] |
Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 |
[19] |
Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327 |
[20] |
Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286 |
2019 Impact Factor: 0.5
Tools
Metrics
Other articles
by authors
[Back to Top]