A Note on Two Classical Theorems of the Fourier Transform for Bounded Variation Functions

Francisco J. Mendoza-Torres


Employing the Henstock-Kurzweil integral, we make simple proofs of the Riemann-Lebesgue lemma and the Dirichlet-Jordan theorem for functions of bounded variation which vanish at infinity.


Riemann-Lebesgue lemma; Dirichlet-Jordan theorem; Bounded variation function; Henstock-Kurzweil integral

Full Text:



M.T. Apostol, Mathematical Analysis, Addison-Wesley Publishing Company, Reading, Massachusetts (1991).

R.G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, Vol. 32, American Mathematical Society, Providence (2001).

Z. Kadelburg and M. Marjanovic, Interchanging two limits, Enseign. Math. 8 (2005), 15–29.


F.J. Mendoza Torres, On pointwise inversion of the Fourier transform of BV0 functions, Ann. Funct. Anal. 2 (2010), 112–120. http://www.emis.de/journals/AFA/AFA-tex_v1_n2_a12.pdf

F.J. Mendoza Torres, J.A. Escamilla Reyna and S. Sánchez Perales, Some results about the Henstock-Kurzweil Fourier transform, Math. Bohem. 134 (4) (2009), 379-386. http://mb.math.cas.cz/full/134/4/mb134_4_5.pdf

F.J. Mendoza Torres and M.G. Morales Macías, On the Convolution Theorem for the Fourier transform of BV0 Functions, J. Class. Anal. 7 (1) (2015), 63–71. doi:10.7153/jca-07-06

I.P. Natanson, Theory of Functions of A Real Variable, Vol. 1, Rev. ed., Frederick Ungar Publishing, New York (1961).

M. Riez and A.E. Livingston, A short proof of a classical theorem in the theory of Fourier integrals, Amer. Math. Montly 62 (1955), 434-437. https://www.researchgate.net/publication/266555117

Ch. Swartz, Introduction to Gauge Integrals, World Scientific Publishing Co., Singapore (2001).

E. Talvila, Henstock-Kurzweil Fourier transforms. Illinois J. Math. 46 (4) (2002), 1207–1226. http://www.math.ualberta.ca/~etalvila/publishedversions/TalvilaIJMfourier.pdf

R.M. Trigub and E.S. Bellinsky, Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers, Dordrecht (2004).

A. Zygmund, Trigonometric Series, Vols. I and II, 3rd edition, Cambridge University Press, Cambridge (2002).

DOI: http://dx.doi.org/10.26713%2Fcma.v7i2.505


  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905