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Keywords:
function of bounded variation over $\mathbb R^+$; function of bounded variation over $(\mathbb R^+)^2$; function of bounded variation over $(\mathbb R^+)^N$; order of magnitude; Riemann-Lebesgue lemma; Walsh-Fourier transform
function of bounded variation over $\mathbb R^+$; function of bounded variation over $(\mathbb R^+)^2$; function of bounded variation over $(\mathbb R^+)^N$; order of magnitude; Riemann-Lebesgue lemma; Walsh-Fourier transform
Summary:
For a Lebesgue integrable complex-valued function $f$ defined on $\mathbb R^+:=[0,\infty )$ let $\hat f$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\hat f(y)\to 0$ as $y\to \infty $. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1(\mathbb R^+)$ there is a definite rate at which the Walsh-Fourier transform tends to zero. We determine this rate for functions of bounded variation on $\mathbb R^+$. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on $(\mathbb R^+)^N$, $N\in \mathbb N$.
References:
[1] Adams, C. R., Clarkson, J. A.: Properties of functions $f(x,y)$ of bounded variation. Trans. Am. Math. Soc. 36 (1934), 711-730 correction ibid. 46 page 468 1939. DOI 10.2307/1989819 | MR 1501762 | Zbl 0010.19902
[2] Clarkson, J. A., Adams, C. R.: On definitions of bounded variation for functions of two variables. Trans. Am. Math. Soc. 35 (1933), 824-854. DOI 10.2307/1989593 | MR 1501718 | Zbl 0008.00602
[3] Edwards, R. E.: Fourier Series. A Modern Introduction. Vol. 1. Graduate Texts in Mathematics 64. Springer, New York (1979). DOI 10.1007/978-1-4612-6208-4 | MR 0545506 | Zbl 0424.42001
[4] Fine, N. J.: On the Walsh functions. Trans. Am. Math. Soc. 65 (1949), 372-414. DOI 10.2307/1990619 | MR 0032833 | Zbl 0036.03604
[5] Fülöp, V., Móricz, F.: Order of magnitude of multiple Fourier coefficients of functions of bounded variation. Acta Math. Hung. 104 (2004), 95-104. DOI 10.1023/b:amhu.0000034364.78876.af | MR 2069964 | Zbl 1067.42006
[6] Ghodadra, B. L.: Order of magnitude of multiple Fourier coefficients of functions of bounded $p$-variation. Acta Math. Hung. 128 (2010), 328-343. DOI 10.1007/s10474-010-9202-y | MR 2670992 | Zbl 1240.42025
[7] Ghodadra, B. L.: Order of magnitude of multiple Walsh-Fourier coefficients of functions of bounded $p$-variation. Int. J. Pure Appl. Math. 82 (2013), 399-408. MR 3100396 | Zbl 1275.42044
[8] Ghodadra, B. L.: An application of Jensen's inequality in determining the order of magnitude of multiple Fourier coefficients of functions of bounded $\phi$-variation. Math. Inequal. Appl. 17 (2014), 707-718. DOI 10.7153/mia-17-52 | MR 3235041 | Zbl 1290.42022
[9] Ghodadra, B. L., Fülöp, V.: On the order of magnitude of Fourier transform. Math. Inequal. Appl. 18 (2015), 845-858. DOI 10.7153/mia-18-61 | MR 3344730 | Zbl 1321.42012
[10] Ghorpade, S. R., Limaye, B. V.: A Course in Multivariable Calculus and Analysis. Undergraduate Texts in Mathematics. Springer, London (2010). DOI 10.1007/978-1-4419-1621-1 | MR 2583676 | Zbl 1186.26001
[11] Ghodadra, B. L., Patadia, J. R.: A note on the magnitude of Walsh Fourier coefficients. JIPAM, J. Inequal. Pure Appl. Math. 9 (2008), Article No. 44, 7 pages. MR 2417326 | Zbl 1160.42012
[12] Hardy, G. H.: On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Quart. J. 37 (1905), 53-79 \99999JFM99999 36.0501.02.
[13] Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups. Die Grundlehren der mathematischen Wissenschaften 152. Springer, New York (1970). DOI 10.1007/978-3-662-26755-4 | MR 0262773 | Zbl 0213.40103
[14] Hobson, E. W.: The Theory of Functions of a Real Variable and the Theory of Fourier's Series. Vol. I. University Press, Cambridge (1927),\99999JFM99999 53.0226.01. MR 0092828
[15] Lebesgue, H.: Sur la répresentation trigonométrique approchée des fonctions satisfaisant à une condition de Lipschitz. Bull. Soc. Math. Fr. 38 French (1910), 184-210 \99999JFM99999 41.0476.02. DOI 10.24033/bsmf.859 | MR 1504642
[16] Móricz, F.: Order of magnitude of double Fourier coefficients of functions of bounded variation. Analysis, München 22 (2002), 335-345. DOI 10.1524/anly.2002.22.4.335 | MR 1955735 | Zbl 1039.42009
[17] Móricz, F.: Pointwise convergence of double Fourier integrals of functions of bounded variation over $\Bbb R^2$. J. Math. Anal. Appl. 424 (2015), 1530-1543. DOI 10.1016/j.jmaa.2014.12.007 | MR 3292741 | Zbl 1321.42020
[18] Natanson, I. P.: Theory of Functions of Real Variable. Frederick Ungar Publishing, New York (1955). MR 0067952 | Zbl 0064.29102
[19] Paley, R. E. A. C.: A remarkable series of orthogonal functions I. Proc. Lond. Math. Soc., II. Ser. 34 (1932), 241-264. DOI 10.1112/plms/s2-34.1.241 | MR 1576148 | Zbl 0005.24806
[20] Schipp, F., Wade, W. R., Simon, P.: Walsh Series. An Introduction to dyadic Harmonic Analysis. With the Assistance of J. Pál. Adam Hilger, Bristol (1990). MR 1117682 | Zbl 0727.42017
[21] Taibleson, M.: Fourier coefficients of functions of bounded variation. Proc. Am. Math. Soc. 18 (1967), page 766. DOI 10.2307/2035460 | MR 0212477 | Zbl 0181.34004
[22] Zygmund, A.: Trigonometric Series. Volumes I and II combined. With a foreword by Robert Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2002),\99999DOI99999 10.1017/CBO9781316036587 \newpage. MR 1963498 | Zbl 1084.42003
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