| Title:
|
On the order of magnitude of Walsh-Fourier transform (English) |
| Author:
|
Ghodadra, Bhikha Lila |
| Author:
|
Fülöp, Vanda |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
145 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
265-280 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $(\mathbb R^+)^N$, $N\in \mathbb N$. (English) |
| Keyword:
|
function of bounded variation over $\mathbb R^+$ |
| Keyword:
|
function of bounded variation over $(\mathbb R^+)^2$ |
| Keyword:
|
function of bounded variation over $(\mathbb R^+)^N$ |
| Keyword:
|
order of magnitude |
| Keyword:
|
Riemann-Lebesgue lemma |
| Keyword:
|
Walsh-Fourier transform |
| MSC:
|
26A12 |
| MSC:
|
26A45 |
| MSC:
|
26B30 |
| MSC:
|
26D15 |
| MSC:
|
42C20 |
| idZBL:
|
07250710 |
| idMR:
|
MR4221834 |
| DOI:
|
10.21136/MB.2019.0075-18 |
| . |
| Date available:
|
2020-09-14T15:01:23Z |
| Last updated:
|
2021-04-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148349 |
| . |
| Reference:
|
[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. Zbl 0010.19902, MR 1501762, 10.2307/1989819 |
| Reference:
|
[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. Zbl 0008.00602, MR 1501718, 10.2307/1989593 |
| Reference:
|
[3] Edwards, R. E.: Fourier Series. A Modern Introduction. Vol. 1.Graduate Texts in Mathematics 64. Springer, New York (1979). Zbl 0424.42001, MR 0545506, 10.1007/978-1-4612-6208-4 |
| Reference:
|
[4] Fine, N. J.: On the Walsh functions.Trans. Am. Math. Soc. 65 (1949), 372-414. Zbl 0036.03604, MR 0032833, 10.2307/1990619 |
| Reference:
|
[5] Fülöp, V., Móricz, F.: Order of magnitude of multiple Fourier coefficients of functions of bounded variation.Acta Math. Hung. 104 (2004), 95-104. Zbl 1067.42006, MR 2069964, 10.1023/b:amhu.0000034364.78876.af |
| Reference:
|
[6] Ghodadra, B. L.: Order of magnitude of multiple Fourier coefficients of functions of bounded $p$-variation.Acta Math. Hung. 128 (2010), 328-343. Zbl 1240.42025, MR 2670992, 10.1007/s10474-010-9202-y |
| Reference:
|
[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. Zbl 1275.42044, MR 3100396 |
| Reference:
|
[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. Zbl 1290.42022, MR 3235041, 10.7153/mia-17-52 |
| Reference:
|
[9] Ghodadra, B. L., Fülöp, V.: On the order of magnitude of Fourier transform.Math. Inequal. Appl. 18 (2015), 845-858. Zbl 1321.42012, MR 3344730, 10.7153/mia-18-61 |
| Reference:
|
[10] Ghorpade, S. R., Limaye, B. V.: A Course in Multivariable Calculus and Analysis.Undergraduate Texts in Mathematics. Springer, London (2010). Zbl 1186.26001, MR 2583676, 10.1007/978-1-4419-1621-1 |
| Reference:
|
[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. Zbl 1160.42012, MR 2417326 |
| Reference:
|
[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. |
| Reference:
|
[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). Zbl 0213.40103, MR 0262773, 10.1007/978-3-662-26755-4 |
| Reference:
|
[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 |
| Reference:
|
[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. MR 1504642, 10.24033/bsmf.859 |
| Reference:
|
[16] Móricz, F.: Order of magnitude of double Fourier coefficients of functions of bounded variation.Analysis, München 22 (2002), 335-345. Zbl 1039.42009, MR 1955735, 10.1524/anly.2002.22.4.335 |
| Reference:
|
[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. Zbl 1321.42020, MR 3292741, 10.1016/j.jmaa.2014.12.007 |
| Reference:
|
[18] Natanson, I. P.: Theory of Functions of Real Variable.Frederick Ungar Publishing, New York (1955). Zbl 0064.29102, MR 0067952 |
| Reference:
|
[19] Paley, R. E. A. C.: A remarkable series of orthogonal functions I.Proc. Lond. Math. Soc., II. Ser. 34 (1932), 241-264. Zbl 0005.24806, MR 1576148, 10.1112/plms/s2-34.1.241 |
| Reference:
|
[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). Zbl 0727.42017, MR 1117682 |
| Reference:
|
[21] Taibleson, M.: Fourier coefficients of functions of bounded variation.Proc. Am. Math. Soc. 18 (1967), page 766. Zbl 0181.34004, MR 0212477, 10.2307/2035460 |
| Reference:
|
[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. Zbl 1084.42003, MR 1963498 |
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