On the order of magnitude of Walsh-Fourier transform

Bhikha Lila Ghodadra; Vanda Fülöp

Mathematica Bohemica (2020)

  • Volume: 145, Issue: 3, page 265-280
  • ISSN: 0862-7959

Abstract

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For a Lebesgue integrable complex-valued function f defined on ℝ + : = [ 0 , ∞ ) let f ^ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that f ^ ( y ) → 0 as y → ∞ . 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 ( 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 ℝ + . We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on ( ℝ + ) N , N ∈ ℕ .

How to cite

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Ghodadra, Bhikha Lila, and Fülöp, Vanda. "On the order of magnitude of Walsh-Fourier transform." Mathematica Bohemica 145.3 (2020): 265-280. <http://eudml.org/doc/297010>.

@article{Ghodadra2020,
abstract = {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 $\hat\{f\}(y)\rightarrow 0$ as $y\rightarrow \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 $(\mathbb \{R\}^+)^N$, $N\in \mathbb \{N\}$.},
author = {Ghodadra, Bhikha Lila, Fülöp, Vanda},
journal = {Mathematica Bohemica},
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},
language = {eng},
number = {3},
pages = {265-280},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the order of magnitude of Walsh-Fourier transform},
url = {http://eudml.org/doc/297010},
volume = {145},
year = {2020},
}

TY - JOUR
AU - Ghodadra, Bhikha Lila
AU - Fülöp, Vanda
TI - On the order of magnitude of Walsh-Fourier transform
JO - Mathematica Bohemica
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 145
IS - 3
SP - 265
EP - 280
AB - 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 $\hat{f}(y)\rightarrow 0$ as $y\rightarrow \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 $(\mathbb {R}^+)^N$, $N\in \mathbb {N}$.
LA - eng
KW - 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
UR - http://eudml.org/doc/297010
ER -

References

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