Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume Issue (2012), Pages 116-119
A NOTE ON WALSH-FOURIER COEFFICIENTS
(COMMUNICATED BY SERGEY TIKHONOV)
K. N. DARJI AND R. G. VYAS
Abstract. In this note we have estimate the order of magnitude of WalshFourier coefficients for functions of the class ΛBV (p(n) ↑ ∞, φ).
1. Introduction
In 1949 N. J. Fine [1], using second mean value theorem, proved that if f is of
bounded variation over [0,1] then its Walsh-Fourier coefficients fˆ(n) = O( n1 ). U.
Goginava [2] has studied uniform convergence of Walsh-Fourier series of a function
of BV (p(n), φ). In 2008 [3], the order of magnitude of Walsh-Fourier coefficients of
functions of ΛBV (p) and ϕΛBV are estimated. Here we have estimate the order of
magnitude of Walsh-Fourier coefficients for a function of ΛBV (p(n) ↑ ∞, φ).
Let f be a function defined on (−∞, ∞) with period 1. P is said to be a partition with period 1 if
P : ... < x−1 < x0 < x1 < ... < xm < ...
satisfies xk+m = xk + 1 for k = 0, ±1, ±2, ..., where m is a positive integer.
lim
Definition 1.1. Let φ(n) be a real sequence such that φ(1) ≥ 2 and n→∞
φ(n) =
∞. For a given sequence Λ∑= {λm } (m = 1, 2, . . .) of non-decreasing positive
∞
1
real numbers λm such that
m=1 λm diverges and 1≤p(n)↑ p as n→ ∞, where
1 ≤ p ≤ ∞, we say that f ∈ ΛBV (p(n) ↑ p, φ) (that is, f is a function of p(n)-Λbounded variation over [0,1]) if
VΛ (f, p(n), φ) =
sup sup
1
{ VΛ (P, f, p(n), φ) : ρ{P} ≥
} < ∞,
n≥1 P
φ(n)
where
VΛ (P, f, p(n), φ) = (
m
∑
| f (xk ) − f (xk−1 ) |p(n)
λk
k=1
)1/p(n)
2000 Mathematics Subject Classification. 42C10, 26D15.
Key words and phrases. Walsh-Fourier coefficients and the generalized Wiener class
ΛBV (p(n) ↑ ∞, φ).
c
⃝2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted January 3, 2012. Accepted March 9, 2012.
116
A NOTE ON WALSH-FOURIER COEFFICIENTS
117
and
inf
| xk − xk−1 | .
k
For p = ∞, we denote the class ΛBV (p(n) ↑ ∞, φ) by simply ΛBV (p(n), φ).
ρ{P} =
Note that, if φ(n) = 2n , ∀n, and p = ∞ then one gets the class ΛBV (p(n) ↑ ∞); if
λm =1, ∀m, then one gets the class BV (p(n) ↑ p, φ); if p(n) = p, ∀n, one gets the
class ΛBV (p) .
Let {ϕn } (n ∈ N0 = {0, 1, 2, ...}) denotes the complete orthonormal Walsh system defined on the interval [0,1] in the Paley enumeration, where the subscript
denote the number of zeros (that is, sign-changes) in the interior of the interval
[0,1].
Any x ∈ [0, 1) can be written as
∞
∑
x=
xk 2−(k+1) , each xk = 0 or 1.
k=0
For any x ∈ [0, 1)\Q, there is only one expression of this form, where Q is the class
of dyadic rationals in [0, 1). When x ∈ Q there are two expression of this form, one
which terminates in 0’s and one which terminates in 1’s. For any x, y ∈ [0, 1) their
dyadic sum is defined as
∞
∑
xuy =
|xk − yk |2−(k+1) .
k=0
Observed that, for each n ∈ N0 , ϕn (x u y) = ϕn (x)ϕn (y), x u y ∈
/ Q.
For a 1-periodic function f ∈ L1 [0, 1], its Walsh-Fourier series is defined by
∑
f (x) ∼
fˆ(n)ϕn (x),
where fˆ(n) =
∫1
0
(1.1)
n∈N0
f (x) ϕn (x) dx, ∀ n ∈ N0 , are the Walsh-Fourier coefficients of f .
2. Statement of the result
Here, we prove the following theorem.
Theorem 2.1. If 1-periodic f ∈ ΛBV (p(n) ↑ ∞, φ, [0, 1]), 1 ≤ p(n) ↑ ∞ as n → ∞,
then
1
fˆ(m) = O( ∑m 1 1/p(τ (m)) ),
( j=1 λj )
where
τ (m) = min{k : k ∈ N, φ(k) ≥ m}, m ≥ 1.
(2.1)
Remark 1. Here λn = ∑
1, f or all n, reduces the class ΛBV (p(n), φ) to the class
m
BV (p(n), φ), and O(1/( i=1 λ1i )1/p(τ (m)) ) reduces to O(1/m)1/p(τ (m)) .
We need the following lemma to prove the result.
Lemma 2.1. ([5, Lemma 3.1]). The class ΛBV (p(n) ↑ p, φ, [0, 1]) (1 ≤ p ≤ ∞) ⊆
B[0, 1].
118
K. N. DARJI AND R. G. VYAS
3. Proof of result
Proof of Theorem 2.1 . In view of Lemma 2.1, f ∈ ΛBV (p(n), φ) over [0,1]
implies f is bounded and hence f ∈ L1 [0, 1].
Fix k ∈ N0 and h =
1
.
2k+1
If we put
g(x) = f (x u
1
1
u k+1 ) − f (x), f or all x.
2k
2
Then g ∈ L1 [0, 1]. For m = 2k , ϕm (h) = −1 and ϕm ( 21k ) = 1 implies
1
ĝ(m) = fˆ(m)ϕm ( k )ϕm (h) − fˆ(m) = −2fˆ(m)
2
and
∫
1
2|fˆ(m)| ≤
|f (x u
0
∫
1
1
1
1
1
) u ( k u k+1 )) − f (x u k+1 )|dx
2k+1
2
2
2
|f ((x u
=
0
∫
1
1
u k+1 ) − f (x)|dx
2k
2
1
|f (x u
=
0
1
1
) − f (x u k+1 )|dx.
k
2
2
Similarly, we get
∫
1
2|fˆ(m)| ≤
|f (x u
0
4
3
) − f (x u k+1 )|dx
2k+1
2
and in general we have
∫ 1
2j
(2j − 1)
ˆ
2|f (m)| ≤
|f (x u k+1 ) − f (x u
)|dx, f or all j = 1 to 2k − 1.
2
2k+1
0
Dividing both the sides of the above inequality by λj and summing over j = 1 to
2k − 1, we get
k
2∑
−1
2|fˆ(2k )|(
j=1
1
)≤(
λj
∫
k
−1
1 2∑
0
|fj (x)|
(
j=1
λj
1
1
+
)
p(τ (2k ))
q(τ (2k ))
dx),
2j
) − f (x u (2j−1)
) and q(τ (2k )) is the index conjugate of
where fj (x) = f (x u 2k+1
2k+1
p(τ (2k )). Then by applying Holder’s inequality on the right side we have
∑2k −1
2|fˆ(2k )|( j=1
1
λj )
∫
k
−1
1 2∑
≤
(
0
j=1
|fj (x)|p(τ (2
λj
k
))
)
1/p(τ (2k ))
k
2∑
−1
(
j=1
1 1/q(τ (2k ))
)
dx.
λj
For any x ∈ R, all these points x u 2jh, x u (2j − 1)h, f or j = 1, 2, ..., 2k −
1 lie in the interval of length 1. Thus, f ∈ ΛBV (p(n), φ) over [0,1] implies
A NOTE ON WALSH-FOURIER COEFFICIENTS
∑2k −1 |f (x)|p(τ (2k )) 1/p(τ (2k ))
∑2k
( j=1 j λj
)
= O(1). This together with j=1
and the above inequality implies
119
1
λj
≈
∑2k −1
j=1
1
λj
2
∑
1 −1/p(τ (2k ))
)
).
λ
j=1 j
k
|fˆ(2k )| = O((
This proves the theorem.
Acknowledgements. Authors are grateful to the referee for his valuable comments and suggestion.
References
[1] N.J. Fine, On the Walsh functions, Trans. Amer. Math. Soc., 65 (1949), 372-414.
[2] U. Goginava, On the uniform convergence of Walsh-Fourier series, Acta Math. Hungar, 93
(2001), No.1-2, 59-70.
[3] B. L. Ghodadra and J. R. Patadia, A note on the magnitude of Walsh Fourier coefficients,
J. Inequal. Pure Appl. Math., 9 (2008), No.2, Art.44, 7 pp.
[4] Ferenc Moricz, Absolute convergence of Walsh Fourier series and related results, Analysis
Mathematica, 36 (2010), 275-286.
[5] R. G. Vyas, A note on functions of p(n) − Λ-bounded variation, J. Indian Math. Soc., V.78,
1-4 (2011), 215-220.
K. N. Darji
Department of Science and Humanity, Tatva Institute of Technological Studies, Modasa,
Sabarkantha, Gujarat, India.
E-mail address: darjikiranmsu@gmail.com
R. G. Vyas
Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University
of Baroda, Vadodara-390002, Gujarat, India.
E-mail address: drrgvyas@yahoo.com