A NOTE ON THE MAGNITUDE OF WALSH
FOURIER COEFFICIENTS
B.L. GHODADRA AND J.R. PATADIA
Department of Mathematics
Faculty of Science
The Maharaja Sayajirao University of Baroda
Vadodara - 390 002 (Gujarat), India.
EMail: bhikhu_ghodadra@yahoo.com
vol. 9, iss. 2, art. 44, 2008
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jamanadaspat@gmail.com
Received:
11 March, 2008
Accepted:
07 May, 2008
Communicated by:
L. Leindler
2000 AMS Sub. Class.:
42C10, 26D15.
Key words:
Functions of p−bounded variation, φ−bounded variation, p − Λ−bounded variation and of φ − Λ−bounded variation, Walsh Fourier coefficients, Integral modulus continuity of order p.
Abstract:
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
In this note, the order of magnitude of Walsh Fourier coefficients for functions
of the classes BV (p) (p ≥ 1), φBV , ΛBV (p) (p ≥ 1) and φΛBV is studied.
For the classes BV (p) and φBV, Taibleson-like technique for Walsh Fourier
coefficients is developed.
However, for the classes ΛBV (p) and φΛBV this technique seems to be not
working and hence classical technique is applied. In the case of ΛBV, it is also
shown that the result is best possible in a certain sense.
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1
Introduction
3
2
Results
6
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
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1.
Introduction
It appears that while the study of the order of magnitude of the trigonometric Fourier
coefficients for the functions of various classes of generalized variations such as
BV (p) (p ≥ 1) [9], φBV [2], ΛBV [8], ΛBV (p) (p ≥ 1) [5], φΛBV [4], etc.
has been carried out, such a study for the Walsh Fourier coefficients has not yet
been done. The only result available is due to N.J. Fine [1], who proves, using the
second mean value theorem that, if f ∈ BV [0, 1] then its Walsh Fourier coefficients
fˆ(n) = O( n1 ). In this note we carry out this study. Interestingly, here, no use of
the second mean value theorem is made. We also prove that for the class ΛBV, our
result is best possible in a certain sense.
Definition 1.1. Let I = [a, b], p ≥ 1 be a real number, {λP
k }, k ∈ N, be a se1
quence of non-decreasing positive real numbers such that ∞
k=1 λk diverges and
φ : [0, ∞) → R, be a strictly increasing function. We say that:
1. f ∈ BV (p) (I) (that is, f is of p−bounded variation over I) if

! p1 
 X

V (f, p, I) = sup
|f (bk ) − f (ak )|p
< ∞,

{Ik } 
k
2. f ∈ φBV (I) (that is, f is of φ− bounded variation over I) if
(
)
X
V (f, φ, I) = sup
φ(|f (bk ) − f (ak )|) < ∞,
{Ik }
k
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
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3. f ∈ ΛBV (p) (I) (that is, f is of p − Λ− bounded variation over I) if

!1 
 X |f (b ) − f (a )|p p 
k
k
VΛ (f, p, I) = sup
< ∞,


λ
k
{Ik }
k
4. f ∈ φΛBV (I) (that is, f is of φ − Λ− bounded variation over I) if
)
(
X φ(|f (bk ) − f (ak )|)
< ∞,
VΛ (f, φ, I) = sup
λ
k
{Ik }
k
in which {Ik = [ak , bk ]} is a sequence of non-overlapping subintervals of I.
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
Title Page
In (2) and (4), it is customary to consider φ a convex function such that
φ(0) = 0,
φ(x)
→0
x
(x → 0+ ),
φ(x)
→ ∞ (x → ∞);
x
such a function is necessarily continuous and strictly increasing on [0, ∞).
Let {ϕn } (n = 0, 1, 2, 3, . . . ) denote the complete orthonormal Walsh system
[7], where the subscript denotes the number of zeros (that is, sign-changes) in the
interior of the interval [0, 1]. For a 1-periodic f in L[0, 1] its Walsh Fourier series is
given by
∞
X
f (x) ∼
fˆ(n)ϕn (x),
n=0
where the nth Walsh Fourier coefficient fˆ(n) is given by
Z 1
ˆ
f (n) =
f (x)ϕn (x)dx
(n = 0, 1, 2, 3, . . . ).
0
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The Walsh system can be realized [1] as the full set of characters of the dyadic
group G = Z2∞ , in which Z2 = {0, 1} is the group under addition modulo 2. We
denote the operation of G by +̇. (G, +̇) is identified with ([0, 1], +) under the usual
convention for the binary expansion of elements of [0, 1] [1].
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
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2.
Results
We prove the following theorems. In Theorem 2.5 it is shown that Theorem 2.3 with
p = 1 is best possible in a certain sense.
. 1 Theorem 2.1. If f ∈ BV (p) [0, 1] then fˆ(n) = O 1
np
.
Note. Theorem 2.1 with p = 1 gives the result of Fine [1, Theorem VI].
Theorem 2.2. If f ∈ φBV [0, 1] then fˆ(n) = O(φ−1 (1/n)).
Theorem 2.3. If 1−periodic f ∈ ΛBV (p) [0, 1] (p ≥ 1) then
 ,
! p1 
n
X
1
.
fˆ(n) = O 1
λ
j
j=1
Theorem 2.4. If 1−periodic f ∈ φΛBV [0, 1] then
, n
!!!
X 1
fˆ(n) = O φ−1 1
.
λ
j=1 j
Theorem 2.5. If ΓBV [0, 1] ⊇ ΛBV [0, 1] properly then
, n
!!
X 1
∃f ∈ ΓBV [0, 1] 3 fˆ(n) 6= O 1
.
λ
j
j=1
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
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Proof of Theorem 2.1. Let n ∈ N. Let k ∈ N ∪ {0} be such that 2k ≤ n < 2k+1 and
put ai = (i/2k ) for i = 0, 1, 2, 3, . . . , 2k . Since ϕn takes the value 1 on one half of
each of the intervals (ai−1 , ai ) and the value −1 on the other half, we have
Z ai
ϕn (x)dx = 0, for all i = 1, 2, 3, . . . , 2k .
ai−1
Define a step function g by g(x) = f (ai−1 ) on [ai−1 , ai ), i = 1, 2, 3, . . . , 2k . Then
Z
J.R. Patadia
k
1
g(x)ϕn (x)dx =
0
2
X
Z
ai
f (ai−1 )
i=1
ϕn (x)dx = 0.
vol. 9, iss. 2, art. 44, 2008
ai−1
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Therefore,
|fˆ(n)| =
Z
Contents
1
[f (x) − g(x)]ϕn (x)dx
Z
(2.1)
Walsh Fourier Coefficients
B.L. Ghodadra and
0
1
≤
|f (x) − g(x)|dx
0
≤ ||f − g||p ||1||q
 p1

2k Z a i
X
|f (x) − f (ai−1 )|p dx
=
i=1
ai−1
by Hölder’s inequality as f, g ∈ BV (p) [0, 1] and BV (p) [0, 1] ⊂ Lp [0, 1].
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Hence,
k
|fˆ(n)| ≤
p
2 Z
X
i=1
k
≤
|f (x) − f (ai−1 )|p dx.
ai−1
2 Z
X
i=1
ai
ai
(V (f, p, [ai−1 , ai ]))p dx
ai−1
Walsh Fourier Coefficients
B.L. Ghodadra and
2k
=
X
(V (f, p, [ai−1 , ai ]))p
i=1
1
≤
(V (f, p, [0, 1]))p
2k
2
≤
(V (f, p, [0, 1]))p ,
n
1
2k
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
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which completes the proof of Theorem 2.1.
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I
Proof of Theorem 2.2. Let c > 0. Using Jensen’s inequality and proceeding as in
Theorem 2.1, we get
Z 1
Z 1
φ c
|f (x) − g(x)|dx ≤
φ(c|f (x) − g(x)|)dx
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0
k
=
2 Z
X
i=1
k
≤
φ(c|f (x) − f (ai−1 )|)dx
ai−1
2 Z
X
i=1
Close
ai
ai
ai−1
V (cf, φ, [ai−1 , ai ])dx
k
=
2
X
V (cf, φ, [ai−1 , ai ])
i=1
1
2k
2
V (cf, φ, [0, 1]).
≤
n
Since φ is convex and φ(0) = 0, for sufficiently small c ∈ (0, 1), V (cf, φ, [0, 1]) <
1/2.
This completes the proof of Theorem 2.2 in view of (2.1).
p
(p)
Remark 1. If φ(x) = x , p ≥ 1, then the class φBV coincides with the class BV
and Theorem 2.2 with Theorem 2.1.
Remark 2. Note that in the proof of Theorems 2.1 and 2.2, we have used the fact that
if a = a0 < a1 < · · · < an = b, then
n
X
(V (f, p, [ai−1 , ai ]))p ≤ (V (f, p, [a, b]))p
i=1
and
n
X
V (f, φ, [ai−1 , ai ]) ≤ V (f, φ, [a, b]),
i=1
for any n ≥ 2 (see [2, 1.17, p. 15]). Such inequalities for functions of the class
ΛBV (p) (p ≥ 1) (resp., φΛBV ), which contain BV (p) (resp., φBV ) properly, do not
hold true.
In fact, the following proposition shows that the validity of such inequalities for
the class ΛBV (p) (resp., φΛBV ) virtually reduces the class to BV (p) (resp., φBV ).
Hence we prove Theorem 2.3 and Theorem 2.4 applying a technique different from
the Taibleson-like technique [6] which we have applied in proving Theorem 2.1 and
Theorem 2.2.
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
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Proposition 2.6. Let f ∈ φΛBV [a, b]. If there is a constant C such that
n
X
VΛ (f, φ, [ai−1 , ai ]) ≤ CVΛ (f, φ, [a, b])),
i=1
for any sequence of points {ai }ni=0 with a = a0 < a1 < · · · < an = b, then
f ∈ φBV [a, b].
Proof. For any partition a = x0 < x1 < · · · < xn = b of [a, b], we have
n
X
φ(|f (xi ) − f (xi−1 )|) = λ1
i=1
n
X
i=1
≤ λ1
n
X
φ(|f (xi ) − f (xi−1 )|)
λ1
VΛ (f, φ, [xi−1 , xi ])
≤ λ1 CVΛ (f, φ, [a, b]),
which shows that f ∈ φBV [a, b].
Remark 3. φ(x) = xp (p ≥ 1) in this proposition will give an analogous result for
ΛBV (p) .
To prove Theorem 2.3 and Theorem 2.4, we need the following lemma.
Lemma 2.7. For any n ∈ N, |fˆ(n)| ≤ ωp (1/n; f ), where ωp (δ; f ) (δ > 0, p ≥ 1)
denotes the integral modulus of continuity of order p of f given by
|h|≤δ
1
p
|f (x + h) − f (x)| dx
ωp (δ; f ) = sup
0
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
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i=1
Z
Walsh Fourier Coefficients
B.L. Ghodadra and
p1
.
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Proof. The inequality [1, Theorem IV, p. 382] |fˆ(n)| ≤ ω1 (1/n; f ) and the fact that
ω1 (1/n; f ) ≤ ωp (1/n; f ) for p ≥ 1 immediately proves the lemma.
P
Proof of Theorem 2.3. For any n ∈ N, put θn = nj=1 1/λj . Let f ∈ ΛBV (p) [0, 1].
For 0 < h ≤ 1/n, put k = [1/h]. Then for a given x ∈ R, all the points x + jh, j =
0, 1, . . . , k lie in the interval [x, x + 1] of length 1 and
Z 1
Z 1
p
|f (x) − f (x + h)| dx =
|fj (x)|p dx,
j = 1, 2, . . . , k,
0
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
0
where fj (x) = f (x + (j − 1)h) − f (x + jh), for all j = 1, 2, . . . , k. Since the left
hand side of this equation is independent of j, multiplying both sides by 1/(λj θk )
and summing over j = 1, 2, . . . , k, we get
Z
1
p
|f (x) − f (x + h)| dx ≤
0
1
θk
Z
0
1
k X
|fj (x)|p
j=1
λj
dx
(VΛ (f, p, [0, 1]))p
θk
(VΛ (f, p, [0, 1]))p
≤
,
θn
because {λj } is non-decreasing and 0 < h ≤ 1/n. The case −1/n ≤ h < 0 is
similar and we get using Lemma 2.7,
This proves Theorem 2.3.
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≤
(VΛ (f, p, [0, 1]))p
p
p
ˆ
.
|f (n)| ≤ (ωp (1/n; f )) ≤
θn
vol. 9, iss. 2, art. 44, 2008
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Proof of Theorem 2.4. Let f ∈ φΛBV [0, 1]. Then for h, k and fj (x) as in the proof
of Theorem 2.3 and for c > 0 by Jensen’s inequality,
Z 1
Z 1
φ c
|f (x) − f (x + h)|)dx ≤
φ(c|f (x) − f (x + h)|)dx
0
0
Z 1
=
φ(c|fj (x)|)dx, j = 1, 2, . . . , k.
Walsh Fourier Coefficients
B.L. Ghodadra and
0
Multiplying both sides by 1/(λj θk ) and summing over j = 1, 2, . . . , k, we get
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
Z
φ c
1
|f (x) − f (x + h)|)dx
≤
0
1
θk
Z
0
1
k X
φ(c|fj (x)|)
λj
j=1
dx
VΛ (cf, φ, [0, 1])
θk
VΛ (cf, φ, [0, 1])
≤
.
θn
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≤
Since φ is convex and φ(0) = 0, φ(αx) ≤ αφ(x) for 0 < α < 1. So we may choose
c sufficiently small so that VΛ (cf, φ, [0, 1]) ≤ 1. But then we have
Z 1
1 −1 1
|f (x) − f (x + h)|)dx ≤ φ
.
c
θn
0
Thus it follows in view of Lemma 2.7 that
1
|fˆ(n)| ≤ ω1 (1/n; f ) ≤ φ−1
c
which proves Theorem 2.4.
1
θn
,
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Proof of Theorem 2.5. It is known [3]P
that if ΓBV contains ΛBV properly with Γ =
{γn } then θn 6= O(ρn ), where ρn = nj=1 γ1j for each n. Also, if c0 = 0, cn+1 = 1
and c1 < c2 < · · · < cn denote all the n points of (0, 1) where the function ϕn
changes its sign in (0, 1), n0 ∈ N is such that ρn ≥ 21 for all n ≥ n0 and E = {n ∈
N : n ≥ n0 is even}, then for each n ∈ E, for the function
fn =
n+1
X
(−1)k−1
k=1
4ρn
Walsh Fourier Coefficients
B.L. Ghodadra and
χ[ck−1 ,ck )
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
extended 1-periodically on R,
VΓ (fn , [0, 1]) =
n+1
X
|fn (ck ) − fn (ck−1 )|
k=1
γk
because
fn (cn+1 ) = fn (1) = fn (0) =
=
n
X
1
1
1
·
=
γk 2ρn
2
k=1
1
= fn (cn )
4ρn
as ϕn ≡ 1 on [c0 , c1 ). Hence ||fn || = 4ρ1n + 12 ≤ 1 for each n ∈ E in the Banach
space ΓBV [0, 1] with ||f || = |f (0)| + VΓ (f, [0, 1]). Observe that for f ∈ ΓBV [0, 1]
Z 1
|f (x) − f (0)|
||f ||1 ≤
γ1 + |f (0)| dx ≤ C||f ||,
C = max{1, γ1 },
γ1
0
and hence, for each n ∈ N the linear map Tn : ΓBV [0, 1] → R defined by Tn (f ) =
θn fˆ(n) is bounded as
|Tn (f )| = θn |fˆ(n)| ≤ θn ||f ||1 ≤ θn C||f ||,
∀f ∈ ΓBV [0, 1].
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Next, for each n ∈ E since fn · ϕn =
Tn (fn ) = θn fˆn (n) = θn
Z
0
1
1
4ρn
on [0, 1), we see that
1
fn (x)ϕn (x)dx =
4
θn
ρn
6= O(1)
and hence
sup{||Tn || : n ∈ N} ≥ sup{||Tn || : n ∈ E} ≥ sup{|Tn (fn )| : n ∈ E} = ∞.
Therefore, an application of the Banach-Steinhaus theorem gives an f ∈ ΓBV [0, 1]
such that sup{|Tn (f )| : n ∈ N} = ∞. It follows that θn fˆ(n) = Tn (f ) 6= O(1) and
hence the theorem is proved.
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
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References
[1] N.J. FINE, On the Walsh functions, Trans. Amer. Math. Soc., 65 (1949), 372–
414.
[2] J. MUSIELAK AND W. ORLICZ, On generalized variations (I), Studia Math.,
18 (1959), 11–41.
[3] S. PERLMAN AND D. WATERMAN, Some remarks on functions of
Λ−bounded variation, Proc. Amer. Math. Soc., 74(1) (1979), 113–118.
Walsh Fourier Coefficients
B.L. Ghodadra and
J.R. Patadia
vol. 9, iss. 2, art. 44, 2008
[4] M. SCHRAMM AND D. WATERMAN, On the magnitude of Fourier coefficients, Proc. Amer. Math. Soc., 85 (1982), 407–410.
Title Page
[5] M. SHIBA, On absolute convergence of Fourier series of function of class Λ −
BV (p) , Sci. Rep. Fac. Ed. Fukushima Univ., 30 (1980), 7–12.
[6] M. TAIBLESON, Fourier coefficients of functions of bounded variation, Proc.
Amer. Math. Soc., 18 (1967), 766.
[7] J.L. WALSH, A closed set of normal orthogonal functions, Amer. J. Math., 55
(1923), 5–24.
[8] D. WATERMAN, On convergence of Fourier series of functions of generalized
bounded variation, Studia Math., 44 (1972), 107–117.
[9] N. WIENER, The quadratic variation of a function and and its Fourier coefficients, Mass. J. Math., 3 (1924), 72–94.
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