Integrability and uniform convergence of
multiplicative Fourier transforms
Sergey Volosivets
Boris Golubov
Saratov State University
Moscow Institute of Physics and Technology
Astrakhanskaya Str., 83, Saratov, 410028, Russia
Institute Lane 9, Dolgoproudny, 141700,
Email: volosivetsSS@mail.ru
Moscow region, Russia
Email: golubov@mail.mipt.ru
Abstract – For multiplicative Fourier transforms the analogues
of some known results (see [1]-[4]) on uniform convergence and
integrability with power weights of classical Fourier transform are
obtained. Some generalizations of the results of C.W. Onneweer
[5], [6] on belonging of multiplicative Fourier transforms to BesovLipschitz or Herz spaces are formulated.
I. I NTRODUCTION
The problems of integrability and uniform
convergence of classical Fourier transform are
studied in monographs [7] - [11]. The same
problems for the Walsh transform and more general multiplicative transforms are considered in
the monographs [12], [13] and [14]. This article
concerned with these problems for multiplicative
transforms, which where introduced by N.Ya.
Vilenkin [15].
At first we formulate some known results from
the papers [1]-[4] on uniform convergence and
integrability of classical Fourier transform.
R
By ab |df (t)| we denote the variation of the function f on the interval [a, b]. We say that the pair of
functions (f, β) defined on (0, +∞) belongs to the
class GM , if f ∈ V [a, b] for any interval [a, b] ⊂ (0, +∞)
R
and the inequality x2x |df (t)| ≤ Cβ (x) holds for each
x ∈ (0, +∞), where the constant C > 0 is not depend
R
on x. Let us set fˆc (x) = R+ f (y) cos xy dy,
In the paper [1] the following theorem was
proved.
Theorem A. If f ∈ Lloc (R+ ) and f (x) > 0
on (0, +∞) or (f, β) ∈ GM and lim xβ (x) =
x→+∞
R +∞
0, then the integral 0 f (y) cos xy dy converges uniformly
on R+ = [0, +∞) iff the
R +∞
integral 0 f (x) dx converges. In the last case
the following estimate
fc − SM (fˆc )
∞
0
M
Z
f (t)dt+C sup
x≥M/2
2x
|df (t)|
x
R
belongs to the class Lip α on R for α ∈ (0, 1).
We combine their results in the following theorem.
Theorem B. 1) Let α ∈ (0, 1) and the
function f ∈ L1loc (R) satisfies the conditions
Z +∞
f (y) exp (−ixy) dy = O (hα )
1/h
−1/h
t
f (y) cos xy dy.
x
is valid, where the constant C > 0 is not depend
on M > 0.
The similar result for the Fourier sinetransform holds [1].
In the papers [2] and [3] the authors investigated the condition on the function f under
which its Fourier transform
Z
F (f ) (x) =
f (y) exp (−ixy) dy
Z
Z
St fˆc (x) =
Z
≤ sup x>M
−∞
f (y) exp (−ixy) dy = O (hα ) , h > 0 (1)
uniformly on R, where the integrals in (1) are
improper. Then F (f ) ∈ Lip α on R. 2) Let α ∈
(0, 1), f ∈ Lloc (R) and
Z
|xf (x)| dx = O y 1−α , y > 0. (2)
|x|<y
Then f ∈ L (R) and F (f ) ∈ Lip α on R. 3)
If F (f ) ∈ Lip α on R and xf (x) ≥ 0 for all
x ∈ R, then the condition (2) holds.
The statement of the item 1) of this theorem
was proved by G. Sampson and G. Tuy [2] and
the items 2) and 3) by F. Moricz [3].
In the paper [4] E. Liflynd and S. Tikhonov
introduced the class GM ∗ of functions f ∈
Vloc (0, +∞) vanishing at +∞ and satisfying the
condition
Z bx
Z 2x
u−1 |f (u)| du, x > 0,
|df (t)| ≤ C
x
x/b
for some b > 1 and C > 0. They proved the
following theorem.
Theorem C. Let 1 < p, q < ∞, 1 − 1/p0 <
γ < 1/p , where 1/p + 1/p0 = 1, the function
f ∈ GMR∗ is non negative and the integral
+∞
F+ (f ) = 0 f (y) exp (−ixy) dy is considered
in improper sense with singular points 0 and
+∞. Then the following statements are valid:
1) if q ≤ p and x1+γ−1/p−1/q f (x) ∈ Lq (R+ ) ,
then x−γ F+ (f )(x) ∈ Lp (R+ ) ;
2) if p ≤ q and x−γ F+ (f )(x) ∈ Lp (R+ ) ,
then x1+γ−1/p−1/q f (x) ∈ Lq (R+ ) .
In the paper [16] we proved an analog of
Theorem C for multiplicative Fourier transforms
of monotone functions. In this report we generalize that result on the functions f satisfying the
condition
Z +∞
Z +∞
θ−1
|df (t)| ≤ Cx
u−θ |f (u)| du, x > 0,
Onneweer [5], [6] proved the following results.
(We combine the results of these papers in one
theorem).
Theorem D. Let α ∈ R, 1 ≤ p ≤ 2 and
0 < q < +∞. Then the following statements are
valid:
1) if f ∈ Λ (α, p, q), then fˆ ∈ K (α, p0 , q),
where 1/p + 1/p0 =T1;
2) if f ∈ Lp (R+ ) K (α, p, q), then fˆ ∈
Λ (α, p0 , q) ;
3) if f ∈ Λ (α, p, q) and 1 ≤ r ≤ p0 , then
fˆ ∈ K (α + 1/p0T
− 1/r, r, q) ;
4) if f ∈ Lr (R+ ) K (α, p, q) and p0 ≤ r ≤
+∞, then fˆ ∈ Λ (α + 1/r − 1/p, r, q) .
In our report we formulate some analogues of
Theorems A, B and a generalization of Theorem
C for multiplicative Fourier P-transforms. Also
we generalize the Theorem D on Besov-Lipschitz
P-spaces with power weights and Herz P-spaces.
II. BASIC DEFINITIONS
Multiplicative Fourier transform was introduced by N. Ya. Vilenkin [15] as a generalization of the Walsh transform, which had been
defined by N.J. Fine [17]. We shall consider only
symmetric multiplicative Fourier transforms (see
[13], p. 127).
Let be given two-sided symmetric sequence of
natural numbers P = {pj }|j|∈N , where pj ∈ N,
pj ≥ 2, and p−j = pj for j ∈ N. We set mj =
p1 · · · pj , m−j = 1/mj for j ∈ N and p0 = 1.
Then each number x ∈ R+ = [0, +∞) can be
expressed in the form
x=
k(x)
X
j=1
x−j mj−1 +
∞
X
xj m−j ,
(3)
j=1
T
where xj ∈ Z [0, pj ) =: Z (pj ) , T|j| ∈ N.
x
x/b
For x = k/mn , k ∈ Z+ =: Z [0, +∞) we
where b > 1, θ ∈ (0, 1), and the constant C > 0 take the expansion (3) with finite number of nonzero coordinates xj . Then the expansion (3) is
is not depend on x > 0.
For functions belonging to Besov-Lipshitz P- unique for all x ∈ R+ . Below we shall suppose
spaces Λ (α, p, q) or Hertz P-spaces K (α, p, q) that the sequence P = {pj }|j|∈N is bounded, that
(see Definitions 1 or 2 in the Section II) C.W. is pj ≤ N, |j| ∈ N, for some N ∈ N.
Let us define the algebraic operation ⊕ on R+ , is finite, where
Z
which we call addition, by the following way.
g ∗ f (x) =
g (y) f (x ⊕ y) dy
For the numbers x, y ∈ R+ with the expansions
R+
of the form (3) we set x ⊕ y = z, where zj =
xj +yj (mod pj ) , zj ∈ Z (pj ) , |j| ∈ N. Moreover is P-convolution of the functions g and f . For
=
we introduce the kernel χ (x, y) by the equality q = +∞ it
isα assumed that kf kΛ(α,p,+∞)
kf kp + sup mk Dmk − Dmk−1 ∗ f p < ∞.




∞

X
(xj y−j + x−j yj ) /pj  .
χ(x, y) = exp 2πi 


j=1
k∈Z
Definition 2. Let α ∈ R, 1 ≤ p ≤ ∞ and 0 <
q < ∞. We say that the function f ∈ Lp (R+ )
belongs to Herz P-space K (α, p, q), if
!1/q
X
mαk X[m ,m ) f q
kf k
=
< ∞,
The multiplicative Fourier P-transform fˆ of
the function f ∈R L1 (R+ ) is defined by the
k−1
k
K(α,p,q)
p
k∈Z
ˆ
equality f (x) = R+ f (y) χ (x, y) dy. For the
function f ∈
Lp (RR+ ) , 1 < p ≤ 2, we set where XE is indicator function of the set E. For
0
a
fˆ (x) = Lp lim 0 f (y) χ (x, y) dy, where q = +∞ it is assumed that
a→+∞
1/p + 1/p0 = 1. The existence of this limit is
kf kK(α,p,+∞) = sup mαk X[mk−1 ,mk ) f p < ∞.
k∈Z
well-known (see, for example, [13], p. 132).
p
Definitions 1 and 2 were introduced by C.W.
For the function f ∈ L (R+ ) , 1 ≤ p ≤ ∞ ,
its Lp -modulus of continuity is defined by the Onneveer in the paper [5].
Let |x|P = mk for mk−1 ≤ x < mk , k ∈ Z.
equality ω ∗ (f, δ)p = sup kf (·) − f (· ⊕ h)kp ,
0<h≤δ
Then
it is easy to see that in the case q = p we
1/p
R
p
where kf kp = R+ |f (x)|
for 1 ≤ p < ∞, have
1/p
Z
and kf k∞ =
inf
sup |f (x)|.
p
αp
{E:mesE=0} x∈R+ \E
.
|x|P |f (x)| dx
kf kK(α,p,p) =
+∞
R+
Let ω = {ωn }n=0 be monotone decreasing
sequence tending to zero. By Hpω we denote the
III. M AIN RESULTS
class of functions f ∈ Lp (R+ ) , 1 ≤ p ≤ ∞ ,
satisfying the condition ωn (f )p = O (ωn ) , n ∈
Theorem 1. If (f, β)
∈
GM and
Z+ , where ωn (f )p = ω ∗ (f, 1/mn )p . We shall lim tβ (t) = 0, then the improper integral
t→+∞
∞
R +∞
say that the sequence
P∞ ω = {ωn }n=0 belongs to
f (t) χ(x, t) dt converges uniformly on R+
0
the class B, if k=n ωk = O (ωn ) for n ∈ Z+
if and only if f ∈ L (R+ ). In the last case the
and it belongs to the class Bl , l > 0, if estimate
P
n
l
l
!
k=0 mk ωk = O mn ωn , n ∈ Z+ .
Z y
Z +∞
Z 2s
Let us introduce
the
multiplicative
Dirichlet
sup f (t) dt + sup s |df (t)|
Ry
f (t) χ (x, t)dt ≤ C y≥A
s≥A/2N s
A
A
kernel Dy (x) = 0 χ (x, t) dt.
Definition 1. Let α ∈ R, 1 ≤ p ≤ ∞ and 0 <
q < ∞. We say that the function f ∈ Lp (R+ ) is valid, where N is upper bound of {pn }∞
n=1 ,
belongs to Besov-Lipschitz P-space Λ (α, p, q), if the constant C > 0 is not depend on A > 0 and
its norm
x ∈ R+ .
!1/q
This theorem is multiplicative analogue of
X
q
α
Theorem A (see Introduction)
kf kΛ(α,p,q)= kf kp +
mk Dmk−Dmk−1 ∗ f p
k∈Z
Theorem 2. 1) Let f ∈ Lloc (R+ ), the
sequence {ωn }∞
n=1 be decreasing and tending
R
+∞
to zero and the estimate ωn f (t) χ(x, t) dt =
O(ωn ) for n ∈ N, x ∈ R+ holds. Then f ∈
L (R+ ) and fˆ ∈ H1ω . (Here the integral defining
fˆ is assumed as improper with singular point
+∞ ). 2) Let f (t) ≥ 0,
R t ∈ R+ , f ∈ L (R
+)
+∞
ω
ˆ
and f ∈ H1 . Then ωn f (t) χ(x, t) dt =
O(ωn ) for n ∈ N, x ∈ R+ , if ω ∈ B .
In the case ωn = m−α
n , 0 < α < 1, this
theorem is multiplicative analogue of the results
of G. Sampson and G. Tuy [2] and F. Moricz [3]
on classical Fourier transform (see Theorem B
in Introduction).
Corollary 1. LetTf (t) ≥ 0, t ∈ R+ , f ∈
L (R+ ) and ω ∈ B Bl , l ∈ (0, +∞). Then the
following three conditions
R +∞ are equivalent:
1) fR ∈ H1ω ; 2) mn f (t) dt =
O (ωn ) ;
mn l
l
3) 0 t f (t) dt = O mn ωn , n ∈ Z+ .
The function f defined on R+ is said to
be admissible, if the following three conditions
hold: 1) f ∈ L [0, 1) ; 2) f ∈ V [1, +∞),
i.e. f has bounded variation on [1, +∞) ; 3)
lim f (x) = 0. For admissible function f
x→+∞
ˆ
the
R +∞multiplicative Fourier P-transform f (x) =
f (y) χ (x, y)dy exists as improper integral
0
(see [18]). The following theorem generalizes
Theorems 4 and 5 from our paper [16].
Theorem 3. Let f be an admissible non
negative function satisfying the condition
Z +∞
Z +∞
θ−1
t−θ f (t) dt, x > 0,
|df (t)| ≤ Cx
x
x/b
In the case θ = 1 Theorem 3 and Corollary 2
are analogues of the results of E. Liflyand and
S. Tikhonov [4] for Fourier cosine- and sinetransforms.
Let us consider Besov-Lipschitz Pspace Λ (α, p, q, β) with power weight
wa (x) = |x|aP , where |x|P = mi for
mi−1 ≤ x < mi , i ∈ N. The space
Λ (α, p, q, β) consists of functions f ∈ L p (R+ )
with finite norm kf kΛ(α,p,q,β) ≡ kf kp,ωβp +
q/p 1/q
P R
p
α
|ml ∆m l ∗ f (x)| wβp (x) dx
,
l∈Z
R+
where 1 ≤ p < ∞, 0 < q ≤ ∞,
α, β ∈ R, ∆m
R ly(x) = Dml (x) − Dml−1 (x),
Dy (x) =
χ (x, t) dt is multiplicative
0
Dirichlet kernel and
1/p
Z
p
|f (x)| wa (x) dx
.
kf kp,wa =
R+
Theorem 4. Let us assume α ∈ R, 1 ≤
p ≤ 2, 0 < q ≤ ∞, β ∈ [0, 1 − 1/p].
Then the following statements are valid: 1) if
f ∈ Λ (α, p, q, β) , then
fˆ ∈ K (α − β, p0 , q)
T
; 2) if f ∈ Lpwβp (R+ ) K (α, p, q), then fˆ ∈
Λ (α, p0 , q, −β).
Corollary 3. 1) If f ∈ Λ (α, p0 , p, α), where
p ∈ [1, 2], α ∈ [0, 1 − 1/p], then fˆ ∈ Lp (R+ ).
2) If f ∈ Lpwβp (R+ ) , 1 ≤ p ≤ 2, and β ∈
[0, 1 − 1/p], then fˆ ∈ Λ (0, p0 , p, −β).
Corollary 4. If α ∈ [0, 1 − 1/p], p ∈ [1, 2] and
f ∈ Lpwαp (R+ ), then fˆ ∈ K (−α, p0 , 2).
Theorem 5. Let 1 ≤ p ≤ 2, 0 < q <
∞, α ∈ R. Then the following statements are
valid. 1) If p0 ≤ r < T∞, β ∈ [0, 1/r]
0
and f ∈ K (α + β, p, q) Lrwβr0 , then fˆ ∈
Λ (α − 1/p0 + 1/r, r, q, −β). 2) If 1 ≤ r ≤
p0 , β ∈ [0, 1/p0 ] and f ∈ Λ (α, p, q, β), then
fˆ ∈ K (α + 1/p0 − 1/r0 − β, r, q).
Theorems 4 and 5 generalize the results of
C.W. Onneweer [5], [6] (see Theorem D in
Introduction).
(4)
for some θ ∈ (0, 1] , b > 1. 1) If 1 < q ≤ p < ∞,
1/p − θ < γ < 1/p and f (x) xγ+1−1/p−1/q ∈
Lq (R+ ), then x−γ fˆ (x) ∈ Lp (R+ ). 2) If f ∈
L r (R+ ) for some r ∈ [1, 2] , 1 < p ≤ q < ∞,
1/p − θ < γ < 1/p and x−γ fˆ (x) ∈ Lp (R+ ),
then f (x) xγ+1−1/p−1/q ∈ Lq (R+ ).
Corollary 2. Let f be an admissible non
negative function satisfying the condition (4) for
some θ ∈ (0, 1] and f ∈ L r (R+ ), r ∈ [1, 2].
ACKNOWLEDGEMENT
Then for 1 < p < ∞, 1/p − θ < γ <
−γ ˆ
p
1/p, the conditions x f (x) ∈ L (R+ ) and
The work of the first author is supported
γ+1−2/p
p
f (x) x
∈ L (R+ ) are equivalent.
by the Russian Foundation for Basic Research
under grant No 14-01-00417. The work of the
second author is supported by the Program No
1.1320.2014/K of the Ministry of Education and
Science of Russian Federation.
R EFERENCES
[1] M. Dyachenko, E. Liflyand and S. Tikhonov, Uniform
convergence and integrability of Fourier integrals, J. Math.
Anal. Appl., vol. 372, pp. 328-338, 2010.
[2] G. Sampson and G. Tuy, Fourier transforms and their
Lipschitz classes, Pacific J. Math., vol. 75, pp. 519-537,
1978.
[3] F. Moricz. Best possible sufficient conditions for the
Fourier transform to satisfy Lipshitz or Zygmund condition, Studia Math., vol. 199, pp. 199-205, 2010.
[4] E. Liflyand and S. Tikhonov. Extended solution of Boas‘
conjecture on Fourier transforms, C.R. Acad. Sci. Paris,
Ser. 1., vol. 346, pp. 1137-1142, (2008).
[5] C.W. Onneweer. Generalized Lipschitz spaces and Herz
spaces on certain totally disconnected groups, Lecture
Notes in Math., 939, pp. 106-121, 1981.
[6] C.W. Onneweer. The Fourier transform and Herz spaces
on certain groups, Monatsh. Math., vol. 97, pp. 297-310,
1984.
[7] E.C. Titchmarsh, Introduction to the Theory of Fourier
Integrals, Oxford Univ. Press, 1948.
[8] A. Zygmund, Trigonometric series, vol. II, Cambridge
Univ. Press, 1959.
[9] S. Bochner, Lectures on Fourier Integrals, Princeton Univ.
Press, 1959.
[10] P.L. Butzer and R.J. Nessel, Fourier Analysis and Approximation, vol. I, Birkhauser Verlag, Basel & Stuttgart,
1971.
[11] E. Stein and G. Weiss, Introduction to Fourier Analysis
on Euclidean Spaces, Princeton Univ. Press, 1971.
[12] F. Shipp, W.R. Wade and P. Simon, Walsh Series, An
Introduction to Dyadic Harmonic Analysis, Akademiai
Kiado, Budapest, 1990.
[13] B. Golubov, A. Efimov and V. Skvortsov, Walsh Series and
Transforms. Theory and Applications, Kluwer Academic
Publishers, Dordrecht, Boston, London, 1991.
[14] G.N. Agaev, N.Ya. Vilenkin, G. M. Jafarly and A.I. Rubinstein, Muliplicative Function Systems and Harmonic
Analysis on Zero Dimensional Groups (Russian), Elm
Publisher, Baku, 1981.
[15] N.Ya. Vilenkin. On the theory of Fourier integrals on
topological groups (Russian), Mat. sb., vol. 30, no. 2, pp.
233-244, 1952.
[16] B.I. Golubov and S.S. Volosivets. Weighted integrability
of multiplicative Fourier transforms (Russian), Trudy Mat.
Inst. im. V.A. Steklova, 269, pp. 71-81, 2010.
[17] N.G. Fine. The generalized Walsh functions, Trans. Amer.
Math. Soc., vol. 69 pp. 66-77, 1950.
[18] B.I. Golubov and S.S. Volosivets. On the integrability and
uniform convergence of multiplicative Fourier transforms,
Georgian Math. J., vol. 16, no. 3, pp. 533-546, 2009.