Russian Mathematics (Iz. VUZ)
Vol. 50, No. 3, pp.7{21, 2006
Izvestiya VUZ. Matematika
UDC 517.444
OPERATORS OF THE MULTIPLICATIVE FOURIER TRANSFORM
M.S. Bespalov
1. Introduction
Let p1 p2 : : : pn : : : be a sequence of integer positive numbers, pn 2 m0 = 1 and mn =
pn mn;1 . Then one can represent any real number x 0 in the form
1
1
X
X
(1)
x = x;k mk;1 + mxk
k=1
k=1 k
where x;k mkx;1 ] (mod pk ), xk xmk ] (mod pk ), 0 x;k , xk pk ; 1, a] is the integer part of
a number a, and the rst sum in (1) contains a nite number of terms. Writing y 0 analogously,
we introduce the group operation as the operation of coordinate-wise summation with respect
to the corresponding module
xy =
1
X
k=1
1 z
X
z;k mk;1 + mk zk xk + yk (mod pk ) z;k x;k + y;k (mod pk ):
k=1
k
We introduce the inverse operation of coordinate-wise subtraction analogously. We dene the
generalized multiplicative functions (x y) with the help of the equality (see 1])
1
X
(x y) = exp 2i x;k yk + xk y;k :
(2)
k=1
R
pk
Denote by D(x ) = (x u)du the Dirichlet kernel of the generalized multiplicative functions
0
(x y). If f 2 L
0 1) then (see 2], p. 129 3]) the function
F (y) =
Z1
0
f (x)(x y)dx
(3)
exists everywhere on 0 1), is g-continuous, and for y ! 1 tends to zero. Denote the space of
these functions, whose norm is dened as the modulo maximum, by L1 0 1). We call the function
F (y) the multiplicative Fourier transform of the function f (x) and denote F f ] = F , where the
operator acts from the space L
0 1) into that L1 0 1).
Remark. In 2], one proposes a denition of generalized multiplicative functions which is more
general than (2). Among the properties of the functions (x y) described in 2]{
4], let us mention
the following: (x y) = x](y)y] (x). Here fn (x)g1
n=0 is a periodic multiplicative orthonormalized
on 0 1] system of functions (one can dene the latter for integer n by formula (2): n (x) = (x n))
The work was supported by the Russian Foundation for Basic Research (project 04-01-00717).
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