Sampling Bessel functions and Bessel sampling
Dragana Jankov Maˇ sirevi´ c
★
, Tibor K. Pog´ any
†,‡
,
´
Arp´ ad Baricz
§,‡
and Aur´ el Gal´ antai
‡
★
University of Osijek/ Department of Mathematics, Osijek, Croatia
†
University of Rijeka/ Faculty of Maritime Studies, Rijeka, Croatia
‡
´
Obuda University/ John von Neumann Faculty of Informatics, Budapest, Hungary
§
Babes ¸-Bolyai University/ Department of Economics, Cluj–Napoca, Romania
e-mails: djankov@mathos.hr (D. Jankov Maˇ sirevi´ c); tkpogany@gmail.com (T. K. Pog´ any);
bariczocsi@yahoo.com (
´
A. Baricz) and galantai.aurel@nik.uni-obuda.hu (A. Gal´ antai)
Abstract—The main aim of this article is to establish sum-
mation formulae in form of sampling expansion series for Bessel
functions
, and , and obtain sharp truncation error upper
bounds occurring in the –Bessel sampling series approximation.
The principal derivation tools are the famous sampling theo-
rem by Kramer and various properties of Bessel and modified
Bessel functions which lead to the so–called Bessel sampling when
the sampling nodes of the initial signal function coincide with a
set of zeros of different cylinder functions.
Index Terms—Kramer’s sampling theorem, Bessel functions
of the first and second kind
, , modified Bessel functions of
the first and second kind
, , sampling series expansions, –
Bessel sampling, sampling series truncation error upper bound.
I. I NTRODUCTION
Development of sampling theory has been rapid and contin-
uous since the middle of the 20th century [1]. It is one of the
most important mathematical techniques used in communica-
tion engineering and information theory, and it is also widely
represented in many branches of physics and engineering, such
as signal analysis, image processing, physical chemistry etc.
[2], [3]. Generally speaking, sampling theory can be used in
any discipline where functions need to be reconstructed from
sampled data, usually from the values of the functions and/or
their derivatives at certain points.
In this article we are interested in sampling of Bessel
functions motivated by the immense research on sampling
theory of special functions, especially given by Zayed (see
e.g. [3], [4], [5], [6]). The article is organized as follows. In
the next section we present some new summation formulae for
Bessel functions
,
and
, which we derived by using
Kramer’s sampling theorem and some Zayed’s results. The –
Bessel sampling method was known already by J. Whittaker
[7], Helms and Thomas [8] and Yao [9]. However, we derive
general –Bessel sampling approximation result, using Bessel
function of the second kind
as the building block of the
kernel function, see Theorem 4.
Truncation error upper bound approach in finite uniform
and nonuniform sampling sum approximation, when the input
signal function has negative power polynomial decay rate,
was intensively studied e.g. by Li [10] and by Olenko and
Pog´ any [11], [12], [13]. Helms and Thomas [8] and Jerri
and Joslin [14] reported on truncation error upper bounds
considered for –Bessel sampling for band–limited Hankel
transform exclusively. Applying the –Bessel sampling result
derived in Theorem 4 of the previous section, we establish
sharp truncation error upper bounds for polynomially decaying
input signal functions, compare Theorem 5.
II. SUMMATION FORMULAE FOR
,
AND
In this section, we recall two theorems which will help us to
derive our first set of summation formulae for Bessel functions
,
and
.
First, we recall the theorem established by Kramer in
1959, which is a generalization of the celebrated Whittaker–
Shannon–Kotel’nikov (WKS) sampling theorem [15], [16],
[7].
Theorem A [17]: Let (,) be in
2
( ) as a function of
for each real number , where =[,] is some finite closed
interval, and let = {
}
∈ℤ
be a countable set of real
numbers such that {(,
)}
∈ℤ
is a complete orthogonal
family of functions in
2
( ). If
()=
∫
()(,)d, (1)
for some ∈
2
[,], then admits the sampling expansion
()=
∑
∈ℤ
(
)
★
(),
where
★
()=
∫
(,) (,
)d
∫
∣(,
)∣
2
d
.
Remark 1: The points {
}
∈ℤ
, which are for practical
reasons preferred to be real, can also be a complex numbers
[18, p. 25].
Let us also mention that it is usual, in the sampling literature
to say that a function , having integral representation property
(1) is bandlimited on [,] or has a band–region contained in
[,]. We next recall a summation formula given by Zayed.
– 79 –
8th IEEE International Symposium on Applied Computational Intelligence and Informatics • May 23–25, 2013 • Timisoara, Romania
978-1-4673-6400-3/13/$31.00 ©2013 IEEE