Sampling Bessel functions and Bessel sampling ´ ad Baricz§,‡ and Aur´el Gal´antai‡ Dragana Jankov Maˇsirevi´c⋆ , Tibor K. Pog´any†,‡ , Arp´ ⋆
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:
[email protected] (D. Jankov Maˇsirevi´c);
[email protected] (T. K. Pog´any); ´ Baricz) and
[email protected] (A. Gal´antai)
[email protected] (A. †
Abstract—The main aim of this article is to establish summation formulae in form of sampling expansion series for Bessel functions Yν , Iν and Kν , and obtain sharp truncation error upper bounds occurring in the Y –Bessel sampling series approximation. The principal derivation tools are the famous sampling theorem 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 Jν , Yν , modified Bessel functions of the first and second kind Iν , Kν , sampling series expansions, Y – Bessel sampling, sampling series truncation error upper bound.
I. I NTRODUCTION Development of sampling theory has been rapid and continuous since the middle of the 20th century [1]. It is one of the most important mathematical techniques used in communication 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 Yν , Iν and Kν , which we derived by using Kramer’s sampling theorem and some Zayed’s results. The J– Bessel sampling method was known already by J. Whittaker [7], Helms and Thomas [8] and Yao [9]. However, we derive general Y –Bessel sampling approximation result, using Bessel function of the second kind Yν 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 J–Bessel sampling for band–limited Hankel transform exclusively. Applying the Y –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. S UMMATION FORMULAE FOR Yν , Iν
AND
Kν
In this section, we recall two theorems which will help us to derive our first set of summation formulae for Bessel functions Yν , Iν and Kν . 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 K(x, t) be in L2 (I) as a function of x for each real number t, where I = [a, b] is some finite closed interval, and let E = {tk }k∈Z be a countable set of real numbers such that {K(x, tk )}k∈Z is a complete orthogonal family of functions in L2 (I). If Z b g(x)K(x, t) dx, (1) f (t) = a
2
for some g ∈ L [a, b], then f admits the sampling expansion X f (t) = f (tk )Sk⋆ (t), k∈Z
where Sk⋆ (t) =
Z
a
b
K(x, t)K(x, tk ) dx . Z b 2 |K(x, tk )| dx a
Remark 1: The points {tk }k∈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 f , having integral representation property (1) is bandlimited on [a, b] or has a band–region contained in [a, b]. We next recall a summation formula given by Zayed.
Theorem B [4, p. 701]: If for some g ∈ L2 (0, a) Z a g(x) cos(xt) dx, f (t) =
Theorem 2: For |ν| < Kν (b) =
0
Moreover (2)
e
k∈Z
uniformly on compact t–subsets of R.
At this point we introduce the normalized sinc function: sin(πx) x 6= 0 sinc(x) := . πx 1 x=0
In [4] Zayed also proved a new summation formula involving the Bessel function of the first kind Jν of order ν, using the previous theorem. Precisely, he used an integral representation [19, p. 716, Eq. 6.681.1] Z π x J2ν 2b cos cos(tx) dx = πJν+t (b)Jν−t (b), 2 0 where ℜ{ν} > − 12 and (2) to derive the following summation formula [4, p. 703, Eq. (2.8)] X Jν+t (b)Jν−t (b) = Jν+k+ 12 (b)Jν−k− 12 (b) sinc(t − k − 21 ) k∈Z
1 4.
valid for all ν > For a real number ν, the functions Jν (z) and Yν (z) each have an infinite number of real zeros, all of which are simple with possible exception of z = 0 [20, p. 370]. For non– negative ν the kth positive zero of these functions are denoted by jν,k , yν,k , respectively. Theorem 1: For all ν > − 41 we have X Iν+t (b)Iν−t (b) = Iν+k+ 21 (b)Iν−k− 21 (b) sinc(t − k − 21 ), k∈Z
uniformly on every compact t–subset of R. Moreover, it holds 4 X (−1)k 1 (b)I I Iν2 (b) = ν−(k+ 21 ) (b), π 2k + 1 ν+(k+ 2 )
cos π( ν2 − k) 2X . K2k+1 (b) π 2k + 1 − ν
(4)
k≥0
Proof: Firstly, consider the integral [19, p. 716, Eq. 6.681.3] Z π x cos(tx)I2ν 2b cos dx = πIν−t (b)Iν+t (b), 2 0 which holds for ℜ{ν} > − 12 . Taking g(x) = I2ν 2b cos x2 , which is in L2 (0, π) for ν > − 12 and f (t) = πIν−t (b)Iν+t (b), from (2) it follows the first statement. here t = 0, bearing in mind that here there holds PIf we setP k∈Z = 2 k≥0 , immediately turns out the assertion (3). Finally, setting ν = 21 in (3), we obtain the Bessel function representation of the hyperbolic sine’s square. 2
√ 16 b X (−1)k (2k + 1) K2k+1 (b) , = √ π 3 k≥0 (4k + 1) (4k + 3)
(5)
Proof: By virtue of the formula [19, p. 716, Eq. 6.681.4] Z π2 π cos(νx)Kν (2b cos x) dx = I0 (b)Kν (b), ℜ{ν} < 1, 2 0
choosing g(x) = Kν (2b cos x) and f (ν) = π2 I0 (b)Kν (b), Theorem B ensures the formula (4). Moreover, for ν = 12 by (4) we can see that √ 4 b X (−1)k K2k+1 (b) −b . e =√ 4k + 1 π3 k∈Z
Because of the parity with respect to the order of Kν (t), i.e. K−ν (t) = Kν (t), this implies (5). Finally, it remains the discussion of (5) in the case when b is growing to infinity. Denote √ 16 beb X (−1)k (2k + 1) K2k+1 (b) . R(b) := √ (4k + 1) (4k + 3) π3 k≥0
It is well known [21, p. 202] that r π 1 + O(z −1 ) , Kν (z) = e−z 2z
|z| → ∞ .
Letting b → ∞ above, we have X (−1)k (2k + 1) 16 R(b) = √ 1 + O(b−1 ) (4k + 1) (4k + 3) π 2 k≥0 X 1 1 1 −1 k (−1) + = √ 1 + O(b ) k + 41 k + 43 π 2 k≥0 =
(3)
X (−1)k Ik+1 (b)I−k (b) . sinh2 b = 2b 2k + 1
−b
where (5) holds for large b → ∞ too.
k≥0
and
we have
k∈Z
then
X 1 π sin(at − (k + 21 )π) , f (t) = f k+ 2 a at − (k + 21 )π
1 2
where
1 √ 1 + O(b−1 ) Φ −1, 1, 14 + Φ −1, 1, 34 , π 2 Φ(z, s, a) :=
X
k≥0
zk , (a + k)s
stands for the familiar Lerch transcendent (or Hurwitz–Lerch Zeta–function), defined for all |z| < 1, or |z| = 1, ℜ{s} > 1; a 6∈ Z− 0 = {0, −1, −2, · · · }, see e.g. [22, p. 27, Eq. 1.11 (1)]. Being √ π π π tan + cot = π 2, Φ −1, 1, 14 + Φ −1, 1, 34 = 2 8 8 the proof of Theorem is complete. 2 Below, we derive another summation formula for modified Bessel function of the second kind Iν , which we believe to be new.
Theorem 3: The modified Bessel function of the first kind Iν admits the sampling expansion X (−1)k (π + 2kπ)1−ν Iν (t) = 2 (2t)ν cosh t (π + 2kπ)2 + 4t2 k∈Z (6) × Jν π k + 21 , t ∈ R; ℜ{ν} > 0 .
Moreover
8t X 1 , tanh(πt) = π (2k − 1)2 + 4t2 Proof: For cos(tx) ∈ L2 (0, 1), taking
the orthogonal family of functions {K(x, tk )}k∈Z is complete in L2 (0, 1). Set K(x, t) = cos(tix) ≡ cosh(tx). Now, rewriting the formula [21, p. 79, Eq. (9)] Z 1 1 21−ν tν (1 − x2 )ν− 2 cosh(tx) dx Iν (t) = √ π Γ(ν + 12 ) 0 (where ℜ{ν} > − 21 ) into √ Z 1 π Γ(ν + 12 ) 1 (1 − x2 )ν− 2 cosh(tx) dx, Iν (t) = 21−ν tν 0 1
X (−1)k t1−ν i k
t2k − t2
Iν (tk ) .
where yν,k , k ∈ N are the positive real zeros of the Bessel function Yν (t). √ Proof: For K(x, t) = x Yν (tx) and tk = a−1 yν,k , k positive integer, we have complete orthogonal family of functions {K(x, tk )}k∈N in L2 (0, a). Further, using the integrals [19, p. 624, Eqs. 5.54.1, 5.54.2] Z xYp (αx)Yp (βx) dx
βxYp (αx)Yp−1 (βx) + αxYp−1 (αx)Yp (βx) α2 − β 2 Z o 2 n x 2 2 x [Yp (αx)] dx = [Yp (αx)] − Yp−1 (αx)Yp+1 (αx) , 2 by Kramer’s Theorem A it follows that =
and taking g(x) = (1 − x2 )ν− 2 , which belongs to L2 (0, 1) when ℜ{ν} > 0, by Theorem A it follows that k∈Z
0
k≥1
E = {tk = π(k + 12 ), k ∈ Z} ,
t−ν Iν (t) = cosh t
Theorem 4: Let for some g ∈ L2 (0, a), a > 0 the function f possesses an integral representation Z a √ g(x) x Yν (tx) dx . f (t) =
Then, for all t ∈ R and ν ∈ [0, 1), f admits the sampling expansion X yk f (a−1 yν,k ) f (t) = 2 Yν (at) , (8) 2 (yν,k − a2 t2 ) Yν+1 (yν,k )
t ∈ R.
k≥1
So, it is easy to deduce a new summation formula for Kν , using a summation formulae for Iν and Yν . For that purpose, we derive a new summation formula for Yν .
(7)
Sk⋆ (t) =
a (t2k
2 tk Yν (at) , − t2 ) Yν+1 (atk )
|ℜ{ν}| < 1 .
Using the well–known identities Iν (t) = i−ν Jν (it) and Jν (−t) = (−1)ν Jν (t), after some simplification, from (7), it follows (6). For ν = 12 , expressing J 21 and I 21 via sine and hyperbolic sine, respectively, the expansion (6) becomes X X 4t 8t sinh t = = , 2 2 cosh t (π + 2kπ) + 4t (π + 2kπ)2 + 4t2
Now, bearing in mind the previous considerations, again by Theorem A we deduce the formula (8), where we set ν to be a non–negative real number, because of the positivity of the zeros yν,k . 2
i.e. there holds the stated sampling expansion formula for hyperbolic tangent. 2
Moreover, for t 6= k − 21 , k ∈ N 8t X 1 tan (πt) = . π (2k − 1)2 − 4t2
k∈Z
k≥0
Notice that one can derive a new summation formula for modified Bessel function of the second kind Kν , by using the previous result and connection between Iν and Kν : π (I−ν (t) − Iν (t)) , Kν (t) = 2 sin(νπ) which, for ν = 12 , becomes the well–known identity e
−t
= cosh t − sinh t .
Modified Bessel function of the second kind Kν can also be represented through Iν and Bessel function of the second kind Yν as for noninteger ν 6∈ Z: π(i t)ν Iν (t) π (i t)2ν cos(πν) −1 − Yν (i t) . Kν (t) = 2ν 2 t sin(πν) 2tν
Corollary 4.1: For ℜ{ν} > 0, t ∈ R we have X (2t)ν+1 Yν (2t) Jν ( yν,k ) Yν ( yν,k ) 2 2 Yν (t) Jν (t) = 2 . (9) ν−1 2 2) Y y (y − 4t (y ) ν+1 ν,k ν,k ν,k k≥1
k≥1
Proof: Considering the formula [19, p. 672, Eq. 6.567.10], we conclude Z 1 1 21−ν tν t t Jν ( 2 )Yν ( 2 ) = √ xν (1 − x2 )ν− 2 Yν (tx) dx, π Γ ν + 12 0 1
where t > 0, ℜ{ν} > − 21 , and choose g(x) = (x − x3 )ν− 2 ∈ L2 (0, 1). From the previous theorem, setting a = 2, we obtain the asserted result. For ν = 12 , having in mind that zeros of Y 21 are of the form y 21 ,k = π2 + π(k − 1), k ∈ N, the expansion formula (9) one reduces to well–known partial fraction series expansion of the tangent function. 2
III. T RUNCATION ERROR UPPER BOUNDS IN Y –B ESSEL SAMPLING EXPANSIONS
In this section we would derive uniform upper bound for the truncation error for the Bessel–sampling expansion (8) setting a = 1 for the sake of simplicity. The truncated sampling reconstruction sum of the size N ∈ N, for Bessel–sampling formula (8) we define as SNY (f ; t) = 2 Yν (t)
N X
f (yν,k )
k=1
yν,k , 2 − t2 ) Y (yν,k ν+1 (yν,k )
where t ∈ R, ν ∈ [0, 1) and the function f has a band–region contained in (0, 1). Also, the truncation error of the order N is the quantity TNY (f ; t) = f (t) − SNY (f ; t) , that is X 2 y Y (t) ν,k ν Y . f (yν,k ) 2 TN (f ; t) = 2 (yν,k − t ) Yν+1 (yν,k )
in turn by increasing t 7→ sinh t for t > 0. Hence 2 sup |Yν (t)| ≤ 1 + =: H(t) . πν t ν
1 2
, t 6= 0.
(10)
Thus, for all ν ∈ [0, 1), it is X |Yν (t)| TNY (f ; t) ≤ 2A ; (11) r 2 yν,k |yν,k − t2 | |Yν+1 (yν,k )| k≥N +1
indeed, being ν ≥ 0 all zeros yν,k are positive [20, p. 370]. Having in mind the well known interlacing inequalities for ′ ′ the positive zeros jν,k , jν,k , yν,k and yν,k of Bessel functions ′ ′ Jν (t), Jν (t), Yν (t) and Yν (t), respectively [20, p. 370]
sup TNY (f ; t) ≤ 2A < 2A
|Yν (t)|
X ν 12 and all N ≥ 2 there holds the truncation error upper bound 1
TNY
(f ; t)
