Approximation of analytic functions by Bessel functions of fractional order
We solve the inhomogeneous Bessel differential equation $$x^2y''(x) + xy'(x) + (x^2 - \nu^2)y(x) = \sum^{\infty}_{m=0} a_mx^m$$, where $\nu$ is a positive nonintegral number, and use this result for the approximation of analytic functions of a special type by the Bessel fu...
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2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508819435552768 |
|---|---|
| author | Jung, S.-M. Юнг, С. М. |
| author_facet | Jung, S.-M. Юнг, С. М. |
| author_sort | Jung, S.-M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:37:39Z |
| description | We solve the inhomogeneous Bessel differential equation
$$x^2y''(x) + xy'(x) + (x^2 - \nu^2)y(x) = \sum^{\infty}_{m=0} a_mx^m$$,
where $\nu$ is a positive nonintegral number, and use this result for the approximation of analytic functions of a special type by the Bessel functions of fractional order. |
| first_indexed | 2026-03-24T02:31:16Z |
| format | Article |
| fulltext |
UDC 517.5
S.-M. Jung (College Sci. and Technology, Hongik Univ., Korea)
APPROXIMATION OF ANALYTIC FUNCTIONS
BY BESSEL FUNCTIONS OF FRACTIONAL ORDER*
НАБЛИЖЕННЯ АНАЛIТИЧНИХ ФУНКЦIЙ
ФУНКЦIЯМИ БЕССЕЛЯ ДРОБОВОГО ПОРЯДКУ
We solve the inhomogeneous Bessel differential equation
x2y′′(x) + xy′(x) + (x2 − ν2)y(x) =
∞∑
m=0
amx
m,
where ν is a positive nonintegral number, and use this result for the approximation of analytic functions of a
special type by the Bessel functions of fractional order.
Розв’язано неоднорiдне диференцiальне рiвняння Бесселя
x2y′′(x) + xy′(x) + (x2 − ν2)y(x) =
∞∑
m=0
amx
m,
де ν — нецiле додатне число. Отриманi результати застосовано до наближення аналiтичних функцiй
спецiального виду функцiями Бесселя дробового порядку.
1. Introduction. The stability problem for functional equations starts from the famous
talk of Ulam and the partial solution of Hyers to the Ulam’s problem (see [6, 27]).
Thereafter, Rassias [24] attempted to solve the stability problem of the Cauchy additive
functional equation in a more general setting.
The stability concept introduced by Rassias’ theorem significantly influenced a num-
ber of mathematicians to investigate the stability problems for various functional equa-
tions (see [2 – 4, 5, 7, 8, 25] and the references therein).
Assume that Y is a normed space and I is an open subset of R. If for any function
f : I → Y satisfying the differential inequality
‖an(x)y(n)(x) + an−1(x)y
(n−1)(x) + . . .+ a1(x)y
′(x) + a0(x)y(x) + h(x)‖ ≤ ε
for all x ∈ I and for some ε ≥ 0, there exists a solution f0 : I → Y of the differential
equation
an(x)y
(n)(x) + an−1(x)y
(n−1)(x) + . . .+ a1(x)y
′(x) + a0(x)y(x) + h(x) = 0
such that ‖f(x) − f0(x)‖ ≤ K(ε) for any x ∈ I, where K(ε) depends on ε only, then
we say that the above differential equation satisfies the Hyers – Ulam stability (or the
local Hyers – Ulam stability if the domain I is not the whole space R). We may apply
these terminologies for other differential equations. For more detailed definition of the
Hyers – Ulam stability, refer to [3, 4, 6 – 9, 24, 25].
Obłoza seems to be the first author who has investigated the Hyers – Ulam stability
of linear differential equations (see [22, 23]). Here, we will introduce a result of Alsina
and Ger (see [1]): If a differentiable function f : I → R is a solution of the differential
inequality |y′(x) − y(x)| ≤ ε, where I is an open subinterval of R, then there exists a
constant c such that |f(x)− cex| ≤ 3ε for any x ∈ I.
*This research was supported by Basic Science Research Program through the National Research Foundation
of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2010-0007143).
c© S.-M. JUNG, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12 1699
1700 S.-M. JUNG
This result of Alsina and Ger has been generalized by Takahasi, Miura and Miyajima:
They proved in [26] that the Hyers – Ulam stability holds for the Banach space valued
differential equation y′(x) = λy(x) (see also [19]).
Using the conventional power series method, the author has investigated the general so-
lution of the inhomogeneous Legendre differential equation under some specific condition
and this result was applied to prove the Hyers – Ulam stability of the Legendre differential
equation (see [10]). In a recent paper, he has also investigated an approximation property
of analytic functions by the Legendre functions (see [14]). This study has been continued
to various special functions including the Airy functions, the exponential functions, the
Hermite functions, and the power functions (see [11 – 13, 15]).
Recently, the author and Kim tried to prove the Hyers – Ulam stability of the Bessel’s
differential equation
x2y′′(x) + xy′(x) + (x2 − ν2)y(x) = 0. (1)
However, the obtained theorem unfortunately does not describe the Hyers – Ulam stability
of the Bessel’s differential equation in a strict sense (see [16]).
In Section 2 of this paper, by using the ideas from [14], we will determine the general
solution of the inhomogeneous Bessel’s differential equation
x2y′′(x) + xy′(x) + (x2 − ν2)y(x) =
∞∑
m=0
amx
m, (2)
where the parameter ν is a positive nonintegral number. Section 3 will be devoted to the
investigation of an approximation property of the Bessel functions.
Throughout this paper, we denote by [x] the largest integer not exceeding x for any
x ∈ R and we define Iρ = (−ρ, 0) ∪ (0, ρ) for any ρ > 0.
2. Inhomogeneous Bessel’s differential equation. A function is called a Bessel
function (of fractional order) if it is a solution of the Bessel’s differential equation (1),
where ν is a positive nonintegral number. The Bessel’s differential equation plays a great
role in physics and engineering. In particular, this equation is most useful for treating the
boundary value problems exhibiting cylindrical symmetries.
The convergence of the power series
∑∞
m=0
amx
m seems not to guarantee the exis-
tence of solutions to the inhomogeneous Bessel’s differential equation (2). Thus, we adopt
an additional condition to ensure the existence of solutions to the equation.
Theorem 2.1. Let ν be a positive nonintegral number and let ρ be a positive
constant. Assume that the radius of convergence of power series
∑∞
m=0
amx
m is at
least ρ and there exists a constant σ > 0 satisfying the condition
|am+2| ≤
m2
σ2
|cm| (3)
for all sufficiently large integers m, where
cm =
−
[m/2]∑
i=0
a2i
[m/2]∏
j=i
1
ν2 − (2j)2
( for even m),
−
[m/2]∑
i=0
a2i+1
[m/2]∏
j=i
1
ν2 − (2j + 1)2
( for odd m)
(4)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
APPROXIMATION OF ANALYTIC FUNCTIONS BY BESSEL FUNCTIONS OF FRACTIONAL . . . 1701
for all m ∈ N0. Let ρ0 = min{ρ, σ}. Then every solution y : Iρ0 → C of the Bessel’s
differential equation (2) can be expressed by
y(x) = yh(x) +
∞∑
m=0
cmx
m (5)
for all x ∈ Iρ0 , where yh(x) is a solution of the homogeneous Bessel’s equation (1).
Proof. We assume that y : Iρ0 → C is a function given in the form (5) and we define
yp(x) = y(x)− yh(x) =
∑∞
m=0
cmx
m. Then, it follows from (3) and (4) that
lim
m→∞
∣∣∣∣cm+2
cm
∣∣∣∣ = lim
m→∞
1
(m+ 2)2 − ν2
∣∣∣∣am+2
cm
− 1
∣∣∣∣ ≤ 1
σ2
,
since we can deduce the relation cm+2 =
am+2 − cm
(m+ 2)2 − ν2
from (4) by some manipulations.
That is, the power series for yp(x) converges for all x ∈ Iρ0 .Hence, we see that the domain
of y(x) is well defined.
We now prove that the function yp(x) satisfies the inhomogeneous equation (2). Indeed,
it follows from (4) that
x2y′′p (x) + xy′p(x) + (x2 − ν2)yp(x) =
=
∞∑
m=2
m(m− 1)cmx
m +
∞∑
m=1
mcmx
m +
∞∑
m=0
cmx
m+2 −
∞∑
m=0
ν2cmx
m =
= c1x− ν2c0 − ν2c1x+
∞∑
m=2
[
cm−2 + (m2 − ν2)cm
]
xm =
= a0 + a1x+
∞∑
m=2
amx
m,
since we obtain
c0 = − 1
ν2
a0, c1 =
1
1− ν2
a1, cm−2 + (m2 − ν2)cm = am for m ≥ 2,
which proves that yp(x) is a particular solution of the inhomogeneous equation (2).
On the other hand, since every solution to (2) can be expressed as a sum of a solution
yh(x) of the homogeneous equation and a particular solution yp(x) of the inhomogeneous
equation, every solution of (2) is certainly of the form (5).
Theorem 2.1 is proved.
3. Approximate Bessel’s differential equation. In this section, assume that ν is a
positive nonintegral number and ρ is a positive constant. For a given K ≥ 0, we denote
by CK the set of all functions y : Iρ → C with the properties (a) and (b):
(a) y(x) is expressible by a power series
∑∞
m=0
bmx
m whose radius of convergence
is at least ρ;
(b)
∑∞
m=0
|amxm| ≤ K
∣∣∣∑∞
m=0
amx
m
∣∣∣ for any x ∈ Iρ, where am = bm−2 +
+ (m2 − ν2)bm for all m ∈ N0 and set b−2 = b−1 = 0.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1702 S.-M. JUNG
For a positive nonintegral number ν, define
Me(x) = max
k∏
j=i
x2
|ν2 − (2j)2|
: 0 ≤ i ≤ k ≤ µ
,
Mo(x) = max
k∏
j=i
x2
|ν2 − (2j + 1)2|
: 0 ≤ i ≤ k ≤ µ
,
M(x) = max{Me(x), Mo(x), 1},
where µ =
[√
ν2 + x2/2
]
, and
Lν =
∞∑
m=0
1
(m− ν)2
<∞.
We remark that M(x)→ 1 as |x| → 0.
We will now solve the approximate Bessel’s differential equations in a special class of
analytic functions, CK .
Theorem 3.1. Let ν be a positive nonintegral number and let p be a nonnegative
integer with p < ν < p + 1. Assume that a function y ∈ CK satisfies the differential
inequality ∣∣x2y′′(x) + xy′(x) + (x2 − ν2)y(x)
∣∣ ≤ ε (6)
for all x ∈ Iρ and for some ε ≥ 0. If the sequence {bm} satisfies the condition
bm+2 = O(bm) as m→∞ (7)
with a Landau constant C ≥ 0, then there exists a solution yh(x) of the Bessel’s differ-
ential equation (1) such that∣∣y(x)− yh(x)∣∣ ≤ KLνM(x)ε
for any x ∈ Iρ0 , where ρ0 = min
{
ρ, 1/
√
C∗
}
and C∗ is a positive number larger than
C. If C and ρ are sufficiently small and large respectively, then
M(x) ≤ max
{
|x||x|+2
|ν2 − p2||x|/2+1
,
|x||x|+2
|ν2 − (p+ 1)2||x|/2+1
}
for all sufficiently large |x|.
Proof. Since y belongs to CK , it follows from (a) and (b) that
x2y′′(x)+xy′(x)+(x2−ν2)y(x) =
∞∑
m=0
[
bm−2+(m2−ν2)bm
]
xm =
∞∑
m=0
amx
m (8)
for all x ∈ Iρ. By considering (6) and (8), we get∣∣∣∣∣
∞∑
m=0
amx
m
∣∣∣∣∣ ≤ ε
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
APPROXIMATION OF ANALYTIC FUNCTIONS BY BESSEL FUNCTIONS OF FRACTIONAL . . . 1703
for any x ∈ Iρ. This inequality, together with (b), yields that
∞∑
m=0
|amxm| ≤ K
∣∣∣∣∣
∞∑
m=0
amx
m
∣∣∣∣∣ ≤ Kε (9)
for each x ∈ Iρ.
Now, it follows from (b) that
n∑
i=0
a2i
n∏
j=i
1
ν2 − (2j)2
=
=
n∑
i=0
b2i−2
n∏
j=i
1
ν2 − (2j)2
−
n∑
i=0
b2i
n∏
j=i+1
1
ν2 − (2j)2
=
=
n−1∑
i=−1
b2i
n∏
j=i+1
1
ν2 − (2j)2
−
n∑
i=0
b2i
n∏
j=i+1
1
ν2 − (2j)2
=
= b−2
n∏
j=0
1
ν2 − (2j)2
− b2n = −b2n,
since b−2 = 0. Similarly, we obtain
n∑
i=0
a2i+1
n∏
j=i
1
ν2 − (2j + 1)2
= −b2n+1
for all n ∈ N0, i.e.,
bm =
−
[m/2]∑
i=0
a2i
[m/2]∏
j=i
1
ν2 − (2j)2
(for even m),
−
[m/2]∑
i=0
a2i+1
[m/2]∏
j=i
1
ν2 − (2j + 1)2
(for odd m)
(10)
for all m ∈ N0.
On the other hand, by (b) and (7), we have
|am+2| =
∣∣bm +
[
(m+ 2)2 − ν2
]
bm+2
∣∣ ≤
≤ |bm|+ (m+ 2)2|bm+2| ≤
m2
(1/
√
C∗)2
|bm| (11)
for all sufficiently large integers m, where C∗ is a positive number larger than C. Hence,
in view of (10) and (11), the condition (3) is satisfied with σ = 1/
√
C∗. Moreover, we
know that the radius of convergence of power series
∑∞
m=0
amx
m is at least ρ because
the convergence radius of the power series expression for y(x) is at least ρ (see (a) and (8)).
According to (8) and Theorem 2.1, there exists a solution yh(x) of the homogeneous
Bessel’s differential equation (1) satisfying (5) for all x ∈ Iρ0 . Thus, it follows from (4)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1704 S.-M. JUNG
and (10) that
|y(x)− yh(x)| =
∣∣∣∣∣
∞∑
n=0
c2nx
2n +
∞∑
n=0
c2n+1x
2n+1
∣∣∣∣∣ =
=
∣∣∣∣∣∣−
∞∑
n=0
x2n
n∑
i=0
a2i
n∏
j=i
1
ν2 − (2j)2
−
∞∑
n=0
x2n+1
n∑
i=0
a2i+1
n∏
j=i
1
ν2 − (2j + 1)2
∣∣∣∣∣∣ =
=
∣∣∣∣∣∣
∞∑
n=0
1
ν2 − (2n)2
n∑
i=0
a2ix
2i
n−1∏
j=i
x2
ν2 − (2j)2
+
+
∞∑
n=0
1
ν2 − (2n+ 1)2
n∑
i=0
a2i+1x
2i+1
n−1∏
j=i
x2
ν2 − (2j + 1)2
∣∣∣∣∣∣
for any x ∈ Iρ0 . Moreover, we have
|y(x)− yh(x)| ≤
≤
µ+1∑
n=0
1
|ν2 − (2n)2|
n∑
i=0
∣∣a2ix2i∣∣ n−1∏
j=i
x2
|ν2 − (2j)2|
+
+
∞∑
n=µ+2
1
|ν2 − (2n)2|
µ∑
i=0
∣∣a2ix2i∣∣ n−1∏
j=i
x2
|ν2 − (2j)2|
+
+
∞∑
n=µ+2
1
|ν2 − (2n)2|
n∑
i=µ+1
∣∣a2ix2i∣∣ n−1∏
j=i
x2
|ν2 − (2j)2|
+
+
µ+1∑
n=0
1
|ν2 − (2n+ 1)2|
n∑
i=0
∣∣a2i+1x
2i+1
∣∣ n−1∏
j=i
x2
|ν2 − (2j + 1)2|
+
+
∞∑
n=µ+2
1
|ν2 − (2n+ 1)2|
µ∑
i=0
∣∣a2i+1x
2i+1
∣∣ n−1∏
j=i
x2
|ν2 − (2j + 1)2|
+
+
∞∑
n=µ+2
1
|ν2 − (2n+ 1)2|
n∑
i=µ+1
∣∣a2i+1x
2i+1
∣∣ n−1∏
j=i
x2
|ν2 − (2j + 1)2|
(12)
for all x ∈ Iρ0 , where µ =
[√
ν2 + x2/2
]
.
We know that
x2
|ν2 − (2j)2|
< 1 and
x2
|ν2 − (2j + 1)2|
< 1 for j ≥ µ + 1 and
x2
|ν2 − (2j)2|
or
x2
|ν2 − (2j + 1)2|
is not perhaps less than 1 for j ≤ µ. Then, we have
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
APPROXIMATION OF ANALYTIC FUNCTIONS BY BESSEL FUNCTIONS OF FRACTIONAL . . . 1705
n−1∏
j=i
x2
|ν2 − (2j)2|
=
µ∏
j=i
x2
|ν2 − (2j)2|
n−1∏
j=µ+1
x2
|ν2 − (2j)2|
≤Me(x) ≤M(x)
for all n ≥ µ+ 2 and i = 0, 1, . . . , µ. Similarly, we get
n−1∏
j=i
x2
|ν2 − (2j + 1)2|
≤M(x)
for all n ≥ µ+ 2 and i = 0, 1, . . . , µ.
Thus, it follows from (9) and (12) that
|y(x)− yh(x)| ≤
µ+1∑
n=0
M(x)
|ν2 − (2n)2|
n−1∑
i=0
∣∣a2ix2i∣∣+
+
µ+1∑
n=0
∣∣a2nx2n∣∣
|ν2 − (2n)2|
+
∞∑
n=µ+2
M(x)
|ν2 − (2n)2|
µ∑
i=0
∣∣a2ix2i∣∣+
+
∞∑
n=µ+2
1
|ν2 − (2n)2|
n∑
i=µ+1
∣∣a2ix2i∣∣+
+
µ+1∑
n=0
M(x)
|ν2 − (2n+ 1)2|
n−1∑
i=0
∣∣a2i+1x
2i+1
∣∣+ µ+1∑
n=0
∣∣a2n+1x
2n+1
∣∣
|ν2 − (2n+ 1)2|
+
+
∞∑
n=µ+2
M(x)
|ν2 − (2n+ 1)2|
µ∑
i=0
∣∣a2i+1x
2i+1
∣∣+
+
∞∑
n=µ+2
1
|ν2 − (2n+ 1)2|
n∑
i=µ+1
∣∣a2i+1x
2i+1
∣∣ ≤
≤M(x)
( ∞∑
n=0
1
|ν2 − (2n)2|
n∑
i=0
∣∣a2ix2i∣∣+
+
∞∑
n=0
1
|ν2 − (2n+ 1)2|
n∑
i=0
∣∣a2i+1x
2i+1
∣∣) ≤
≤M(x)Kε
∞∑
m=0
1
|ν2 −m2|
≤ KLνM(x)ε
for x ∈ Iρ0 .
Finally, assume that C is sufficiently small and ρ is sufficiently large. Then ρ0 is
sufficiently large. For any j ≥ 0 and x ∈ Iρ0 , we have
x2
|ν2 − (2j)2|
≤ max
{
x2
|ν2 − p2|
,
x2
|ν2 − (p+ 1)2|
}
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1706 S.-M. JUNG
If |x| is so large that
µ =
[
1
2
√
ν2 + x2
]
=
[
|x|
2
]
≤ |x|
2
,
then it follows from the definition of M(x) that
M(x) ≤ max
{
|x||x|+2
|ν2 − p2||x|/2+1
,
|x||x|+2
|ν2 − (p+ 1)2||x|/2+1
}
for all sufficiently large |x|.
Theorem 3.1 is proved.
If ν is large enough, then we can prove the (local) Hyers – Ulam stability of the Bessel’s
differential equation (1) as we see in the following corollary.
Corollary 3.1. Let ν be a positive nonintegral number and let p be a nonnegative
integer with p < ν < p + 1. Assume that a function y ∈ CK satisfies the differential
inequality (6) for all x ∈ Iρ and for some ε ≥ 0. Suppose the sequence {bm} satisfies
the condition (7) with a Landau constant C ≥ 0 and define ρ0 = min
{
ρ, 1/
√
C∗
}
for
a positive number C∗ > C. If
x2
|ν2 − p2|
≤ 1 and
x2
|ν2 − (p+ 1)2|
≤ 1
for all x ∈ Iρ0 , then there exists a solution yh(x) of the Bessel’s differential equation
(1) such that
|y(x)− yh(x)| ≤ KLνε
for any x ∈ Iρ0 .
Proof. For any j ≥ 0 and x ∈ Iρ0 , we have
max
{
x2
|ν2 − (2j)2|
,
x2
|ν2 − (2j + 1)2|
}
≤ max
{
x2
|ν2 − p2|
,
x2
|ν2 − (p+ 1)2|
}
≤ 1.
Thus, we get
M(x) ≤ max
{
x2
|ν2 − p2|
,
x2
|ν2 − (p+ 1)2|
, 1
}
= 1,
and the assertion is true due to Theorem 3.1.
4. Examples. We will show that there exist functions y(x) which satisfy all the
conditions given in Theorem 3.1 and Corollary 3.1: Let us define a function y : I10 → R
by
y(x) = J100.5(x) + cx2 =
∞∑
m=0
bmx
m, (13)
where J100.5(x) is the Bessel function of the first kind of order 100.5, n is a positive
integer, and c is a constant satisfying
c =
ε
999625
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
APPROXIMATION OF ANALYTIC FUNCTIONS BY BESSEL FUNCTIONS OF FRACTIONAL . . . 1707
for some ε > 0. It is obvious that the convergence radius of the power series
∑∞
m=0
bmx
m
is infinity. (So we can set ρ = 10.) In fact, the infinite series
∑∞
m=0
bm11m converges.
So we have
lim sup
m→∞
∣∣∣∣11m+1bm+1
11mbm
∣∣∣∣ ≤ 1 and lim sup
m→∞
∣∣∣∣11m+2bm+2
11m+1bm+1
∣∣∣∣ ≤ 1.
Thus, it holds true that
lim sup
m→∞
∣∣∣∣bm+2
bm
∣∣∣∣ = lim sup
m→∞
∣∣∣∣bm+2
bm+1
∣∣∣∣ lim sup
m→∞
∣∣∣∣bm+1
bm
∣∣∣∣ ≤ 1
121
,
which implies that the sequence {bm} satisfies the condition (7) with a Landau constant
C =
1
121
. If we take C∗ =
1
100
, then ρ0 = min
{
ρ, 1/
√
C∗
}
= 10.
Since J100.5(x) is a particular solution of the Bessel differential equation (1) with
ν = 100.5, it follows from (13) that
x2y′′(x) + xy′(x) +
(
x2 − 40401
4
)
y(x) = −40385
4
cx2 + cx4
for any x ∈ I10.
If we set
am =
−40385
4
c for m = 2,
c for m = 4,
0 otherwise,
then we have
x2y′′(x) + xy′(x) +
(
x2 − 40401
4
)
y(x) =
∞∑
m=0
amx
m
and ∣∣∣∣x2y′′(x) + xy′(x) +
(
x2 − 40401
4
)
y(x)
∣∣∣∣ =
=
∣∣∣∣∣
∞∑
m=0
amx
m
∣∣∣∣∣ = c
(
40385
4
x2 − x4
)
< 999625c = ε
for all x ∈ I10.
Moreover, we have∣∣∣∑∞
m=0
amx
m
∣∣∣∑∞
m=0
|amxm|
=
40385
4
cx2 − cx4
40385
4
cx2 + cx4
>
39985
40785
,
and hence, we get
∞∑
m=0
|amxm| <
8157
7997
∣∣∣∣∣
∞∑
m=0
amx
m
∣∣∣∣∣
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1708 S.-M. JUNG
for all x ∈ I10. That is, {am} satisfies the property (b) with K =
8157
7997
.
It holds true that
x2
|ν2 − p2|
< 1,
x2
|ν2 − (p+ 1)2|
< 1
for all x ∈ I10, and since
L100.5 =
∞∑
m=0
1
(m− 100.5)2
=
=
1
(−100.5)2
+
1
(−99.5)2
+
1
(−98.5)2
+ . . .+
1
(−0.5)2
+
+
1
0.52
+
1
1.52
+
1
2.52
+
1
3.52
+ . . . ≤
≤ 1
1002
+
1
992
+
1
982
+ . . .+
1
12
+
(
1
0.52
+
1
0.52
)
+
+
1
12
+
1
22
+
1
32
+ . . . ≤
≤ 2ζ(2) + 8 =
π2
3
+ 8,
it follows from Corollary 3.1 that there exists a solution yh(x) of the Bessel’s differential
equation (1) such that
|y(x)− yh(x)| ≤
8157
7997
L100.5ε <
8157
7997
(
π2
3
+ 8
)
ε
for any x ∈ I10.
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ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
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| id | umjimathkievua-article-2835 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:31:16Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/51/d748dc3b3c3c0bd6f9fc5d77978f9851.pdf |
| spelling | umjimathkievua-article-28352020-03-18T19:37:39Z Approximation of analytic functions by Bessel functions of fractional order Наближення аналiтичних функцiй функцiями Бесселя дробового порядку Jung, S.-M. Юнг, С. М. We solve the inhomogeneous Bessel differential equation $$x^2y''(x) + xy'(x) + (x^2 - \nu^2)y(x) = \sum^{\infty}_{m=0} a_mx^m$$, where $\nu$ is a positive nonintegral number, and use this result for the approximation of analytic functions of a special type by the Bessel functions of fractional order. Розв’язано неоднорiдне диференцiальне рiвняння Бесселя $$x^2y''(x) + xy'(x) + (x^2 - \nu^2)y(x) = \sum^{\infty}_{m=0} a_mx^m$$, де $\nu$ — нецiле додатне число. Отриманi результати застосовано до наближення аналiтичних функцiй спецiального виду функцiями Бесселя дробового порядку. Institute of Mathematics, NAS of Ukraine 2011-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2835 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 12 (2011); 1699-1709 Український математичний журнал; Том 63 № 12 (2011); 1699-1709 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2835/2423 https://umj.imath.kiev.ua/index.php/umj/article/view/2835/2424 Copyright (c) 2011 Jung S.-M. |
| spellingShingle | Jung, S.-M. Юнг, С. М. Approximation of analytic functions by Bessel functions of fractional order |
| title | Approximation of analytic functions by Bessel functions of fractional order |
| title_alt | Наближення аналiтичних функцiй функцiями Бесселя дробового порядку |
| title_full | Approximation of analytic functions by Bessel functions of fractional order |
| title_fullStr | Approximation of analytic functions by Bessel functions of fractional order |
| title_full_unstemmed | Approximation of analytic functions by Bessel functions of fractional order |
| title_short | Approximation of analytic functions by Bessel functions of fractional order |
| title_sort | approximation of analytic functions by bessel functions of fractional order |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2835 |
| work_keys_str_mv | AT jungsm approximationofanalyticfunctionsbybesselfunctionsoffractionalorder AT ûngsm approximationofanalyticfunctionsbybesselfunctionsoffractionalorder AT jungsm nabližennâanalitičnihfunkcijfunkciâmibesselâdrobovogoporâdku AT ûngsm nabližennâanalitičnihfunkcijfunkciâmibesselâdrobovogoporâdku |