RETRACTED ARTICLE: Approximation of analytic functions by bessel functions of fractional order
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APPROXIMATION OF ANALYTIC FUNCTIONS BY BESSEL FUNCTIONS OF FRACTIONAL ORDER S.-M. Jung
UDC 517.5
We solve the inhomogeneous Bessel differential equation x 2 y 00 .x/ C xy 0 .x/ C .x 2
2 /y.x/ D
1 X
am x m ;
mD0
where 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.
1. Introduction The stability problem for functional equations originates from the famous talk of Ulam and the partial solution of Hyers to the Ulam problem (see [6, 27]). Thereafter, Rassias [24] attempted to solve the stability problem for the Cauchy additive functional equation in a more general setting. The stability concept introduced by the Rassias theorem significantly influenced a number of mathematicians to investigate stability problems for various functional equations (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 W I ! Y satisfying the differential inequality kan .x/y .n/ .x/ C an
1 .x/y
.n 1/
.x/ C : : : C a1 .x/y 0 .x/ C a0 .x/y.x/ C h.x/k "
for all x 2 I and some " 0; there exists a solution f0 W I ! Y of the differential equation an .x/y .n/ .x/ C an
1 .x/y
.n 1/
.x/ C : : : C a1 .x/y 0 .x/ C a0 .x/y.x/ C h.x/ D 0
such that kf .x/ f0 .x/k K."/ for any x 2 I; where K."/ depends only on "; then we say that the above differential equation possesses the Hyers–Ulam stability (or the local Hyers–Ulam stability if the domain I is not the whole space R/: We may apply this terminology to other differential equations. For more detailed definition of Hyers–Ulam stability, see [3, 4, 6–9, 24, 25]. Obłoza seems to be the first author who investigated the Hyers–Ulam stability of linear differential equations (see [22, 23]). Here, we present the following result of Alsina and Ger (see [1]): If a differentiable function f W I ! R is a solution of the differential inequality jy 0 .x/ y.x/j "; where I is an open subinterval of R; then there exists a constant c such that jf .x/ ce x j 3" for any x 2 I: College of Science and Technology, Hongik University, Jochiwon, Korea. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 12, pp. 1699–1709, December, 2011. Original article submitted September 2, 2010. 0041-5995/12/6312–1933
c 2012 Springer Science+Business Media, Inc.
1933
S.-M. J UNG
1934
This result of Alsina and Ger was generalized by Takahasi, Miura, and Miyajima: they proved in [26] that the Hyers–Ulam stability holds for the Banach-space-valued differential equation y 0 .x/ D y.x/ (see also [19]). Using the conventional power-series method, the author investigated the general solution of the inhomogeneous Legendre differential equation under some specific conditions, and this result was used to prove the Hyers– Ulam stability of the Legendre differential equation (see [10]). In a recent paper, he also investigated the approximation of analytic functions by Legendre functions (see [14]). This study was ext
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