Approximation of Analytic Functions by Bessel's Functions of Fractional Order
We will solve the inhomogeneous Bessel's differential equation , where is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.
The stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to the Ulam problem (see [1, 2]). Thereafter, Rassias  attempted to solve the stability problem of the Cauchy additive functional equation in a more general setting.
The stability concept introduced by Rassias's theorem significantly influenced a number of mathematicians to investigate the stability problems for various functional equations (see [4–14] and the references therein).
Assume that is a normed space and is an open subset of . If for any function satisfying the differential inequality for all and for some , there exists a solution of the differential equation such that for any , where depends on only, then we say that the above differential equation satisfies the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain is not the whole space ). We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to [2, 3, 6, 8, 10–12, 14].
Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [15, 16]). Here, we will introduce a result of Alsina and Ger (see ): If a differentiable function is a solution of the differential inequality , where is an open subinterval of , then there exists a constant such that for any .
Using the conventional power series method, the author has investigated the general solution 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 ). In a recent paper, he has also investigated an approximation property of analytic functions by the Legendre functions (see ). This study has been continued to various special functions including the Airy functions, the exponential functions, the Hermite functions, and the power functions (see [22–25]).
Recently, the author and Kim tried to prove the Hyers-Ulam stability of the Bessel differential equation However, the obtained theorem unfortunately does not describe the Hyers-Ulam stability of the Bessel differential equation in a strict sense (see ).
In Section 2 of this paper, by using the ideas from , we will determine the general solution of the inhomogeneous Bessel differential equation where the parameter 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 the largest integer not exceeding for any , and we define for any .
2. Inhomogeneous Bessel's Differential Equation
A function is called a Bessel function (of fractional order) if it is a solution of the Bessel differential equation (1.3), where is a positive nonintegral number. The Bessel 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 seems not to guarantee the existence of solutions to the inhomogeneous Bessel differential equation (1.4). Thus, we adopt an additional condition to ensure the existence of solutions to the equation.
Theorem 2.1. Let be a positive nonintegral number, and let be a positive constant. Assume that the radius of convergence of power series is at least and there exists a constant satisfying the condition for all sufficiently large integers , where for all . Let . Then every solution of the Bessel's differential equation (1.4) can be expressed by for all , where is a solution of the homogeneous Bessel equation (1.3).
Proof. We assume that is a function given in the form (2.3), and we define . Then, it follows from (2.1) and (2.2) that
since we can deduce the relation from (2.2) by some manipulations. That is, the power series for converges for all . Hence, we see that the domain of is well defined.
We now prove that the function satisfies the inhomogeneous equation (1.4). Indeed, it follows from (2.2) that since we obtain which proves that is a particular solution of the inhomogeneous equation (1.4).
On the other hand, since every solution to (1.4) can be expressed as a sum of a solution of the homogeneous equation and a particular solution of the inhomogeneous equation, every solution of (1.4) is certainly of the form (2.3).
3. Approximate Bessel's Differential Equation
In this section, assume that is a positive nonintegral number and is a positive constant. For a given , we denote by the set of all functions with the properties (a) and (b): (a) is expressible by a power series whose radius of convergence is at least ; (b) for any , where for all and set .
For a positive nonintegral number , define where and We remark that as .
We will now solve the approximate Bessel differential equations in a special class of analytic functions, .
Theorem 3.1. Let be a positive nonintegral number, and let be a nonnegative integer with . Assume that a function satisfies the differential inequality for all and for some . If the sequence satisfies the condition with a Landau constant , then there exists a solution of the Bessel differential equation (1.3) such that for any , where and is a positive number larger than . If and are sufficiently small and large; respectively, then for all sufficiently large .
Proof. Since belongs to , it follows from (a) and (b) that
for all . By considering (3.3) and (3.7), we get
for any . This inequality, together with (b), yields
for each .
Now, it follows from (b) that since . Similarly, we obtain for all , that is, for all .
On the other hand, by (b) and (3.4), we have for all sufficiently large integers , where is a positive number larger than . Hence, in view of (3.12) and (3.13), the condition (2.1) is satisfied with . Moreover, we know that the radius of convergence of power series is at least because the convergence radius of the power series expression for is at least (see (a) and (3.7)).
According to (3.7) and Theorem 2.1, there exists a solution of the homogeneous Bessel differential equation (1.3) satisfying (2.3) for all . Thus, it follows from (2.2) and (3.12) that for any . Moreover, we have for all , where .
We know that and for and or is not perhaps less than 1 for . Then, we have for all and . Similarly, we get for all and .
Thus, it follows from (3.9) and (3.15) that for .
Finally, assume that is sufficiently small and is sufficiently large. Then is sufficiently large. For any and , we have If is so large that then it follows from the definition of that for all sufficiently large .
If is large enough, then we can prove the (local) Hyers-Ulam stability of the Bessel differential equation (1.3) as we see in the following corollary.
Corollary 3.2. Let be a positive nonintegral number and let be a nonnegative integer with . Assume that a function satisfies the differential inequality (3.3) for all and for some . Suppose the sequence satisfies the condition (3.4) with a Landau constant and define for a positive number . If for all , then there exists a solution of the Bessel differential equation (1.3) such that for any .
Proof. For any and , we have Thus, we get and the assertion is true due to Theorem 3.1.
We will show that there exist functions which satisfy all the conditions given in Theorem 3.1 and Corollary 3.2. Let us define a function by where is the Bessel function of the first kind of order 100.5, is a positive integer, and is a constant satisfying for some . It is obvious that the convergence radius of the power series is infinity. (So we can set .) In fact, the infinite series converges. So we have Thus, it holds true that which implies that the sequence satisfies the condition (3.4) with a Landau constant . If we take , then .
If we set then we have for all .
Moreover, we have and hence, we get for all . That is, satisfies the property (b) with .
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (no. 2010-0007143).
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