An approximation property of simple harmonic functions
© Jung; licensee Springer 2013
Received: 5 September 2012
Accepted: 16 December 2012
Published: 3 January 2013
In this paper, applying the power series method, we approximate analytic functions by simple harmonic functions in a neighborhood of zero.
Differential equations have been studied for more than 300 years since the seventeenth century when the concepts of differentiation and integration were formulated by Newton and Leibniz. By use of differential equations, we can explain many natural phenomena: gravity, projectiles, wave, vibration, nuclear physics, and so on.
Let us consider a closed system which can be explained by the first-order linear differential equation, namely, . The past, present, and future of this system are completely determined if we know the general solution and an initial condition of that differential equation. So, we can say that this system is ‘predictable.’ Sometimes, because of the disturbances (or noises) of the outside, the system may not be determined by but can only be explained by an inequality like . Then it is impossible to predict the exact future of the disturbed system.
Even though the system is not predictable exactly because of outside disturbances, we say the differential equation has the Hyers-Ulam stability if the ‘real’ future of the system follows the solution of with a bounded error. But if the error bound is ‘too big,’ we say that the differential equation does not have the Hyers-Ulam stability. Resonance is the case.
There is another way to explain the Hyers-Ulam stability. Usually the experiment (or the observed) data do not exactly coincide with theoretical expectations. We may express natural phenomena by use of equations, but because of the errors due to measurement or observance, the actual experiment data can almost always be a little bit off the expectations. If we used inequalities instead of equalities to explain natural phenomena, then these errors could be absorbed into the solutions of inequalities, i.e., those errors would no longer be errors. Considering this point of view, the Hyers-Ulam stability (of differential equations) is fundamental. (Hyers-Ulam stability is not same as the concept of the stability of differential equations which has been studied by many mathematicians for a long time.)
and for all , where is an expression of ε with , then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [1–6].
Hyers-Ulam stability of functional equations has been studied for more than 70 years. But the history of the Hyers-Ulam stability of differential equations is less than 20 years. For example, Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations in 1993 (see [7, 8]). Thereafter, Alsina and Ger published their paper , which handles the Hyers-Ulam stability of the linear differential equation : If a differentiable function is a solution of the inequality for any , then there exists a constant c such that for all . We know that the general solution of the linear differential equation is , where c is a constant.
Therefore, we say that the differential equation has the Hyers-Ulam stability. If we can get a similar result with a control function in place of ε, we say that the differential equation has the Hyers-Ulam-Rassias stability.
In 2001 and 2002, Miura et al.  expanded Alsina and Ger’s result by proving that the differential equation has the Hyers-Ulam stability. The author wrote a paper with Miura and Takahasi which expanded the result of Hyers-Ulam stability of that differential equation. To be more precise, we may choose a constant c such that the solution of the inequality is not too far away from in the sense of upper norm (see ).
This paper is an extension and an improvement of the previous paper . We denote by the set of all nonnegative integers.
2 Inhomogeneous simple harmonic oscillator equation
Let ω be a positive constant. A function is called a simple harmonic oscillator function if it satisfies the simple harmonic oscillator equation (1.1), which plays an important role in quantum mechanics.
for all , where is a simple harmonic oscillator function and the value of is given by (2.4).
for all . We omit the proof.
which implies that the radius of convergence of power series is at least . Moreover, we notice that the radius of convergence of a general solution of the simple harmonic oscillator equation (1.1) is .
where is a simple harmonic oscillator function. □
3 Approximate simple harmonic oscillator equation
is expressible by a power series whose radius of convergence is at least ;
There exists a constant such that for any , where for all .
for all and , then is a vector space over complex numbers. We remark that the set is large enough to be a vector space.
Now, we prove the main theorem of this paper.
for all , where C is determined by (3.3) and K is defined in (b).
i.e., it holds that . In view of the proof of Theorem 2.1, it holds that .
for any .
for all .
for any . (That is, the radius of convergence of power series is at least .)
for all , where is a solution of the simple harmonic oscillator equation (1.1).
for all . □
We remark that Theorem 3.1 is true whether the ‘radius’ of the domain interval of y is larger than one or not, while [, Theorem 3.1] holds only when the function y has the domain whose ‘radius’ is not larger than one.
4 An example
Moreover, we set .
for every . Hence, y satisfies the condition (a).
for all , i.e., (b) is satisfied with and , and hence .
for all .
for all .
for all .
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. 2011-0004919).
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