1. Preliminary Knowledge
Definition 1 [1] -[3] Let be an integrable function on. Then the coefficients and are calculated by
and are called the Fourier coefficients of.
Definition 2 [1] -[5] Let with the period be an integrable function on, trigonometric series with the Fourier coefficient are called Fourier series of, denoted by
Lemma 1 [6] Let be an integrable function on with period of, the Fourier coefficients are calculated according to period of. The calculation indicates there are same results between Fourier
series with period of and.
2. Calculating Fourier Series According to the Nature of the Function
Theorem 1 Let be an integrable function on and satisfy the condition, then we have
where, ,.
Proof It was clear that the period of is and we have
Let, then. Therefore
.
So we get
.
.
Let, then. Therefore
Therefore, we obtain
In the same way, we have
In a word, while is an even number, , and
Thus in this case, the expansion reduces to
Theorem 2 Let be an integrable function on and satisfy the conditions, then we have
where,.
Proof The period of is, so we can calculate Fouries series of with period of by Lemma 1. We have
Let, then. Therefore
So we get
Let, then. Therefore
We obtain
In the same way, we have
Thus in this case, the expansion reduces to
where,.
Theorem 3 Let be an integrable function on. When, it satisfies the condition. Then we have
(1) While is an even function in, then we get
where.
(2) While is an odd function in,then we get
where.
Proof (1) We use the method of periodic extension to with period of.
Because is an even function, we have,.
Let, , then we have
Let, therefore
We obtain
Thus in this case, the expansion reduces to
where.
(2) In the same way, we can prove Theorem 3 (2).
Similarly, we can prove the following Theorem 4.
Theorem 4 Let be an integrable function on. When, it satisfies the condition. Then we have
(1) While is an even function in, then we get
where.
(2) While is an odd function in, then we get
where.
3. Conclusion
Suppose the function is defined on, if we use symmetry extension about the point and then
use odd and periodic extension, we can get two forms of Fourier series as Theorem 3. If we use symmetry ex-
tension about the line and then use odd and periodic extension, we can get two forms of Fourier series as Theorem 4. Suppose the function is defined on, we have a similar conclusion.
Acknowledgements
I would like to thank the referees and the editor for their valuable suggestions.