1.5 Operations on Fourier Series

1.5 Operations on Fourier Series

• We will have to perform certain operations in this book.
• The purpose of this section is to find legitimate conditions.
• There are two things that must be noted.
• First of all, the results are not the best because there are weaker hypotheses and the same conclusions.
• In applying mathematics, we carry out operations that are legitimate or not.
• For correctness, the results must be checked.

• The periodic extension must fulfill the hypotheses for functions only on a finite interval.

• This is a simple result of the fact that a constant passes through an integral.
• The fact that the sum of the integrals is the sum of the integrals leads to the following.

• The reader probably already used the two theorems because they are so natural.
• It is more difficult to prove the following theorems.

• The series on the right converges and equals the integral on the left according to the theorems.

• The formulas for the Fourier coefficients in Section 1 were derivations of Theorem 4.
• What follows is an application of Theorems 3 and 4.

• If the correspondence were an equality, the manipulation would be simple.

• The series on the right is equal to the periodic extension of the function on the left.

• The series on the right is worth mentioning.

• The equations can be verified directly, but Theorem 4 and Section 1 also guarantee them.

• We tend to assume that the property is true without checking because it is so natural.

• The Fourier series is unique if f (x) is periodic and sectionally continuous.

• If two Fourier series are equal and correspond to the same function, then the coefficients of similar terms must match.

• The last operation to be discussed is differentiation, which is a role in applications.

• We know a function through its Fourier series later on.
• It is important to get the properties of the function by examining its coefficients.