Let c > 0 be a constant. For F E Cp (-C, C), the Bessel inequality is
ao squared/ 2 + N E n = 1 (an squared + bn squared) is less than or equal to 1/C { c on top, c on bottom [ f (x)] squared of x, N = 1, 2, ...
Where an and bn are the Fourier coefficients for f (x), -c < x < c.
Using this inequality show that lim an = 0, n ---> infinity and lim bn = 0, n ----> infinity.
To prove that lim an = 0 as n approaches infinity, we can use the Bessel inequality and the fact that the sum of squares of the Fourier coefficients is bounded.
First, let's rewrite the Bessel inequality as:
ao^2/2 + Σn=1 to ∞ (an^2 + bn^2) ≤ 1/C ∫[-c,c] f(x)^2 dx.
Since this inequality holds for all C > 0, we can take the limit as C approaches infinity, resulting in:
lim C→∞ [ao^2/2 + Σn=1 to ∞ (an^2 + bn^2)] ≤ lim C→∞ [1/C ∫[-c,c] f(x)^2 dx].
The left side of the inequality is the sum of squares of the Fourier coefficients, and it is bounded by the right side of the inequality. Therefore, the left side must also be bounded.
To show that lim an = 0 as n approaches infinity, suppose the contrary, i.e., that lim an ≠ 0. In that case, there exists some ε > 0 such that for infinitely many values of n, |an| ≥ ε. Since the sum of squares of the Fourier coefficients is bounded, this implies that there are infinitely many terms with |bn| ≥ ε as well.
However, if there are infinitely many terms with nonzero coefficients, the sum of squares of the Fourier coefficients will not be bounded, contradicting the Bessel inequality. Therefore, our assumption that lim an ≠ 0 must be false, and we conclude that lim an = 0 as n approaches infinity.
The same reasoning can be applied to the bn coefficients, leading to the conclusion that lim bn = 0 as n approaches infinity.