Question

Estimate de Fourier coefficients and series for the function g:[−π,π]→R defined by g(x)=∣x∣. Is the Fourier series point-wise or uniform convergent to something?

Answer 1

The Fourier series coefficients for the function g(x) = |x| on the interval [-π,π] can be estimated using the formulas for a₀, aₙ, and bₙ. The resulting Fourier series for g(x) is point-wise convergent to the function |x|, but it is not uniformly convergent.

Step-by-step explanation:

The Fourier coefficients of the function g(x) = |x| on the interval [-π,π] can be determined by using the formula:

a₀ = (1/π) ∫[-π,π] |x| dx = (2/π) ∫[0,π] x dx = 1

aₙ = (1/π) ∫[-π,π] |x| cos(nx) dx = (2/π) ∫[0,π] x cos(nx) dx = (-2/πn²)(-1)ⁿ + (2/πn²)

bₙ = (1/π) ∫[-π,π] |x| sin(nx) dx = 0

To estimate the Fourier series for g(x), we can use the formula:

g(x) = (a₀/2) + Σ(aₙ cos(nx) + bₙ sin(nx))

Since the function g(x) = |x| is an odd function, all the cosine terms (aₙ) will be zero. Therefore, the Fourier series for g(x) simplifies to:

g(x) = (1/2) + Σ((2/πn²)sin(nx))

The Fourier series for g(x) is point-wise convergent to the function |x|, but it is not uniformly convergent.