Pointwise, uniform convergence of fourier series

[lilphy]

Hello;
I'm struggling with pointwise and uniform convergence, I think that examples are going to help me understand

Homework Statement

Consider the Fourier sine series of each of the following functions. In this exercise de not compute the coefficients but use the general convergence theorems to discuss convergence of each of the series pointwise, uniform and L² senses
1. f(x)=x^3 on (0,l)
2.f(x)=lx-x² on (0,l)
3.f(x)=x^-2 on (0,l)

Homework Equations

Theorems:

1-Uniform convergence : The Fourier series Σ An Xn(x) converges to f(x) uniformly on [a, b] provided that
(i) f (x), f' (x), and f"(x) exist and are continuous for a ≤ x ≤ b and
(ii) f (x) satisfies the given boundary conditions

2-Pointwise Convergence of Classical Fourier Series
(i) The classical Fourier series (full or sine or cosine) converges to f(x) pointwise on (a, b) provided that f(x) is a continuous function on a ≤ x ≤ b and f '(x) is piecewise continuous on a ≤ x ≤ b.
(ii) More generally, if f(x) itself is only piecewise continuous on a ≤ x ≤ b and f '(x) is also piecewise continuous on a ≤ x ≤ b, then the classical Fourier series converges at every point x(−∞ < x < ∞).
The sum is
Σ An Xn(x) = 1/2 [ f (x+) + f (x−)] for all a < x < b.

The Attempt at a Solution

a/ f(x)=x^3 is an odd function so the Fourier serie is
∑bn sin(nπx/l)
It satisfies the condition (i) of theorem 1 but not (ii) because at x=0 and x=l the serie is 0 so it doesn't satisfy the boundary conditions
And it satisfies (i) of theorem 2 so it converges pointwise to x^3 on (0,l)

b/ f(x)=lx-x²
∑(ancos(nπx/l)+bn sin(nπx/l)
It satisfies condition (i) of theorem 1 but how can we know for (ii) at x=0 and l f(x)=0 but the serie at x=0 is equal to ∑an and to ∑(-1)ⁿan at x=l.
And it satisfies (i) of theorem 2 so it converges pointwise to f(x) on (0,l)

x/
Doesn't converge at x=0 so it doesn't converge uniformly but converge pointwise

Thanks

 

[pasmith]

Hello;
I'm struggling with pointwise and uniform convergence, I think that examples are going to help me understand

Homework Statement

Consider the Fourier sine series of each of the following functions. In this exercise de not compute the coefficients but use the general convergence theorems to discuss convergence of each of the series pointwise, uniform and L² senses
1. f(x)=x^3 on (0,l)
2.f(x)=lx-x² on (0,l)
3.f(x)=x^-2 on (0,l)

Homework Equations

Theorems:

1-Uniform convergence : The Fourier series Σ An Xn(x) converges to f(x) uniformly on [a, b] provided that
(i) f (x), f' (x), and f"(x) exist and are continuous for a ≤ x ≤ b and
(ii) f (x) satisfies the given boundary conditions

2-Pointwise Convergence of Classical Fourier Series
(i) The classical Fourier series (full or sine or cosine) converges to f(x) pointwise on (a, b) provided that f(x) is a continuous function on a ≤ x ≤ b and f '(x) is piecewise continuous on a ≤ x ≤ b.
(ii) More generally, if f(x) itself is only piecewise continuous on a ≤ x ≤ b and f '(x) is also piecewise continuous on a ≤ x ≤ b, then the classical Fourier series converges at every point x(−∞ < x < ∞).
The sum is
Σ An Xn(x) = 1/2 [ f (x+) + f (x−)] for all a < x < b.

The Attempt at a Solution

a/ f(x)=x^3 is an odd function so the Fourier serie is
∑bn sin(nπx/l)
It satisfies the condition (i) of theorem 1 but not (ii) because at x=0 and x=l the serie is 0 so it doesn't satisfy the boundary conditions
And it satisfies (i) of theorem 2 so it converges pointwise to x^3 on (0,l)

b/ f(x)=lx-x²
∑(ancos(nπx/l)+bn sin(nπx/l)

It satisfies condition (i) of theorem 1 but how can we know for (ii) at x=0 and l f(x)=0 but the serie at x=0 is equal to ∑an and to ∑(-1)ⁿan at x=l.
And it satisfies (i) of theorem 2 so it converges pointwise to f(x) on (0,l)

If a function defined on [0,l] doesn't satisfy f(0) = f(l) then its Fourier series will have poor convergence properties. However if you find that the Fourier series doesn't satisfy that condition then you are doing something wrong. Here your error is that the series should be \sum a_n \cos (2n\pi x/l) + b_n \sin (2n \pi x / l).