MATH 414 Lecture 11

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Point-Wise Convergence of Fourier Series

1. Riemann-Lebesgue
2. Partial Sumes
3. Find Error
4. Prove Convergence

Theorem.

Proof.

Lemma. [Riemann-Lebesgue]. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} be $L'\left(\int _{a}^{b}\left|f(t)\right|\,\mathrm {d} <\infty \right)$ in $\left[a,b\right]$. Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \lim_{\lambda \to \infty} \int_{a}^{b} f(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t &= 0 \\ \lim_{\lambda \to \infty} \int_{a}^{b} f(t) \, \left\{ \begin{matrix} \cos{t} \\ \sin{t} \end{matrix} \right\} &= 0 \end{align}}

Proof. (special case). Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \in C^{(1)} \left[ a,b \right]} . Integrating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{a}^{b} f(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t} by parts gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^b f(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t = \frac{f(b) \, \mathrm{e}^{i \, \lambda \, b - f(a) \, \mathrm{e}^{i \, \lambda \, a}}}{i \, \lambda} - \frac{1}{i \, \lambda} \, \int_{a}^{b} f'(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t}

Taking the absolute value of the above yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \int_{a}^{b} f(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t \right| \le \left| \frac{f(b) \, \mathrm{e}^{i \, \lambda \, b} - f(a) \, \mathrm{e}^{i \, \lambda \, a}}{i \, \lambda} \right| + \frac{1}{ \left| \lambda \right| } \, \left| \int_{a}^{b} f'(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t \right|}

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \frac{f(b) \, \mathrm{e}^{i \, \lambda \, b} - f(a) \, \mathrm{e}^{i \, \lambda \, a}}{i \, \lambda} \right| \le \frac{ \left| f(b) \, \mathrm{e}^{i \, \lambda \, b} \right| + \left| f(a) \, \mathrm{e}^{i \, \lambda \, a} \right|}{ \left| \lambda \right|} \le \frac{ \left| f(b) \right| + \left| f(a) \right|}{ \left| \lambda \right|}}
2. We know Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \int_{a}^{b} g(t) \,\mathrm{d}t \right| \le \int_{a}^b \left| g(t) \right| \,\mathrm{d}t} for both real and complex numbers, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \int_{a}^{b} f'(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t \right| \le \int_{a}^{b} \left| f'(t) \right| \,\mathrm{d}t}

Therefore we are left with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \int_{a}^{b} f(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t \right| \le \frac{\left| f(b) \right| + \left| f(a) \right|}{\left| \lambda \right|} + \frac{1}{\left| \lambda \right|} \, \int_a^b \left| f'(t) \right| \,\mathrm{d}t }

Observe that the integral does not depend on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} , and in fact, as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \to \infty} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{a}^{b} f(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t \to 0} .

quod erat demonstrandum

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Partial sums. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = a_0 + \sum_{n=1}^\infty a_n \, \cos{(n\,x)} + b_n \, \sin{(n\,x)}} be a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} -periodic, piecewise-smooth function. Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} S_N(x) &= a_0 + \sum_{n=1}^N a_n \, \cos{(n\,x)} + b_n \, \sin{(n\,x)} \\ a_n &= \frac{1}{\pi} \, \int_{-\pi}^\pi f(t) \, \cos{(n\,t)} \,\mathrm{d}t \\ b_n &= \frac{1}{\pi} \, \int_{-\pi}^{\pi} f(t) \, \sin{(n\,t)} \,\mathrm{d}t \end{align}}

Therefore

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} a_n \, \cos{(n\,x)} + b_n \, \sin{(n\,x)} &= \frac{1}{\pi} \, \int_{-\pi}^{\pi} f(t) \, \left( \cos{(n\,t)} \, \cos{(n\,x)} + \sin{(n\,t)} \, \sin{(n\,x)} \right) \,\mathrm{d}t \\ &= \frac{1}{\pi} \, \int_{-\pi}^{\pi} f(t) \, \cos{(n \, (t - x))} \,\mathrm{d} \end{align}}

Now our partial sum is of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_N(x) = a_0 + \sum_{n=1}^N a_n \, \cos{(n\,x)} + b_n \, \sin{(n\,x)} = \int_{-\pi}^{\pi} f(x) \, P_N(t-x) \,\mathrm{d}t}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_N(u) = \frac{1}{2 \pi} + \frac{1}{\pi} \, \sum_{n=1}^N \cos{(n\,u)}} is called the Fourier kernel or the Dirichlet kernel

Properties of the Fourier Kernel:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_N(u)} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} -periodic
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_N(-u) = P_N(u)} (it is even)
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\pi}^{\pi} P_N(u) \,\mathrm{d}u = \frac{1}{2\pi} \, \int_{-\pi}^\pi \,\mathrm{d}u = 1}
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\pi} P_N(u) \,\mathrm{d}u = \frac{1}{2}}
5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_N(u) = \frac{1}{2\pi} \, \frac{\sin{ \left( \left( N + \frac{1}{2} \right) \, u \right)}}{\sin{ \left( \frac{u}{2} \right)}}}

Let's change index on the integral:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_N(x) = \int_{-\pi}^{\pi} f(t) \, P_N(t-x) \,\mathrm{d}t = \int_{-\pi-x}^{\pi-x} f(u + x) \, P_N(u) \,\mathrm{d}u}

Observe that the entire integrand is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} -periodic, so we can shift the index back to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\pi, \pi \right]} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_N(x) = \int_{-\pi}^{\pi} f(u+x) \, P_N(u) \, \mathrm{d}u}

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Error Estimation. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_N(x) = S_N(x) - f(x)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a point of continuity. Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_N(x) = S_N(x) - f(x) \, \int_{-\pi}^\pi P_N(u) \,\mathrm{d}u = \int_{-\pi}^\pi \left( f(x+u) - f(x) \right) \, P_N(u) \,\mathrm{d}u}

Using property 5, we can rewrite Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_N(x)} as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_N(x) = \frac{1}{2\pi} \, \int_{-\pi}^\pi \frac{f(u+x) - f(x)}{\sin{\left( \frac{u}{2} \right)}} \, \sin{ \left( \left( N + \frac{1}{2} \right) \, u \right)} \,\mathrm{d}u}

Assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . By L'Hôspital's rule,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{u \to 0} \frac{f(u+x) - f(x)}{\sin{ \left( \frac{u}{2} \right)}} = 2f'(x)}

This means that the integrand of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_N(x)} is continuous at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = 0} . If we actually calculate the integral, we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_N(x) = 0} .

quod erat demonstrandum

Important Identity: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x-1 \right) \, \left( \sum_{k=0}^n x^k \right) = z^{n+1} - 1}