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 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 L'\left( \int_{a}^{b} \left| f(t) \right| \,\mathrm{d} < \infty \right)} in $\left[a,b\right]$. Then

${\begin{aligned}\lim _{\lambda \to \infty }\int _{a}^{b}f(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t&=0\\\lim _{\lambda \to \infty }\int _{a}^{b}f(t)\,\left\{{\begin{matrix}\cos {t}\\\sin {t}\end{matrix}}\right\}&=0\end{aligned}}$

Proof. (special case). Let $f\in C^{(1)}\left[a,b\right]$. Integrating $\int _{a}^{b}f(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t$ by parts gives

$\int _{a}^{b}f(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t={\frac {f(b)\,\mathrm {e} ^{i\,\lambda \,b-f(a)\,\mathrm {e} ^{i\,\lambda \,a}}}{i\,\lambda }}-{\frac {1}{i\,\lambda }}\,\int _{a}^{b}f'(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t$

Taking the absolute value of the above yields

$\left|\int _{a}^{b}f(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t\right|\leq \left|{\frac {f(b)\,\mathrm {e} ^{i\,\lambda \,b}-f(a)\,\mathrm {e} ^{i\,\lambda \,a}}{i\,\lambda }}\right|+{\frac {1}{\left|\lambda \right|}}\,\left|\int _{a}^{b}f'(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t\right|$

1. $\left|{\frac {f(b)\,\mathrm {e} ^{i\,\lambda \,b}-f(a)\,\mathrm {e} ^{i\,\lambda \,a}}{i\,\lambda }}\right|\leq {\frac {\left|f(b)\,\mathrm {e} ^{i\,\lambda \,b}\right|+\left|f(a)\,\mathrm {e} ^{i\,\lambda \,a}\right|}{\left|\lambda \right|}}\leq {\frac {\left|f(b)\right|+\left|f(a)\right|}{\left|\lambda \right|}}$
2. We know $\left|\int _{a}^{b}g(t)\,\mathrm {d} t\right|\leq \int _{a}^{b}\left|g(t)\right|\,\mathrm {d} t$ for both real and complex numbers, so $\left|\int _{a}^{b}f'(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t\right|\leq \int _{a}^{b}\left|f'(t)\right|\,\mathrm {d} t$

Therefore we are left with

$\left|\int _{a}^{b}f(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t\right|\leq {\frac {\left|f(b)\right|+\left|f(a)\right|}{\left|\lambda \right|}}+{\frac {1}{\left|\lambda \right|}}\,\int _{a}^{b}\left|f'(t)\right|\,\mathrm {d} t$

Observe that the integral does not depend on $\lambda$, and in fact, as $\lambda \to \infty$, $\int _{a}^{b}f(t)\,\mathrm {e} ^{i\,\lambda \,t}\,\mathrm {d} t\to 0$.

quod erat demonstrandum

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Partial sums. Let $f(x)=a_{0}+\sum _{n=1}^{\infty }a_{n}\,\cos {(n\,x)}+b_{n}\,\sin {(n\,x)}$ be a $2\pi$-periodic, piecewise-smooth function. Then

${\begin{aligned}S_{N}(x)&=a_{0}+\sum _{n=1}^{N}a_{n}\,\cos {(n\,x)}+b_{n}\,\sin {(n\,x)}\\a_{n}&={\frac {1}{\pi }}\,\int _{-\pi }^{\pi }f(t)\,\cos {(n\,t)}\,\mathrm {d} t\\b_{n}&={\frac {1}{\pi }}\,\int _{-\pi }^{\pi }f(t)\,\sin {(n\,t)}\,\mathrm {d} t\end{aligned}}$

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}