MATH 414 Lecture 17

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End Exam 1 content

Inner Products

• Standard Inner products on $\mathbb {R} ^{n}$, $\mathbb {C} ^{n}$, $L^{2}$, and $\ell ^{2}$
• Is $\left\langle Y,X\right\rangle =Y^{T}\,A\,X$ an inner product (given a matrix $A$)?
  • Positivity: $\left\langle {\vec {v}},{\vec {v}}\right\rangle >0$ for all nonzero ${\vec {v}}$
  • Homogeneity: $\left\langle c\,{\vec {u}},{\vec {v}}\right\rangle =c\,\left\langle {\vec {u}},{\vec {v}}\right\rangle$ for all vectors ${\vec {u}},{\vec {v}}$ and scalars $c\in \mathbb {C}$.
  • (Conjugate) symmetry: ${\overline {\left\langle {\vec {u}},{\vec {v}}\right\rangle }}=\left\langle {\vec {v}},{\vec {u}}\right\rangle$ for all vectors ${\vec {u}},{\vec {v}}$
  • Linearity: $\left\langle {\vec {u}}+{\vec {v}},{\vec {w}}\right\rangle =\left\langle {\vec {u}},{\vec {w}}\right\rangle +\left\langle {\vec {v}},{\vec {w}}\right\rangle$
• Angle between vectors: $\cos {\theta }={\frac {{\vec {u}}\cdot {\vec {v}}}{\left\|{\vec {u}}\right\|\,\left\|{\vec {v}}\right\|}}$
• Length of a vector: $\left\|{\vec {u}}\right\|={\sqrt {\left\langle {\vec {u}},{\vec {u}}\right\rangle }}$
• Distance between two vectors: $d=\left\|{\vec {u}}-{\vec {v}}\right\|$

Fourier Series

Parseval's Equation

Real Version

If $f(x)=a_{0}+\sum _{k=1}^{\infty }a_{k}\,\cos {(k\,x)}+b_{k}\,\sin {(k\,x)}\in L^{2}[-\pi ,\pi ]$, then

${\frac {1}{\pi }}\int _{-\pi }^{\pi }\left|f(x)\right|^{2}\,\mathrm {d} x=2\left|a_{0}\right|^{2}+\sum _{k=1}^{\infty }\left|a_{k}\right|^{2}+\left|b_{k}\right|^{2}$

Complex Version

If $f(x)=\sum _{k=-\infty }^{\infty }\alpha _{k}\,\mathrm {e} ^{i\,k\,x}\in L^{2}[-\pi ,\pi ]$, then

${\frac {1}{2\pi }}\left\|f\right\|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left|f(x)\right|^{2}\,\mathrm {d} x=\sum _{k=-\infty }^{\infty }\left|\alpha _{k}\right|^{2}$

Moreover, for $f,g\in L^{2}[-\pi ,\pi ]$, we get

${\frac {1}{2\pi }}\,\left\langle f,g\right\rangle ={\frac {1}{2\pi }}\,\int _{-\pi }^{\pi }f(t)\,{\overline {g(t)}}\,\mathrm {d} t=\sum _{n=0}^{\infty }\alpha _{n}\,{\overline {\beta _{n}}}$