Functional Analysis

• For the ../Glossary.html#orgbedb211 \(f: [0, 1] \to \mathbb{R}\), and \([0, 1]\) equipped with the :
  • The support of \(f\) is the entire interval \([0, 1]\),
  • but the essential support of \(f\) is empty.

Commutation relations

• \( [D,x] = 1 \)
• \( [D, f] = f' \)
• \( [D, x^n] = nx^{n-1} \)
• \( [D^n, x] = nD^{n-1} \)

• Exponential Shift Theorem

For a polynomial \(P\): \[ P(D)(e^{ax}y) = e^{ax}P(D+a)y. \]

It is the operator that is obtained by performing integration by part on the inner product.

Boundary conditions on one of the operand must be given to constitute an adjoint, which restricts the space of the operand.

The boundary conditions of the other operand are chosen forming its own space, such that the conjunct or bilinear concomitant is zero: \[ J(u, v) := \langle v, Lu\rangle - \langle L^{\dagger} v, u\rangle = 0. \]

Given a linear differential operator \( L \), its Green's function \( G \) satisfies: \[ LG(x, \xi)=\delta(x-\xi). \] where \( \delta \) is the Dirac delta function.

• Green's functions: the genius way to solve DEs - YouTube

  \begin{align*} &f(\xi)LG(x, \xi) = f(\xi)\delta(x-\xi)\\ \implies &Lf(\xi)G(x,\xi) = f(\xi)\delta(x-\xi)\\ \implies &\int_\Omega Lf(\xi)G(x,\xi)\,d\xi = \int_\Omega f(\xi)\delta(x-\xi)\,d\xi\\ \implies & L\int_\Omega f(\xi)G(x,\xi)\,d\xi = \int_\Omega f(\xi)\delta(x-\xi)\,d\xi = f(x).\\ \end{align*}

• For 3D Laplace Operator

\[ -\frac{1}{4\pi |\mathbf{r} - \boldsymbol{\xi}| } \]

• \[ z=Ae^{i\phi} \]

A complex number representing both the amplitude \(A\) and the phase \(\phi\) of a sinusoid.

Phasor can be used to scale and offset a sinusoid: \[ ze^{i\omega t}=Ae^{i\phi} e^{i\omega t}= Ae^{i(\omega t + \phi)}. \]

\[ \mathcal{F}[f](\omega)=\int_P f(t) e^{-i\omega t}\, dt \] where \(e^{-i\omega t}\) part is called the kernel.

The transformed function can be notated with \(\hat{f}(\omega)\). \(\hat{f}\) maps frequency to its phasor.

Sine funciton \[ \mathcal{F}[\sin(\omega_0t)](\omega)=\frac{1}{2i}(\delta(\omega-\omega_0)-\delta(\omega + \omega_0)) \] where \(\delta\) is the Dirac delta function.

Decaying exponential \[ \mathcal{F}[e^{-\alpha t}](\omega) = \frac{1}{\alpha + i\omega}. \]

Square pulse transforms into a sinc function. \[ f(t)=\begin{cases}A&-\frac{\tau}{2}

Gaussian function transforms into a Gaussian function. \[ e^{\frac{-t^2}{2\sigma^2}}\ \overset{\mathcal{F}}{\to}\ \sigma\sqrt{2\pi}e^{-2\pi^2\sigma^2k^2}. \]

• Fourier transform of a real function is Hermitian: \(\hat{f}(-\omega)=\hat{f}^*(\omega)\).
• Even functions have real Fourier transform and odd functions have imaginary Fourier transform.
• \(\mathcal{F}[f*g]=\mathcal{F}[f]\cdot\mathcal{F}[g]\) where \(*\) is the convolution.

• \(k\) is for the spacial frequency.
• Fourier transform of a spacial information results in a \(k\)-space.
• For 2D case, the kernel is \(e^{-i(k_xx+k_yy)}\).
• \(e^{i(k_xx+k_yy)}\) is a sinusoidal surface with the fringe progression in the direction of the vector \([k_x\ k_y]^{\mathrm{T}}\).
• In a discrete case, \(k\)-space is quantized with \(\Delta k_{x_i} = 1/\mathrm{FOV}_{x_i}\).
• The pixel shape is the shape of \(k\)-space and the reciprocal of the image shape is the pixel shape of the \(k\)-space.
• DC centered view of \(k\)-space is the normal result of Fourier transform, and from we get

• Projection-Slide Theorem, Central Slice Theorem

\[ F_1P_1 = S_1F_2 \] where \(F_1\) and \(F_2\) are 1- and 2-dimensional Fourier transform operators, \(P_1\) is a projection operator, and \(S_1\) is a slice operator that slice a 2-dimensional function along the origin perpendicular to the projection direction.

\[ \mathcal{L}\{f\}(s) := \int_0^\infty f(t)e^{-st}dt. \]

It transforms the complex-valued real function to a complex-valued complex function. The domain is changed from time domain to s-domain.

By convention, the Laplace transform of a function is often written in a capital letter.

Linearity \[ \mathcal{L}\{af + bg\} = a\mathcal{L}\{f\} + b\mathcal{L}\{g\}. \]

Frequency-Domain Derivative \[ \mathcal{L}\{t^nf\} = (-1)^n(\mathcal{L}f)^{(n)} \]

(Time-Domain) Derivative \[ \mathcal{L}\{f^{(n)}\} = s^n\mathcal{L}f - \sum_{k=1}^ns^{n-k}f^{(k-1)}(0^-) \]

Frequency Shifting \[ \mathcal{L}\{e^{at}f\} = \mathcal{Lf}(s-a) \]

Convolution \[ \mathcal{L}\{f\ast g\} = (\mathcal{L}f)\cdot(\mathcal{L}g) \]

• \[\mathcal{L}\{e^{at}\} = \frac{1}{s - a}\]

• \[ \mathcal{M}\{f\}(s) = \varphi(s) = \int_0^\infty x^{s-1}f(x)\,dx. \]
• The inverse transform is given by \[ \mathcal{M}^{-1}\{\varphi\}(x) = f(x) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\varphi(s)\,ds. \]

\begin{align*} p(s, \phi) &= \mathcal{R}\{f(x,y)\} \\ &= \int_{L_{s,\phi}} f(x,y)\dd{u} \\ &= \int_{-\infty}^{\infty} f(s\cos\phi - u\sin\phi, s\sin\phi + u\cos\phi)\dd{u}. \end{align*}

\begin{align*} f(x,y) &= \mathcal{F}_{x,y}^{-1}\{F(k_x,k_y)\} = \int_{-\infty}^{\infty} \dd{k_x \int_{-\infty}^{\infty}\dd{k_y}} F(k_x, k_y) e^{2\pi i (k_xx + k_yy)} \\ &= \int_0^{\pi} \dd{\phi}\int_{-\infty}^{\infty}\dd{\omega}|\omega|F(\omega\cos\phi, \omega\sin\phi) e^{2\pi i \omega(x\cos\phi + y\sin \phi)} \\ \end{align*}

where we just change the coordinate system to polar coordinates \((x,y)\to (\omega, \phi)\).

By the Fourier slide theorem, \(F(\omega\cos\phi, \omega\sin\phi) = P(\omega, \phi)\), and substituting \(s = x\cos\phi + y\sin\phi\):

\begin{align*} \dots &= \int_0^{\pi}\dd{\phi}\int_{-\infty}^{\infty} \dd{\omega}|\omega|P(\omega, \phi) e^{2\pi i\omega s} \\ &= \int_0^{\pi}\dd{\phi}\int_{-\infty}^{\infty} \dd{\omega}|\omega| \left( \int_{-\infty}^{\infty} p(s, \phi) e^{-2\pi i \omega s}\dd{s}\right) e^{2\pi i \omega s} \\ &= \int_0^{\pi} \dd{\phi}\mathcal{F}_\omega^{-1}\{P(\omega, \phi)\cdot |\omega|\}. \end{align*}

The integrand is also written as \(\tilde{p}(\omega,s)\).

The restriction on \(h\) can be loosened up to a function that vanished at the endpoints, \(a\) and \(b\), since the lemma directly implies that.

For a function \(L(q, \dot{q}, t)\), \[ \frac{ \partial L }{ \partial q }- \frac{d}{dt}\frac{ \partial L }{ \partial \dot{q} } =0. \]

This implies that the function \(q\) which satisfies this equation is \[ \operatorname*{arg min}_{q} \left( \int_{a}^{b} L(x,q,q') \, dx \right). \]

It implicitly contains \[ \frac{dq}{dx} = q', \] which together with the Euler-Lagrange equation becomes the Hanilton's equations via Legendre transformation. \[ p := \frac{\partial L}{\partial \dot{q}}, \quad \dot{q} \leadsto \frac{\partial H}{\partial p}. \]

Given a functional of the form: \[ \int_a^b L(x, f(x), f'(x))\,dx \] \(L\) can be as well be considered as a generalized Lagrangian.(Dr. Trefor Bazett, ????)

• \[ \varphi_k \stackrel{\mathcal{D}}{\to} \varphi \]
• \(\exists \text{ bounded set }M\colon \forall k, x\not\in M\implies \varphi_k(x) = 0.\)
• \(\varphi_k \to \varphi\text{ unifromly}\) and \(\forall \alpha, D^\alpha \varphi_k \to D^\alpha\varphi\text{ uniformly}\) where \(\alpha\) is the multi-index.

• A distribution \(T\) on \(U\) is a linear functional on \(C^\infty_c(U)\), that is continuous when \(C^\infty_c(U)\) is given a topology called the canonical LF topology.
• \(T\colon \mathcal{D}(\mathbb{R}^n) \to \mathbb{R}\) with
  • Linearity
  • Continuity
    • Sequentially continuous with respect to the convergence of the test functions. \[ \varphi_k\stackrel{D}{\to} \varphi \implies T(\varphi_k) \to T(\varphi). \]

• Product of two distributions is also a distribution: \[ (f\cdot S)(\varphi) = (T_f\cdot S)(\varphi) := S(f\cdot \varphi) \] where \(S\in \mathcal{D}'(\mathbb{R}^n), f\in C^\infty(\mathbb{R}^n)\).
• Equivalently: \[ \langle f\cdot S, \varphi\rangle := \langle S, f\cdot\varphi\rangle. \]
• Motivated by \(T_f\cdot T_g = T_{f\cdot g}\).

• For a invertible linear map \( A\colon \mathbb{R}^n \to \mathbb{R}^n \): \[ \langle T\circ A, \varphi\rangle := \langle T, |\det A|^{-1}\varphi\circ A^{-1}\rangle. \]
• With different notation: \[ \langle T(Ax), \varphi(x)\rangle := \langle T(x), |\det A|^{-1}\varphi(A^{-1}x)\rangle. \]

• Extension of convolution.
• \[ \langle \psi\ast T, \varphi\rangle := \langle T, \check{\psi}\ast \varphi\rangle \] where the check operator simply reverses the argument: \(\check{\psi}(x) = \psi(-x)\).
• The convolution \(*\colon \mathcal{D}(\mathbb{R}^n)\times \mathcal{D}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)\) is a bilinear map, which can be thought of as a multiplication.
• Notably, \[ \psi \ast \delta = \psi. \]

• For every compact subset \(K\subseteq U\), there exists constants \(C > 0\) and \(N\in \mathbb{N}\) such that for all \(f\in C_c^\infty(U)\) with support contained in \(K\): \[ |T(f)| \le C\sup\{|\partial^\alpha f(x)| : x\in U, |\alpha| \le N\}. \]
• For every compact subset \(K\subseteq U\) an every sequence \(\{f_i\}_{i=0}^\infty\) in \(C^\infty_c(U)\) whose supports are contained in \(K\): If \(\{\partial^\alpha f_i\}_{i=1}^\infty\) converges uniformly to zero on \(U\) for every multi-index \(\alpha\), then \(T(f_i) \to 0\).

• Distribution can be added and scalar multiplied.
• Distributions forms a vector space equipped with the bilinear map: \[ \langle \cdot, \cdot \rangle\colon \mathcal{D}'(\mathbb{R}^n)\times \mathcal{D}(\mathbb{R}^n) \to \mathbb{R} (\text{or $\mathbb{C}$}). \]

• Even: \(\delta(x) = \delta(-x)\)
• Dirac delta \(\delta(x-y)\) is the continuous version of the Kronecker delta \(\delta_{ij}\)

The family of function \(\eta_\varepsilon (x)\) such that: \[ \lim_{\varepsilon\to 0+}\eta_\varepsilon(x) = \delta(x). \]