02 Operators

Operator

A function: functions $\to$ functions. All functions are operators. Transform: an operator which simplifies some operations. Adjoint of operator T: \(T^{} : \dprod{Tu,v} = \dprod{u, T^{}v}\).

Eigenfunctions

ev of D is $e^x$.

Differential operator

D or $D_x=\frac{d}{dx}$. A Linear operator.

Operators $T=\sum _k{c}_k{D}^k$. Any polynomial in D with function coefficients also a diff operator.

Vector and matrix calculus operators

See linear algebra ref.

Divergence operator

$divf:=\mathrm{\nabla }.f:=\sum _i\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}$.

Laplacian and Elliptic operators

(Laplace) $\mathrm{\Delta }=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_1^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_2^2}$. Elliptic op: $L=div(a\mathrm{\nabla })$: a is scalar.

Integral operator

Aka Integral transform. $Tf(u)={\int }_{t_1}^{t_2}K(t,u)f(t)dt$: changing the domain of the fn. K is the Kernel function or the nucleus of the transform. A Linear operator.

Symmetric kernels are indifferent to permutation of (t,u).

Inverse Kernel

$K^{-1}(u,t)$ yields inverse transform:\ $f(t)={\int }_{u_1}^{u^2}{K}^{-1}(u,t)(Tf(u))du$.

Convolution

$(f.g)(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)g(x-t)dt={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x-t)g(t)dt$.

You take a $g$, reflect it about 0, translate it by $t$. So, it is convoluted!

Properties: commutative, associative, distributive.

K(t, u) as a basis function

Maybe ${\int }_{t_1}^{t_2}abdt$ specifies an inner product in a function space. Maybe K(t, u) specifies the form of basis functions in that space: so for a fixed u, you get a fixed basis fn. Maybe you are trying to find the form of the component of f(t) along the basis fn K(t, u). The integral transform is the solution. Eg: Fourier transform.

Or maybe, you have the form of the component f(u) along a certain basis fn K(u, t), and ye want to reconstruct the fn form f(t) from it. Then the inverse integral operator is useful. Eg: Inverse Fourier transform.

Some vector differential operators

Jacobian matrix

Generalizes f’(x) for Vector map F. \ Matrix $J=J_F(x_1\dots )=\frac{\mathrm{\partial }(y_1\dots )}{\mathrm{\partial }(x_1\dots )}$: $J_{i,j}=\frac{\mathrm{\partial }{y}_i}{\mathrm{\partial }{x}_j}$.

Hessian matrix

Of scalar f(x) wrt vector x: $H_{i,j}=D_i{D}_{j}f(x)$: Always symmetric. Aka ${\mathrm{\nabla }}^{2}f(x)$.

If ${\mathrm{\nabla }}^{2}f(x)>0$ or +ve definite: f(x) convex, $\mathrm{\exists }$ unique global minimum.