Engineering Analysis/L2 Space

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The L2 space is very important to engineers, because functions in this space do not need to be continuous. Many discontinuous engineering functions, such as the delta (impulse) function, the unit step function, and other discontinuous functions are part of this space.

L2 Functions[edit]

A large number of functions qualify as L2 functions, including uncommon, discontinuous, piece-wise, and other functions. A function which, over a finite range, has a finite number of discontinuities is an L2 function. For example, a unit step and an impulse function are both L2 functions. Also, other functions useful in signal analysis, such as square waves, triangle waves, wavelets, and other functions are L2 functions.

In practice, most physical systems have a finite amount of noise associated with them. Noisy signals and random signals, if finite, are also L2 functions: this makes analysis of those functions using the techniques listed below easy.

Null Function[edit]

The null functions of L2 are the set of all functions φ in L2 that satisfy the equation:

\int_a^b |\phi(x)|^2dx = 0

for all a and b.

Norm[edit]

The L2 norm is defined as follows:


[L2 Norm]

\|f(x)\|_{L_2} = \sqrt{\int_a^b |f(x)|^2dx}

If the norm of the function is 1, the function is normal.

We can show that the derivative of the norm squared is:

\frac{\partial \|x\|^2}{\partial x} = 2x

Scalar Product[edit]

The scalar product in L2 space is defined as follows:


[L2 Scalar Product]

\langle f(x), g(x)\rangle_{L_2} = \int_a^bf(x)g(x)dx

If the scalar product of two functions is zero, the functions are orthogonal.

We can show that given coefficient matrices A and B, and variable x, the derivative of the scalar product can be given as:

\frac{\partial}{\partial x}\langle Ax, Bx\rangle = A^TBx + B^TAx

We can recognize this as the product rule of differentiation. Generalizing, we can say that:

\frac{\partial}{\partial x}\langle f(x), g(x)\rangle = f'(x)g(x) + f(x)g'(x)

We can also say that the derivative of a matrix A times a vector x is:

\frac{d}{dx}Ax = A^T

Metric[edit]

The metric of two functions (we will not call it the "distance" here, because that word has no meaning in a function space) will be denoted with ρ(x,y). We can define the metric of an L2 function as follows:


[L2 Metric]

\rho(x, y)_{L_2} = \sqrt{\int_a^b|f(x) - g(x)|^2dx}

Cauchy-Schwarz Inequality[edit]

The Cauchy-Schwarz Inequality still holds for L2 functions, and is restated here:

|\langle f(x), g(x)\rangle| \le \|f\|\|g\|

Linear Independence[edit]

A set of functions in L2 are linearly independent if:

a_1f_1(x) + a_2f_2(x) + \cdots + a_nf_n(x) = 0

If and only if all the a coefficients are 0.

Grahm-Schmidt Orthogonalization[edit]

The Grahm-Schmidt technique that we discussed earlier still works with functions, and we can use it to form a set of linearly independent, orthogonal functions in L2.

For a set of functions φ, we can make a set of orthogonal functions ψ that space the same space but are orthogonal to one another:


[Grahm-Schmidt Orthogonalization]

\psi_1 = \phi_1
\psi_i = \phi_i - \sum_{n=1}^{i-1}\frac{\langle \psi_n, \phi_{i}\rangle}{\langle \psi_n, \psi_n\rangle}\psi_n

Basis[edit]

The L2 is an infinite-basis set, which means that any basis for the L2 set will require an infinite number of basis functions. To prove that an infinite set of orthogonal functions is a basis for the L2 space, we need to show that the null function is the only function in L2 that is orthogonal to all the basis functions. If the null function is the only function that satisfies this relationship, then the set is a basis set for L2.

By definition, we can express any function in L2 as a linear sum of the basis elements. If we have basis elements φ, we can define any other function ψ as a linear sum:

\psi(x) = \sum_{n = 1}^\infty a_n\phi_n(x)

We will explore this important result in the section on Fourier Series.