Integral Transforms: Origins & Derivations

[amcavoy]

I am curious about the different integral transforms. I found definitions online, but where did they come from? Does anyone know a good site that shows the derivation / need for the common integral transforms?

Thanks.

 

[HallsofIvy]

The LaPlace transform is an integral transform:
$$L(f)= \int_0^{\infty}e^{-st}f(t)dt$$
which has the nice property of transforming (linear) differential equations to algebraic equations.

The Fourier transform:
$$F(f)= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-itx}f(t)dt$$
is a generalization of Fourier series to infinite intervals.

 

[amcavoy]

Alright. So take, for instance, the Laplace Transform. Was it stumbled upon by accident, or is there an actual derivation for it?

 

[aranyakam]

Read "Mathews & Walker - Mathematical methods of Physics"

 

[amcavoy]

Are there any online resources?

Thanks again.

 

[lurflurf]

An Integral transform is a linear transform of the form.
$$L[f(t)]:=\int_a^bK(s,t)f(t)dt$$
where a and b may be + or - infinity and K is called the kernel.
Integral transform kernels are selected using three main traits.
1. Existence/convergence
For functions under consideration does the transform exist? What about inversion? Do integrals that arise converge? In some cases the definition of integral is expanded using Cauchy principle values or distribution theory.
2. Computability
Can the transform be found analytically or numerically? What about inversion? For instance the Laplace, and Fourier transforms are often computed analytically. Numerical inversion of Laplace transform is difficult. The Hankel and Fourier transforms are often inverted numerically.
3. Operational properties
Do functions operated on by some operator have transforms that are easy to deal with?
In practice not many Transforms are of wide applicability they are
-Laplace transform and variations. Useful for temporal problems.
-Fourier transform and variations. Useful for spatial problems.
-Special transforms. Useful in very limited situations. Things like Greens functions that are often useful for only one specific problem.

Alright. So take, for instance, the Laplace Transform. Was it stumbled upon by accident, or is there an actual derivation for it?

Derivation is I think the wrong choice of words. One can be led to the transform. The first way is as a variation on Taylors series.
$$\sum_{n=0}^{\infty} a_ne^{-n s}\Delta n$$
This is Taylors series in a new form that leads the way to the Laplace transform. Delta(n)=1 so it is normaly omited, and normally z is used instead of exp(-s).
Now we allow n to be a sequence rather then just integers
$$\sum_{n=0}^{\infty} a_ne^{-\lambda_n s}\Delta\lambda_n$$
this is a Riemann sum so it is natural to take the limit to an integral
$$\int_0^{\infty}f(t)e^{-st}dt$$
A similar process leads from the Fourier trig series to the Fourier transform.
$$\sum_{n=-\infty}^{\infty} a_ne^{i n s}\Delta n$$
allow nonintegers
$$\sum_{n=-\infty}^{\infty} a_ne^{i\lambda_n s}\Delta\lambda_n$$
Limit to integral
$$\int_{-\infty}^{\infty}f(t)e^{ist}dt$$
We may also be led to the Laplace transform by operational properties. We desire a kernel so that
$$\int_0^{\infty}K(s,t)f'(t)dt=s\int_0^{\infty}K(s,t)f(t)dt$$
integration by parts on the left side leads us to
$$K(s,t)f(t)|_{t=0}^{t=\infty}=\int_0^{\infty}(K'(s,t)+sK(s,t))f(t)dt$$
the ' notation means t partial
Now we make things simple by requiring the left side to be zero. Thus the right side is zero giving a differential equation
K'(s,t)+sK(s,t)=0
the solution is
K(s,t)=A(s)exp(-s*t)
where A(s) is an abitrary function of s
The standard choice is A(s)=1 to make things nice.
The second most popular choice used in some older work is the p-multiplied form where A(s)=s.
Thus we are again led to the Laplace transform.
Now I will list several transforms they are related to one another, so we really have few. Also there are many more than these, but they are mostly either variations of these, or of quite limited usefulness.
-Laplace transform
-p-multiplied Laplace transform (Laplace multiplies by the variable)
-Bilateral Laplace Transform (Laplace with lower limit -infinity)
-Stiltjes transform (Laplace twice)
-Fourier transform
-Fourier cos transform (real part of fourier)
-Fourier sin transform (imaginary part of fourier)
-Hartley transform (F. cos transform - f. sin transform a Fourier like real transform)
-various Finite range Fourier transforms (any Fourier like transform on a finite interval)
-Hilbert transform (similar to Stiltjes the principle value of a singular integral)
-Hankel transform (involves bessel functions like a Fourier transform for polar coordinates)
-various transforms that are like Fourier for some coordinate system.