FOURIER TRANSFORM: PROPERTIES, APPLICATIONS, EXAMPLES - MATHS - 2025

The Fourier transform is an analytical adequacy method oriented to integrable functions that belongs to the family of integral transforms. It consists of a redefinition of functions f (t) in terms of Cos (t) and Sen (t).

The trigonometric identities of these functions, together with their derivation and antiderivation characteristics, serve to define the Fourier transform through the following complex function:

Which is true as long as the expression makes sense, that is, when the improper integral is convergent. Algebraically the Fourier transform is said to be a linear homeomorphism.

Every function that can be worked with a Fourier transform must present null outside a defined parameter.

Properties

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The Fourier transform meets the following properties:

Existence

To verify the existence of the Fourier transform in a function f (t) defined in the reals R, the following 2 axioms must be fulfilled:

1. f (t) is piecewise continuous for all R
2. f (t) is integrable in R

Fourier transformation linearity

Let M (t) and N (t) be any two functions with definite Fourier transforms, with any constants a and b.

F (z) = a F (z) + b F (z)

Which is also supported by the linearity of the integral of the same name.

Fourier transform of a derivative

There is a function f that is continuous and integrable in all reals, where:

And the derivative of f (f ') is continuous and piecewise defined throughout R

The Fourier transform of a derivative is defined by integration by parts, by the following expression:

F (z) = iz F (z)

In the derivations of higher order, it will be applied in a homologous way, where for all n 1 we have:

F (z) = (iz) ⁿ F (z)

Fourier transform differentiation

There is a function f that is continuous and integrable in all reals, where:

Fourier transform of a translation

For every θ that belongs to a set S and T that belongs to the set S ', we have:

F = e ^-iay FF = e ^-iax F

With τ _a working as the translation operator on the vector a.

Translation of the Fourier transform

For every θ that belongs to a set S and T that belongs to the set S ', we have:

τ _a F = F τ _a F = F

For all of which belong to R

Fourier transform of a scale group

For all θ that belongs to a set S. T that belongs to the set S '

λ belonging to R - {0} we have:

F = (1 / -λ-) F ( y / λ)

F = (1 / -λ-) F (y / λ)

If f is a continuous and clearly integrable function, where a> 0. Then:

F (z) = (1 / a) F (z / a)

To demonstrate this result, we can proceed with the change of variable.

When T → + then s = at → + ∞

When T → - then s = at → - ∞

Symmetry

To study the symmetry of the Fourier transform, the identity of Parseval and the Plancherel formula must be verified.

We have θ and δ that belong to S. From there it can be deduced that:

Getting

1 / (2π) ^d { F, F } Parseval identity

1 / (2π) ^{d / 2} - F - _L ² _R ^d Plancherel formula

Fourier transform of a convolution product

Pursuing similar objectives as in the Laplace transform, the convolution of functions refers to the product between their Fourier transforms.

We have f and g as 2 bounded, defined and completely integrable functions:

F (f * g) = F (f). F (g)

F (f). F (g) = F (f. G)

Continuity and fall into infinity

What is the Fourier transform for?

It serves primarily to significantly simplify equations, while transforming derived expressions into power elements, denoting differential expressions in the form of integrable polynomials.

In the optimization, modulation and modeling of results, it acts as a standardized expression, being a frequent resource for engineering after several generations.

The Fourier series

They are series defined in terms of Cosines and Sines; They serve to facilitate work with general periodic functions. When applied, they are part of the techniques for solving ordinary and partial differential equations.

Fourier series are even more general than Taylor series, because they develop periodic discontinuous functions that do not have Taylor series representation.

Other forms of the Fourier series

To understand the Fourier transform analytically, it is important to review the other ways in which the Fourier series can be found, until we can define the Fourier series in its complex notation.

-Fourier series on a function of period 2L

Many times it is necessary to adapt the structure of a Fourier series to periodic functions whose period is p = 2L> 0 in the interval.

-Fourier series in odd and even functions

The interval is considered, which offers advantages when taking advantage of the symmetric characteristics of the functions.

If f is even, the Fourier series is established as a series of Cosines.

If f is odd, the Fourier series is established as a series of Sines.

-Complex notation of the Fourier series

If we have a function f (t), which meets all the developability requirements of the Fourier series, it is possible to denote it in the interval using its complex notation:

Applications

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Calculation of the fundamental solution

The Fourier transform is a powerful tool in the study of partial differential equations of the linear type with constant coefficients. They apply for functions with unbounded domains equally.

Like the Laplace transform, the Fourier transform transforms a partial derivative function into an ordinary differential equation much simpler to operate.

The Cauchy problem for the heat equation presents a field of frequent application of the Fourier transform where the nucleus of heat or Dirichlet nucleus is generated.

Regarding the calculation of the fundamental solution, the following cases are presented where it is common to find the Fourier transform:

Signal theory

The general reason for the application of the Fourier transform in this branch is mainly due to the characteristic decomposition of a signal as an infinite superposition of more easily treatable signals.

It can be a sound wave or an electromagnetic wave, the Fourier transform expresses it in a superposition of simple waves. This representation is quite frequent in electrical engineering.

On the other hand, are examples of application of the Fourier transform in the field of signal theory:

Examples

Example 1

Define the Fourier transform for the following expression:

We can also represent it in the following way:

F (t) = Sen (t)

The rectangular pulse is defined:

p (t) = H _{(t + k)} - H _{(t - k)}

The Fourier transform is applied to the following expression which resembles the modulation theorem.

f (t) = p (t) Sen (t)

Where: F = (1/2) i

And the Fourier transform is defined by:

F = (1/2) i

Example 2

Define the Fourier transform for the expression:

Since f (h) is an even function, it can be stated that

Integration by parts is applied by selecting the variables and their differentials as follows

u = sin (zh) du = z cos (zh) dh

dv = h (e ^-h) ² v = (e ^-h) ² /2

Substituting you have

After evaluating under the fundamental theorem of calculus

Applying prior knowledge regarding first-order differential equations, the expression is denoted as

To obtain K we evaluate

Finally, the Fourier transform of the expression is defined as

Proposed exercises

• 

• 

• Get the transform of the expression W / (1 + w ²)

References

1. Duoandikoetxea Zuazo, J., Fourier analysis. Addison– Wesley Iberoamericana, Autonomous University of Madrid, 1995.
2. Lions, JL, Mathematical Analysis and Numerical Methods for Science and Technology. Springer – Verlag, 1990.
3. Lieb, EH, Gaussian kernels have only gaussian maximizers. Invent. Math. 102, 179-208, 1990.
4. Dym, H., McKean, HP, Fourier Series and Integrals. Academic Press, New York, 1972.
5. Schwartz, L., Théorie des Distributions. Ed. Hermann, Paris, 1966.