Fourier's motivation

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Although Fourier series bare his name, they were well known to the great analysists of the eighteenth century (including Euler), even if it was Fourier who first had the proper insight into the true nature of these expansions. The Fourier integral is entirely Fourier's own discovery---it appeared for the first time in Fourier's monograph Théorie analytique de la chaleur (The Analytic Theory of Heat in French, 1822) and it was he who recognized its fundamental significance. The Fourier integral is a tool to analyze operators with continuous spectrum, in which all frequencies are represented.

The Fourier series expansions are applied to the class of periodic functions defined on finite interval [−ℓ, ℓ]. But not every function is periodic. Of course, we are free to extend the function outside the given interval by any definition we want and no harm is done if we continue the function by periodic repetitions. However, a function may be given in an infinite range. Fourier had the ingenious idea to consider such a function as a periodic function whose period grows to infinity. More precisely, we make ℓ increasingly larger and larger and investigate what happens in the limit when ℓ becomes infinite.

We begin our analysis with considering a function f(x) that is defined on finite interval [−ℓ, ℓ]. If this function is absolutely integrable on this interval, its Fourier coefficients are well-defined:

Of course, we have no reason to assume that an arbitrary function would be reducible to strictly periodic functions. However, we have seen in the study of the Fourier series that an arbitrary continuous and piecewise differentiable function of a finite range could certainly be resolved into an infinite superposition of sine and cosine functions, whose frequencies were not even arbitrary, but multiples of one fundamental frequency: Using Euler's formula, we can rewrite trigonometric functions in exponential form Upon setting we can introduce complex coefficients These coefficients are common to denote as It is more convenient to rewrite the Fourier series in complex form where c^✶ denotes the complex conjugate of c (in pure math, it is customary also denoted as \( \displaystyle \quad \overline{c} = \left( a + {\bf j}b \right)^{\ast} = \overline{a + {\bf j}b} = a - {\bf j}b \) ) and VP means the Cauchy principal value, so For Fourier series, Parseval's identity holds: and for complex form, To analyze the Fourier series, it is particularly instructive to introduce the function in which ξ. can take any value, because for large ℓ the coefficients cₙ demand the function values This means that the values of the function F{ξ) become a very dense in ℝ, although n still jumps in integers. The Fourier coefficients of the standard Fourier series (normalized by setting ℓ - π) can be written in the form which brings into evidence the fact that the Fourier components of the function f(x) are not some arbitrarily capricious numbers, but they are connected by the common bond that they belong to a continuous function F(ξ) that is uniquely tied to the given function f{x). This commonℓ becomes very large, approaching infinity, because in that case we do not restrict ourselves to the integer values of ξ, but to all values of aξ.

Now we extend the given function f{x) f from the range [−ℓ, ℓ] to a larger interval {−2ℓ, 2ℓ] by setting f{x) &equive; 0 outside the given range [−ℓ, ℓ]. In this zase, we need to evaluate integrals Fourier coefficients

Example 1: Linear function It was Fourier's discovery that a large class of functions of an infinite domain follow the same pattern, provided that f(x) remains absolutely integrable in the infinite interval. Let ℓ > 0 and consider a complex‑valued function which is integrable and (for simplicity) piecewise smooth. Extend f_ℓ periodically to all of ℝ with period 2ℓ: The complex Fourier series of this 2ℓ‑periodic function is where the discrete frequencies are and the Fourier coefficients are This is the standard complex Fourier series on [-ℓ,ℓ].

Embedding a nonperiodic function into this framework involves

with sufficient decay and regularity—for instance, assume f ∈ 𝔏¹(ℝ) ∩ ℭ¹(ℝ) and that f(x) → 0 as |x| → ∞. We want to represent f on all of points of ℝ.

Rewrite the series explicitly in a Riemann-sum form:

Introduce the spacing in frequency: Then So we can rewrite Hence Factor the integral: The bracketed term is a discrete approximation to a Fourier integral kernel.

As ℓ → ∞ : frequency becomes continuous.

• The interval [-ℓ,ℓ] expands to ℝ.
• The frequency spacing Δk = π/ℓ → 0.
• The discrete set { kₙ }_n∈ℤ becomes dense in ℝ.

Formally, the sum is a Riemann sum for Thus, under appropriate conditions on f (e.g., f ∈ 𝔏¹(ℝ) and some mild regularity), we expect for each continuity point x of f, and Interchanging limit and integral is justified by dominated convergence or similar theorems when f has sufficient decay and regularity (e.g., f ∈ 𝒮(ℝ)).

Identification of the Fourier transform. Define the Fourier transform (one of the standard normalizations) by

Then Assuming Fubini’s theorem applies (e.g., f ∈ 𝕃¹(ℝ) ∩ 𝔏²(ℝ), we can interchange integrals (in distribution sense): Comparing with the limit obtained from the Fourier series, we arrive at the Fourier inversion formula: for all x where f is continuous (and more generally in 𝔏²-sense).

Thus, the Fourier integral representation

is obtained as the rigorous limit of the Fourier series on [-ℓ,ℓ] as ℓ → ∞.

Summary in “series → integral” language:

• Fourier series on [-ℓ,L]:
  \[ f_{\ell} (x) \sim \sum _{n=-\infty }^{\infty } c_n^{(\ell )}e^{{\bf j}k_n x},\quad c_n^{(\ell )}=\frac{1}{2\ell}\int _{-\ell}^{\ell} f(t)\, e^{-{\bf j}k_n t}\, {\text d}t. \]
• Introduce frequency spacing:
  \[ k_n = \frac{n\pi }{\ell},\quad \Delta k =\frac{\pi }{\ell},\quad \frac{1}{2\ell} = \frac{\Delta k}{2\pi }. \]
• Rewrite as Riemann sum in k:
  \[ f_{\ell}(x)\sim \sum _{n=-\infty }^{\infty }\frac{\Delta k}{2\pi }\left( \int _{-\ell}^{\ell} f(t)\, e^{-{\bf j}k_n t}\, {\text d}t\right) e^{{\bf j}k_n x}. \]
• Let ℓ → ∞:
  \[ \sum _n\frac{\Delta k}{2\pi }\cdots \; \longrightarrow \; \frac{1}{2\pi }\int _{-\infty }^{\infty }\cdots \, {\text d}k, \]
  giving
  \[ f(x) = \frac{1}{2\pi }\int _{-\infty }^{\infty }\hat {f}(k)\, e^{{\bf j}kx}\,{\text d}k. \]

This is the rigorous, academic transition from Fourier series (discrete spectrum on a finite interval) to the Fourier integral / Fourier transform (continuous spectrum on ℝ), with the Riemann-sum limit ℓ → ∞ as the bridge.

 

We present another derivation of the Fourier integral from Fourier series of different flavor for pedagogical reasons because education is mostly a repetition. We start with an absolutely integrable function f on interval [−π, π]. This condition, f ∈ 𝔏(−π, π), guarantees existence of Fourier coefficients The corresponding Fourier series is associated with function f: We use the following notations: Then using \[ \cos (kx) = \frac{1}{2} \left[ e^{{\bf j}kx} + e^{-{\bf j}kx} \right] , \quad \sin (kx) = \frac{1}{2{\bf j}} \left[ e^{{\bf j}kx} - e^{-{\bf j}kx} \right] , \quad {\bf j}^2 = -1, \]