When Mathematics Faced Infinity: The Legacy of Fourier (part 2)
In our first part, we explored how Fourier’s idea of breaking down periodic functions into sine waves helps us understand complex patterns. Now, let’s continue the journey with something very physical: heat flowing through a metal ring. What starts as a warm patch eventually spreads out. But how exactly? The answer naturally leads us from Fourier series to one of the most important tools in mathematics: the Fourier Transform.
The Setup: Heat Around a Ring
Let’s imagine a thin metal ring with uneven temperature — hotter in some places, cooler in others. We denote the initial temperature distribution as \( f(x) \), where \( x \in [0,1] \), and the function is periodic: \( f(0) = f(1) \). Over time, the heat spreads out according to the heat equation:
$$ \frac{\partial u}{\partial t} = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}, $$ where \( u(x,t) \) is the temperature at position \( x \) and time \( t \), and we’ve taken the thermal diffusivity \( \kappa = 1/2 \) just to simplify our calculations.
We’re interested in solving this equation given \( u(x,0) = f(x) \).
A Natural Strategy: Fourier Series
Since the ring is periodic, it makes perfect sense to expand \( u(x,t) \) in a Fourier series:
$$ u(x,t) = \sum_{k=-\infty}^{\infty} c_k(t) e^{2\pi i k x}. $$ This is like saying: "Let’s write the temperature as a sum of harmonic waves." We plug this into the heat equation, and find that each coefficient satisfies:
$$ \frac{d c_k}{dt} = - 2\pi^2 k^2 c_k(t), $$ whose solution is:
$$ c_k(t) = c_k(0) e^{-2\pi^2 k^2 t}. $$ Now we determine the initial values \( c_k(0) \). At \( t = 0 \), we must recover the initial temperature distribution \( f(x) \), so we match:
$$ f(x) = \sum_{k=-\infty}^{\infty} \hat{f}(k) e^{2\pi i k x}, $$ with the coefficients:
$$ \hat{f}(k) = \int_0^1 f(y) e^{-2\pi i k y} \, dy. $$ So finally, our full solution becomes:
$$ u(x,t) = \sum_{k=-\infty}^{\infty} \hat{f}(k) e^{-2\pi^2 k^2 t} e^{2\pi i k x}. $$ Each frequency component of the initial distribution decays exponentially — the sharper the wiggle, the faster it fades. A symphony of heat slowly goes silent, leaving only the smoothest melody behind.
From Fourier Series to a Green’s Function
Let’s define a new function:
$$ G(x,t) = \sum_{k=-\infty}^{\infty} e^{-2\pi^2 k^2 t} e^{2\pi i k x}. $$ This is called the Green’s function or heat kernel. It represents how heat spreads from a single point over time. Think of it as the fingerprint of the system — the echo from a single clap in a circular room.
To link this to our initial data, we recall that \( \hat{f}(k) = \int_0^1 f(y) e^{-2\pi i k y} dy \), so we plug it into our solution:
$$ u(x,t) = \sum_{k=-\infty}^{\infty} \left( \int_0^1 f(y) e^{-2\pi i k y} dy \right) e^{-2\pi^2 k^2 t} e^{2\pi i k x}. $$ We switch the order of summation and integration, leading to:
$$ u(x,t) = \int_0^1 f(y) \left( \sum_{k=-\infty}^{\infty} e^{-2\pi^2 k^2 t} e^{2\pi i k (x - y)} \right) dy. $$ This means:
$$ u(x,t) = \int_0^1 f(y) G(x - y, t) dy = (f * G)(x,t). $$ This is a convolution. Each point of \( f \) spreads out like a ripple, and the result is the superposition of all these effects. The idea of convolution is subtle, but intuitive: every part of the initial distribution contributes to every point in space, blended through the heat kernel like a musical echo slowly filling the room.
How the Spectrum Looks
For a function with period 1, its frequency content appears at evenly spaced points: \( k = 0, \pm1, \pm2, \dots \). These spikes mirror each other: the energy at frequency \( +k \) equals that at \( -k \). It’s a perfect symmetry in the frequency domain. But when the function is no longer periodic, this tidy structure vanishes. Instead of discrete spikes, we get a smooth curve — a continuous spectrum. We move from a row of tuning forks to a fluid violin note that slides without gaps. And that’s the gateway to the Fourier Transform.
Domain of Time vs Frequency Domain
Functions like \( f(t) \) live in the "time domain" (or "space domain", depending on the variable). Their Fourier coefficients \( \hat{f}(k) \), or transforms \( \hat{f}(\xi) \), live in the frequency domain. The magic of Fourier is translating between these worlds — decoding how frequency content builds a signal.
Stretching the Period to Infinity
Now here’s the brilliant trick. When we take a periodic function and stretch its period \( T \to \infty \), we effectively stop repeating it. Even if the original function lives only between \( a \) and \( b \), we can imagine it as a limit of repeating copies. In that limit, the discrete spectrum becomes continuous. The Fourier coefficients \( \hat{f}(k) \) become a continuous function \( \hat{f}(\xi) \). The clear spikes blur into a fluid spectrum. That’s where the Fourier Transform lives: it’s the limit of the Fourier Series as the period goes to infinity.
The Fourier Transform and Its Inverse
We now define the Fourier Transform:
$$ \hat{f}(\xi) = \int_{-\infty}^{\infty} f(t) e^{-2\pi i \xi t} \, dt, $$ and the Inverse Fourier Transform:
$$ f(t) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i \xi t} \, d\xi. $$ The Fourier Transform is the natural generalisation of Fourier coefficients — a continuous picture of a function’s frequency content. And the Inverse Fourier Transform generalises the reconstruction of a signal from its frequencies. These are not just formulas, but the ultimate toolkit for listening to the hidden harmonies in any signal.
A Final Reflection
We began with heat flowing around a ring and ended up with the Fourier Transform — a mathematical microscope that decomposes signals into waves. The Green’s function helped us understand how heat spreads. The convolution showed us how simple effects stack together. And the Fourier Transform revealed that all of this lives in a second, invisible world: the frequency domain.
What started as a physical intuition became a universal language. Whether it’s sound, light, ocean waves, or temperature, the logic is the same: break things down into their hidden waves, then reassemble them with clarity.
In the next post, we will formally derive the Fourier transform as the limiting case of Fourier coefficients, and the inverse Fourier transform as the limiting case of Fourier series, all within the broader context of analysing non-periodic phenomena.
See ya later! César
The Setup: Heat Around a Ring
Let’s imagine a thin metal ring with uneven temperature — hotter in some places, cooler in others. We denote the initial temperature distribution as \( f(x) \), where \( x \in [0,1] \), and the function is periodic: \( f(0) = f(1) \). Over time, the heat spreads out according to the heat equation:
$$ \frac{\partial u}{\partial t} = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}, $$ where \( u(x,t) \) is the temperature at position \( x \) and time \( t \), and we’ve taken the thermal diffusivity \( \kappa = 1/2 \) just to simplify our calculations.
We’re interested in solving this equation given \( u(x,0) = f(x) \).
A Natural Strategy: Fourier Series
Since the ring is periodic, it makes perfect sense to expand \( u(x,t) \) in a Fourier series:
$$ u(x,t) = \sum_{k=-\infty}^{\infty} c_k(t) e^{2\pi i k x}. $$ This is like saying: "Let’s write the temperature as a sum of harmonic waves." We plug this into the heat equation, and find that each coefficient satisfies:
$$ \frac{d c_k}{dt} = - 2\pi^2 k^2 c_k(t), $$ whose solution is:
$$ c_k(t) = c_k(0) e^{-2\pi^2 k^2 t}. $$ Now we determine the initial values \( c_k(0) \). At \( t = 0 \), we must recover the initial temperature distribution \( f(x) \), so we match:
$$ f(x) = \sum_{k=-\infty}^{\infty} \hat{f}(k) e^{2\pi i k x}, $$ with the coefficients:
$$ \hat{f}(k) = \int_0^1 f(y) e^{-2\pi i k y} \, dy. $$ So finally, our full solution becomes:
$$ u(x,t) = \sum_{k=-\infty}^{\infty} \hat{f}(k) e^{-2\pi^2 k^2 t} e^{2\pi i k x}. $$ Each frequency component of the initial distribution decays exponentially — the sharper the wiggle, the faster it fades. A symphony of heat slowly goes silent, leaving only the smoothest melody behind.
From Fourier Series to a Green’s Function
Let’s define a new function:
$$ G(x,t) = \sum_{k=-\infty}^{\infty} e^{-2\pi^2 k^2 t} e^{2\pi i k x}. $$ This is called the Green’s function or heat kernel. It represents how heat spreads from a single point over time. Think of it as the fingerprint of the system — the echo from a single clap in a circular room.
To link this to our initial data, we recall that \( \hat{f}(k) = \int_0^1 f(y) e^{-2\pi i k y} dy \), so we plug it into our solution:
$$ u(x,t) = \sum_{k=-\infty}^{\infty} \left( \int_0^1 f(y) e^{-2\pi i k y} dy \right) e^{-2\pi^2 k^2 t} e^{2\pi i k x}. $$ We switch the order of summation and integration, leading to:
$$ u(x,t) = \int_0^1 f(y) \left( \sum_{k=-\infty}^{\infty} e^{-2\pi^2 k^2 t} e^{2\pi i k (x - y)} \right) dy. $$ This means:
$$ u(x,t) = \int_0^1 f(y) G(x - y, t) dy = (f * G)(x,t). $$ This is a convolution. Each point of \( f \) spreads out like a ripple, and the result is the superposition of all these effects. The idea of convolution is subtle, but intuitive: every part of the initial distribution contributes to every point in space, blended through the heat kernel like a musical echo slowly filling the room.
How the Spectrum Looks
For a function with period 1, its frequency content appears at evenly spaced points: \( k = 0, \pm1, \pm2, \dots \). These spikes mirror each other: the energy at frequency \( +k \) equals that at \( -k \). It’s a perfect symmetry in the frequency domain. But when the function is no longer periodic, this tidy structure vanishes. Instead of discrete spikes, we get a smooth curve — a continuous spectrum. We move from a row of tuning forks to a fluid violin note that slides without gaps. And that’s the gateway to the Fourier Transform.
Domain of Time vs Frequency Domain
Functions like \( f(t) \) live in the "time domain" (or "space domain", depending on the variable). Their Fourier coefficients \( \hat{f}(k) \), or transforms \( \hat{f}(\xi) \), live in the frequency domain. The magic of Fourier is translating between these worlds — decoding how frequency content builds a signal.
Stretching the Period to Infinity
Now here’s the brilliant trick. When we take a periodic function and stretch its period \( T \to \infty \), we effectively stop repeating it. Even if the original function lives only between \( a \) and \( b \), we can imagine it as a limit of repeating copies. In that limit, the discrete spectrum becomes continuous. The Fourier coefficients \( \hat{f}(k) \) become a continuous function \( \hat{f}(\xi) \). The clear spikes blur into a fluid spectrum. That’s where the Fourier Transform lives: it’s the limit of the Fourier Series as the period goes to infinity.
The Fourier Transform and Its Inverse
We now define the Fourier Transform:
$$ \hat{f}(\xi) = \int_{-\infty}^{\infty} f(t) e^{-2\pi i \xi t} \, dt, $$ and the Inverse Fourier Transform:
$$ f(t) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i \xi t} \, d\xi. $$ The Fourier Transform is the natural generalisation of Fourier coefficients — a continuous picture of a function’s frequency content. And the Inverse Fourier Transform generalises the reconstruction of a signal from its frequencies. These are not just formulas, but the ultimate toolkit for listening to the hidden harmonies in any signal.
A Final Reflection
We began with heat flowing around a ring and ended up with the Fourier Transform — a mathematical microscope that decomposes signals into waves. The Green’s function helped us understand how heat spreads. The convolution showed us how simple effects stack together. And the Fourier Transform revealed that all of this lives in a second, invisible world: the frequency domain.
What started as a physical intuition became a universal language. Whether it’s sound, light, ocean waves, or temperature, the logic is the same: break things down into their hidden waves, then reassemble them with clarity.
In the next post, we will formally derive the Fourier transform as the limiting case of Fourier coefficients, and the inverse Fourier transform as the limiting case of Fourier series, all within the broader context of analysing non-periodic phenomena.
See ya later! César
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