Suppose you have a constant function _f_(_x_) = _c_. What is the Fourier transform of _f_? We will show why the direct approach doesn’t work, give two hand-wavy approaches, and a rigorous definition. ## Direct approach Unfortunately there are multiple conventions for defining the Fourier transform. For this post, we will define the Fourier transform of a function _f_ to be If _f_(_x_) = _c_ then the integral diverges unless _c_ = 0. ## Heuristic approach The more concentrated a function is in the time domain, the more it spreads out in the frequency domain. And the more spread out a function is in the time domain, the more concentrated it is in the frequency domain. If you think this sounds like the Heisenberg uncertainty principle, you’re right: there is a connection. A constant function is as spread out as possible, so it seems that its Fourier transform should be as concentrated as possible, i.e. a delta function. The delta function isn’t literally a function, but it can be made rigorous. More on that below. ## Gaussian density approach The Fourier transform of the Gaussian function exp(− _x_ ²/2) is the same function, i.e. the Gaussian function is a fixed point of the Fourier transform. More generally, the Fourier transform of the density function for a normal random variable with standard deviation σ is the density function for a normal random variable with standard deviation 1/σ. As σ gets larger, the density becomes flatter. So we could think of our function _f_(_x_) = _c_ as some multiple of a Gaussian density in the limit as σ goes to infinity. The Fourier transform is then some multiple of a Gaussian density with σ = 0, i.e. a point mass or delta function. ## Rigorous approach If _f_ and φ are two well-behaved functions then In other words, we can move the “hat” representing the Fourier transform from one function to the other. The equation above is a theorem when _f_ and φ are nice functions. We can use it to motivate a definition when the function _f_ is not so nice but the function φ is very nice. Specifically, we will assume φ is an infinitely differentiable function that goes to zero at infinity faster than any polynomial. Given a Lebesgue integrable function _f_ , we can think of _f_ as a linear operator via the map More generally, we can define a _distribution_ to be any continuous [1] linear operator from the space of test functions to the complex numbers. A distribution that can be defined by integral as above is called a _regular_ distribution. When we say we’re taking the Fourier transform of the constant function _f_(_x_) = _c_ , we’re actually taking the Fourier transform of the regular distribution associated with _f_. [2] Not all distributions are regular. The delta “function” δ(_x_) is a distribution that acts on test functions by evaluating them at 0. We define the Fourier transform of (the regular distribution associated with) a function _f_ to be the distribution whose action on a test function φ equals the integral of the product of _f_ and the Fourier transform of φ. When a function is Lebesgue integrable, this definition matches the classical definition. With this definition, we can calculate that the Fourier transform of a constant function _c_ equals Note that with a different convention for defining the Fourier transform, you might get 2π c δ or just c δ. An advantage of the convention that we’re using is that the Fourier transform of the Fourier transform of _f_(_x_) is _f_(− _x_) and not some multiple of _f_(− _x_). This implies that the Fourier transform of √2π δ is 1 and so the Fourier transform of δ is 1/√2π. ## Related posts * How to differentiate a non-differentiable function * Two meanings of distribution * The Dirac comb (Sha) function * Bessel series for a constant [1] To define continuity we need to put a topology on the space of test functions. That’s too much for this post. [2] The constant function doesn’t have a finite integral, but its product with a test function does because test functions decay rapidly. In fact, even the product of a polynomial with a test function is integrable