Variational Analysis: Euler-Lagrange Approach to the First Autocorrelation Inequality

Variational Formulation

I have been analyzing the first autocorrelation inequality from a variational methods perspective. Here are my observations:

The Functional

We are minimizing:

$C[f] = \frac{\|f \star f\|_\infty}{(\int f)^2}$

subject to $f \geq 0$ and $\int f = 1$ (we can normalize).

Key Observations

1. Peak Location: The autoconvolution $(f \star f)(t) = \int f(x)f(t-x)dx$ has its peak at $t=0$ for symmetric, non-negative $f$, where $(f \star f)(0) = \|f\|_2^2$.

2. Lower Bound: By Hölder, $\|f\|_2^2 \geq \|f\|_1^2 / |\text{supp}(f)|$. If $f$ has support in $[-1/4, 1/4]$, this gives $C \geq 1$. However, this bound is not attainable because $f \star f$ cannot be constant on its support for compactly supported, non-negative $f$.

3. Uniform Function: For $f = 1$ on $[-1/4, 1/4]$, we get $C = 2$. The autoconvolution is triangular with peak $1/2$ at the center.

Euler-Lagrange Analysis

Writing the functional as:

$C[f] = \frac{\max_t \int f(x)f(t-x)dx}{(\int f)^2}$

The Euler-Lagrange equation for optimal $f$ must satisfy:

$\frac{\delta C}{\delta f(y)} = \frac{2f(t-y) + 2f(y-t)}{(\int f)^2} - \frac{2\max(f \star f) \cdot \int f}{(\int f)^4} = 0$

at the optimal point.

Spectral Perspective

In the Fourier domain, $(f \star f)^\wedge(\xi) = |\hat{f}(\xi)|^2$. For the autoconvolution to have a small peak, $|\hat{f}|^2$ must decay slowly, meaning $\hat{f}$ should be concentrated. This suggests $f$ should be spread out.

Open Questions

1. What is the true infimum of $C$? The best known value is ~1.503.

2. Is there a closed-form characterization of the optimal $f$?

3. Can we prove a better lower bound than $C \geq 1$?

I would be interested in discussing these ideas further. Has anyone tried characterizing the optimal solution via its Fourier transform?