Quantitative Projections in the Sturm Oscillation Theorem

TL;DR

This paper provides a quantitative version of the Sturm-Hurwitz theorem, linking the number of roots of functions to their Fourier coefficients and eigenfunction projections, with implications for Sturm-Liouville problems.

Contribution

It establishes explicit lower bounds on Fourier coefficients and eigenfunction projections based on the number of roots, extending classical topological results.

Findings

Quantitative bounds on Fourier coefficients from topological root conditions

Extension of Sturm-Hurwitz theorem to eigenfunctions of Sturm-Liouville problems

Explicit estimates on projections onto eigenfunction spans

Abstract

There is $c_{} > 0$ such that for all $f \in C[0,\pi]$ with at most $d-1$ roots inside $(0,\pi)$ $$ \sum_{1 \leq n \leq d}{ \left| \left\langle f, \sin\left( n x\right) \right\rangle \right|} \geq \kappa^{-\kappa^2 \log{\kappa}}\|f\|_{L^2} \qquad \mbox{where} \quad \kappa = \frac{c_{} \| \nabla f\|_{L^2}}{\|f\|_{L^2}}.$$ This quantifies the Sturm-Hurwitz Theorem and connects a purely topological condition (number of roots) to the Fourier spectrum. It is also one of few estimates on Fourier coefficients from below. The result holds more generally for eigenfunctions of regular Sturm-Liouville problems $$ - (p(x) y'(x))' + q(x) y(x) = \lambda w(x) y(x) \qquad \mbox{on}~(a,b).$$ Sturm-Liouville theory shows the existence of a sequence of solutions $(\phi_n)_{n=1}^{\infty}$ that form an orthogonal basis of $L^2(a,b)$ with respect to $w(x)dx$. Sturm himself proved that if $f:(a,b) \rightarrow \mathbb{R}$ is a finite linear combinations of $\phi_n$ having $d-1$ roots inside $(a,b)$, then $f$ cannot be orthogonal to $A = \mbox{span}\left\{\phi_1, \dots, \phi_{d}\right\}$. We prove a lower bound on the size of the projection $\| \pi_A f\|_{}$.