There is $c>0$ such that for all $f\in C[0,\pi ]$ with at most $d-1$ roots inside $(0,\pi )$$\sum _{1\le n\le d}\left|〈f,\sin \left(nx\right)〉\right|\ge {\kappa }^{-{\kappa }^2\log \kappa }{\Vert f\Vert }_{L^2}\text{where}\kappa =\frac{c{\Vert \mathrm{\nabla }f\Vert }_{L^2}}{{\Vert f\Vert }_{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$-{(p(x)y^{\prime }(x))}^{\prime }+q(x)y(x)=\lambda w(x)y(x)\text{on}(a,b).$ Sturm-Liouville theory shows the existence of a sequence of solutions ${({\varphi }_n)}_{n=1}^{\infty }$ that form an orthogonal basis of $L^2(a,b)$ with respect to $w(x)dx$. Sturm himself proved that if $f:(a,b)\to \mathbb{R}$ is a finite linear combinations of ${\varphi }_n$ having $d-1$ roots inside $(a,b)$, then f cannot be orthogonal to $A=\text{span}\{{\varphi }_1,\dots ,{\varphi }_d\}$. We prove a lower bound on the size of the projection $\Vert {\pi }_{A}f\Vert$.

有 $C>0$ 这样对于所有人 $F\in C[0，\pi ]$ 最多 $d-\mathrm{1个}$ 扎根 $（0，\pi ）$$\sum _{\mathrm{1个}\le ñ\le d}\left|〈F，\mathrm{罪}（ñX）〉\right|\ge {\kappa }^{-{\kappa }^2\mathrm{日志}\kappa }{\Vert F\Vert }_{{\mathrm{大号}}^2}\text{哪里}\kappa =\frac{C{\Vert \mathrm{\nabla }F\Vert }_{{\mathrm{大号}}^2}}{{\Vert F\Vert }_{{\mathrm{大号}}^2}}。$这量化了Sturm-Hurwitz定理，并将纯拓扑条件（根数）与傅立叶谱联系起来。这也是从下面对傅立叶系数进行的少数估计之一。该结果更适用于常规Sturm-Liouville问题的本征函数$-{（p（X）{ÿ}^{\prime }（X））}^{\prime }+q（X）ÿ（X）=\lambda w（X）ÿ（X）\text{上}（\mathrm{一种}，b）。$ Sturm-Liouville理论表明了一系列解的存在 ${（{\varphi }_{ñ}）}_{ñ=\mathrm{1个}}^{\infty }$ 形成一个正交的基础 ${\mathrm{大号}}^2（\mathrm{一种}，b）$ 关于 $w（X）dX$。斯特姆自己证明，如果$F：（\mathrm{一种}，b）\to \mathbb{[R}$ 是以下项的有限线性组合 ${\varphi }_{ñ}$ 有 $d-\mathrm{1个}$ 扎根 $（\mathrm{一种}，b）$，则f不能正交于$\mathrm{一种}=\text{跨度}\{{\varphi }_{\mathrm{1个}}，\dots ，{\varphi }_d\}$。我们证明了投影尺寸的下界$\Vert {\pi }_{\mathrm{一种}}F\Vert$。