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Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator $F$, defined on $L^2([-1,1])$ by $$ F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here constant}\, a>0. $$ Functions $(f_n)_{n\geq 0}$ are normalized by using the following rule, $$ \int_{-1}^1 |f_{n}(x)|^2\,\mathrm{d}x = 1,\quad \int_{\mathbb R} |f_{n}(x)|^2\,\mathrm{d}x =\frac{1}{\lambda_n(c)},\quad n\geq 0, $$ where $(\lambda_n(a))_n$ is the infinite sequence of the eigenvalues of $F$, arranged in the decreasing order $1> \lambda_0(a)> \lambda_1(a)>\cdots>\lambda_n(a)>\cdots$ and such that $\displaystyle{C_1\left(\frac{a}{ n}\right)^{n-1}\leq \lambda_n(a)\leq C_2 a\left(\frac{a}{n}\right)^{2n}}$ ($C_1, C_2>0$ are absolute constants).

Question: Show that $f_{n}$ satisfies the conditions of the following Lemma:

Lemma: Let $g\in L^2(R)$. Fix $\epsilon>0$ and define $L$ so that $$ \int_{-\infty}^{-L}|g(x)|^2\,\mathrm{d}x<\varepsilon^2, \quad \int_{L}^{\infty}|g(x)|^2\,\mathrm{d}x<\varepsilon^2. $$ Let $S_K$ be the $K$-th partial Hermite expansion of $g$, that is $S_K=\sum_{k=0}^{K}<g,h_k>h_k$ where $h_k$ is the $k$-th Hermite function. Then, $$ \left|S_K(x)-\frac{1}{\pi}\int_{-L}^{L}g(y)\frac{\sin N(x-y)}{x-y}\,\mathrm{d}y\right|<\varepsilon(1+\Gamma), $$ where $\displaystyle{N=\frac{\sqrt{2K+3}+\sqrt{2K+1}}{2}}$ and $\Gamma=O\left(\frac{1}{K^{1/2}}\right)$.

Thank you.

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