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Let $U$ be an $n$-dimensional subspace of $L:=L_2([-1,1])$. Let $F$ be an acting on $L$, given at $f \in L$ $$ (Ff)(x):=\int_{-1}^1 \frac{\sin a(x-y)}{(x-y)}f(y) dy, \quad x \in [-1,1], \quad a>0. $$ Let $\lambda_n(a), \, n=0,1,2\ldots,$ be the eigenvalues of $F$. Using Rayleigh-Ritz Theorem show that for all $n>a$ $$ \lambda_n(a)\geq \min_{U/\{0\}}\frac{\|Ff\|_L}{\|f\|_L}. $$

Thank you.

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You haven't specified whether $F$ acts on $U$ or all of $L^2$. – A Blumenthal Feb 7 '13 at 17:19
Thank you. It acts on $L^2$. – Michael Feb 7 '13 at 17:22
So this is supposed to hold for every $U$ of dimension $n$, and every $n>a$, right? Are the $\lambda_n(a)$ in some order? – 1015 Feb 10 '13 at 1:08
Thank you. Yes, $\lambda_m$ are in a non-increasing order. And it hold for every $U$. – Michael Feb 10 '13 at 3:03
In the last equation, what is $U$? – Jas Ter Feb 13 '13 at 20:43
up vote 1 down vote accepted

Since no one more knowledgeable has answered this, let me make an attempt. I will look at a closely related operator namely $$ (Ff)(x) = \int_{-1}^1 \frac{\sin (a (x-y))}{ (x-y)} 1_{[-2,2]}(x-y) f(y) dy $$ which seems to be exactly what you have (check?) Here $1_{[-2,2]}$ is the indicator of the set $[-2,2]$. Introducing it does not change the integral over $x \in [-1,1]$.

I will use $\|\cdot\|_2$ for the norm of $L^2([-1,1]) =: L^2[-1,1]$.

Now, we can use the corollary 4.28 of Brezis's functional analysis (p. 114 on English version) which says that if $g$ is a fixed function in $L^1(\mathbb{R})$ and $\mathcal{B}$ is a bounded set in $L^p(\mathbb{R})$ for $p \in [1,\infty)$, and we let $\mathcal{F} = g * \mathcal{B}$, then $\mathcal{F}|_{\Omega}$ has compact closure in $L^p(\Omega)$ for any measurable set $\Omega$ with finite measure.

Apply the above with $g(y) = \frac{\sin(ay)}{y} 1_{[-2,2]}(y)$ and $\mathcal{B}= \{ f: \| f\|_2 \le 1\}$, $p = 2$ and $\Omega = [-1,1]$ to conclude that $F:L^2[-1,1] \to L^2[-1,1]$ is a compact operator.

Now, here is a general statement of the Rayleigh-Ritz from Garling's Inequalities (p. 246)

Suppose that $T = \sum_{n=1}^\infty s_n(T) \langle \cdot,x_n\rangle y_n \in K(H_1,H_2)$ (that is compact from $H_1$ to $H_2$) where $(x_n)$ and $(y_n)$ are orthonomral sequences in $H_1$ and $H_2$, respectively, and $(s_n(T))$ is a decreasing sequence of non-negative real numbers, then $$ s_n(T) = \inf \big\{ \| T_{|J^\perp} \|:\; \text{dim} J < n\big\} $$ where $ \|T_{|J^\perp}\|$ is the norm of $T$ restricted to $J^\perp$, that is $$ \|T_{|J^\perp}\| = \sup\{ \|T(x)\|:\; \|x\| \le 1, x \in J^\perp\} $$ Note that this is also the same as $$ \|T_{|J^\perp}\| = \sup\Big\{ \frac{\|T(x)\|}{\|x\|}:\; x \in J^\perp \setminus\{0\}\Big\} $$

Take $T$ to be our $F$ operator (and $H_1 = H_2 = L^2[-1,1]$). Since $F$ is compact it has such a representation and putting the pieces together should give you the result you want for the singular values, and it seems that you are missing a $\perp$ in your statement. You might also try to show that the operator is positive (self-adjoint) in which case the result holds for eigenvalues too.

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