Hilbert's inequalities on $\ell_2$


Notes by G.J.O. Jameson

As the title implies, these notes are confined to the discrete version of Hilbert's inequalities and (largely) to the case $p = 2$.

Statements of the inequalities

Hilbert's inequalities are usually presented as a pair of statements applying to sequences $x = (x_1, x_2, \ldots )$. However, in the case $p = 2$ they are unified and clarified by considering two-sided sequences, i.e. elements of $\ell_2(\mathbb{Z})$. For such a sequence $x = (x_n)_{n \in \mathbb{Z}}$ (real or complex), we write, as usual, $\Vert x\Vert^2 = \sum_{n \in \mathbb{Z}} \vert x_n\vert^2$.

For $n \in \mathbb{Z}$, write

$$c_n = \left\{ \begin{array}{ll} 1/n & \mbox{if } n \neq 0 \\ 0 & \mbox{if } n = 0. \end{array} \right.$$

For $m,n \in \mathbb{Z}$, define  $a_{m,n} = c_{m+n}$ and $b_{m,n}= c_{m-n}$. Let $A$, $B$ be the doubly infinite matrices $(a_{m,n})$ and $(b_{m,n})$ with $m,n \in \mathbb{Z}$.

With this notation, the basic result is as follows.

THEOREM 1. If A,B are regarded as operators on $\ell_2(\mathbb{Z})$, then $\Vert A\Vert \leq \pi$ and $\Vert B\Vert \leq \pi$.

Of course, the trivial substitution $z_n = x_{-n}$ shows that $\Vert A\Vert = \Vert B\Vert$.

We shall describe two proofs of this theorem. But first we show how Hilbert's one-sided inequalities (indeed, a slightly stronger statement) follow from it. Let

$$a'_{m,n} = c_{m+n-1} = a_{m-1,n},$$

and let $A_1$, $B_1$ be the matrices $(a'_{m,n})_{m,n \geq 1}$ and $(b_{m,n})_{m,n \geq 1}$, corresponding to operators on $\ell_2(\mathbb{N})$.

THEOREM 2. For $x$ in $\ell_2(\mathbb{N})$, we have

$$\Vert A_1x\Vert^2 + \Vert B_1x\Vert^2 \leq \pi^2 \Vert x\Vert^2.$$

Consequently, $\Vert A_1\Vert \leq \pi$ and $\Vert B_1\Vert \leq \pi$ (the ``symmetric" and ``skew-symmetric" inequalities).

Proof. Extend $x$ to a two-sided sequence $x^*$ by putting $x^*_n = 0$ for all $n \leq 0$. We write $(Ax^*)(m)$ for term $m$ of the sequence $Ax^*$. For $m \geq 0$, we have

$$(Ax^*)(m) = \sum_{n \geq 1} c_{m+n} x_n = \sum_{n \geq 1} a'_{m+1,n} x_n = (A_1x)(m+1),$$

and for $m \geq 1$ we have

$$(Ax^*)(-m) = \sum_{n \geq 1} c_{-m+n}x_n = - \sum_{n \geq 1}c_{m-n}x_n = -(B_1x)(m).$$

Hence  $\Vert A_1x\Vert^2 + \Vert B_1x\Vert^2 = \Vert Ax^*\Vert^2$, and the statement follows. $\Box$

Let $A_2$ be the matrix $(a_{m,n})_{m,n \geq 1}$. The proof of Theorem 2, without the shift from $m$ to $m+1$, clearly gives:

PROPOSITION 3. We have

$$\Vert A_2x\Vert^2 + \Vert B_1x\Vert^2 \leq \pi^2 \Vert x\Vert^2.$$

Note that it is obvious that $\Vert A_2\Vert \leq \Vert A_1\Vert$, since $0 < a_{m,n} < a'_{m,n}$ for all $m,n \geq 1$.

Proof 1: the Fourier series method

This short and elegant method is in fact a refinement of Hilbert's original proof. (cf. [HLP, section 9.6]). This version is attributed to Toeplitz in [Montg1], p.554. It does not require any theorems of Fourier analysis; it just uses elementary integrals involving $e^{int}$. An abstract version of the method applies to the Hankel and Toeplitz matrices associated with a multiplication operator on $L_2[0, 2\pi ]$ [Young, sect. 13.4, 15.2].

Theorem 1, proof 1. We shall use bilinear forms to show that $\Vert A\Vert \leq \pi$. For $x,y \in \ell_2(\mathbb{Z})$, write (presupposing convergence)

$$u(x,y) = \sum_{m \in \mathbb{Z}} \sum_{n \in \mathbb{Z}} a_{m,n} x_m y_n.$$

We have to show that  $\vert u(x,y)\vert \leq \pi \Vert x\Vert\: \Vert y\Vert$  for all $x,y$.

To avoid reliance on results like the completeness of $L_2$, we consider first finite sequences $(x_m)_{m \in E}$ and $(y_n)_{n \in F}$, where $E = [-M, M]$ and $F = [-N, N]$. Now define

$$X(t) = \sum_{m \in E} x_m e^{imt}, \qquad Y(t) = \sum_{n \in F} y_n e^{int}.$$

Then, clearly,

$$\frac{1}{2\pi } \int_0^{2\pi } \vert X(t)\vert^2 \: dt = \sum_{m \in E} \vert x_m\vert^2 = \Vert x\Vert^2 ,$$

and similarly for $\vert Y(t)\vert^2$ (denote these two integrals by $I_X$ and $I_Y$).

Now  $\int_0^{2\pi } (t- \pi )\: dt = 0$, while for integers $n \neq 0$,

$$\frac{1}{2\pi }\int_0^{2\pi } (t - \pi ) e^{int} \; dt = \frac{1}{in}.$$

Hence

$$I = : \frac{1}{2\pi } \int_0^{2\pi }(t - \pi ) X(t)Y(t) \: dt... ...{i} \sum_{m\in E} \sum_{n\in F} a_{m,n} x_m y_n = -i u(x,y).$$

But by the fact that $\vert t - \pi \vert \leq \pi$ on $[0, 2\pi ]$ and the Cauchy-Schwarz inequality, we have

$$\vert I\vert \leq \frac{1}{2\pi } \pi \int_0^{2\pi } \vert ... ...\leq \pi I_X^{1/2} I_Y^{1/2} = \pi \Vert x \Vert \Vert y\Vert.$$

Now consider an infinite (two-sided) sequence $y$. Convergence of $\sum_{n\in \mathbb{Z}}a_{m,n}y_n$ (for each $m$) is assured by the Cauchy-Schwarz inequality. By the result already proved, for any $M,N$,

$$\sum_{m=-M}^M \left\vert \sum_{n=-N}^N a_{m,n}y_n \right\vert^2 \leq \pi ^2 \Vert y\Vert^2.$$

In this inequality, we can first let $N \to \infty$ and then let $M \to \infty$ to deduce that  $\Vert Ay\Vert \leq \pi \Vert y\Vert$. $\Box$

Note 1. Clearly, the same estimate applies if $m, \: n$ are replaced by $q_m, \: q_n$ for any sequence $(q_n)$ of distinct integers.

Note 2. Let $\alpha \notin \mathbb{Z}$. Similar reasoning, with $t - \pi$ replaced by $e^{i\alpha t}$, shows that the operator on $\ell_2(\mathbb{Z})$ with matrix $[1/(m+n+\alpha )]$ has norm $\pi /\vert\sin \alpha \pi \vert$. The relevant integral is

$$\frac{1}{2\pi } \int_0^{2\pi } e^{i(n+\alpha )t}\: dt = \frac{1}{\pi (n+ \alpha )}e^{i \alpha \pi} \sin \alpha \pi .$$

Note 3. The proof can be presented in terms of integrals on $[-\pi ,\pi ]$, with $t$ instead of $(t - \pi )$ and $(-1)^n e^{int}$ instead of $e^{int}$.

Proof 2: Schur's method (cf. [HLP, section 8.12])

Though slightly longer, this method is also (in the writer's view) elegant, and it leads to different generalizations. Again, two-sided sequences are its natural context.

LEMMA 4 (``Schur's lemma"). Let $D = (d_{m,n})$ be a matrix (finite or infinite) such that  $\sum_n \vert d_{m,n}\vert \leq R$  for all $m$ and  $\sum_m \vert d_{m,n}\vert \leq C$  for all $n$. Then (as an operator on $\ell_2$) $\Vert D\Vert \leq (RC)^{1/2}$.

Proof. Assume that $d_{m,n} \geq 0$ (or replace $d_{m,n}$ by $\vert d_{m,n}\vert$ in the following). Choose elements $x,y$ of $\ell_2$, and let

$$S = \sum_m \sum_n d_{m,n} x_m y_n .$$

Write  $d_{m,n} x_m y_n$  as  $({d_{m,n}}^{1/2}x_m)({d_{m,n}}^{1/2}y_n)$. By Cauchy-Schwarz (applied to the double sum), $\vert S\vert \leq (UV)^{1/2}$, where

$$U = \sum_m \vert x_m\vert^2 \sum_n d_{m,n} \leq R \Vert x\Ver... ...rt y_n\vert^2 \sum_m d_{m,n} \leq C \Vert y\Vert^2. \quad \Box$$

It is clear that this lemma cannot be applied to $A$, since the row and column sums diverge. Instead, we apply it to $A^*A$ (for neatness of notation, we actually consider $B^*B$ instead).

Theorem 1, proof 2. Let $x \in \ell_2(\mathbb{Z})$ and let $y = Bx$, so that $y_p = \sum_m b_{p,m}x_m$ and

$$\vert y_p\vert^2 = \sum_m \sum_n b_{p,m} b_{p,n} \overline{x}_m x_n,$$

hence 

$$\sum_p \vert y_p\vert^2 = \sum_m \sum_n d_{m,n} \overline{x}_m x_n,$$

where 

$$d_{m,n} = \sum_p b_{p,m} b_{p,n} = \sum_p c_{p-m} c_{p-n}$$

(this just says that $\Vert Bx\Vert^2 = \langle B^*Bx, x \rangle$).

Clearly, $d_{n,m} = d_{m,n}$. Also,

$$d_{n,n} = \sum_{p \in \mathbb{Z}, p \neq n} \frac{1}{(p-n)^2} = \frac{\pi^2}{3}.$$ (1)

Now let $n = m+r$, where $r > 0$. Then  $d_{m,n} = S_1 + S_2 + S_3$, where

$$\begin{eqnarray*} S_1 & = & \sum_{p > n} \frac{1}{(p-m)(p-n)} = \sum_{q \geq 1} ... ...ac{1}{r} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{r} \right) , \end{eqnarray*}$$

$$S_2 = \sum_{p < m} \frac{1}{(m-p)(n-p)} = \sum_{q \geq 1} \frac{1}{q(q+r)} = S_1 \qquad \mbox{(put $q = m-p$)},$$

$$\begin{eqnarray*} S_3 & = & \sum_{p=m+1}^{n-1} \frac{1}{(p-m)(p-n)} = -\sum_{q... ...c{2}{r} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{r-1} \right). \end{eqnarray*}$$

The cancellation of positive and negative terms leaves us with

$$d_{m,n} = \frac{2}{r^2} = \frac{2}{(n-m)^2}.$$ (2)

By (1) and (2), we have for each $m$

$$\sum_{n \in \mathbb{Z}} d_{m,n} = \frac{\pi^2}{3} + \frac{2\pi^2}{3} = \pi^2,$$

and the statement follows, by Schur's lemma. $\Box$

Note. For the one-sided case, this proof can be seen to apply to $A_1^*A_1 + B_1^*B_1$ (cf. [HLP]). However, it does not work for $A_1^*A_1$ and $B_1^*B_1$ separately, since the cancellation is lost and the row sums diverge. More precisely, if $A_1^*A_1 = (e_{m,n})$, one sees easily that $e_{m,m+r} \geq 1/(m+r-1)$.

Lower bounds

In infinite dimensions, $\pi$ is the best constant in all the statements above. In other words, each of $A, A_1, A_2, B, B_1$ has norm equal to $\pi$. This is true equally for real and complex scalars.

Clearly, $\Vert A_2\Vert \leq \Vert A_1\Vert \leq \Vert A\Vert$ and $\Vert B_1\Vert \leq \Vert B\Vert$. It is also clear from Theorem 2 that different choices of $x$ will be needed to show this for $A_1$ and for $B_1$.

Direct proof for $\Vert A_2\Vert$. Fix $N$ and take $x_n = 1/n^{1/2}$ for $1 \leq n \leq N$,  $x_n = 0$ for $n > N$. Let $y = A_2x$. By routine integral estimation, one finds that

$$y_m \geq \frac{\pi }{m^{1/2}} - \frac{2}{m} - \frac{2}{N^{1/2}},$$

and hence that $\sum_{m=1}^N y_m^2 \geq \pi ^2 \log N - C$  for a certain $C$ independent of $N$. $\Box$

Fourier-series proof for $A, B$ and $B_1$. In proof 1 of Theorem 1, fix $N$ and take $X(t)= (t - \pi )^N$ and $Y(t) = (t - \pi )^{N-1}$. One checks easily that $I = \pi \left( \frac{2N-1}{2N+1} \right)^{1/2} I_X^{1/2}I_Y^{1/2}$. The statement for $A$ follows (of course, we are now assuming that $X(t)$ and $Y(t)$ are the $L_2$-sums of their Fourier series.)

To adapt this for $B_1$, truncate the Fourier series for $X(t)$ and $Y(t)$ to finite series of the form $\sum_{n=-N}^N x_ne^{int}$, defining functions $X_1(t)$ and $Y_1(t)$. Multiply $X_1(t)$ by $e^{-iNt}$ and $Y_1(t)$ by $e^{iNt}$. Then $I, I_X$ and $I_Y$ are unchanged, and (as seen in Theorem 2) one is effectively evaluating $\Vert B_1y\Vert$. $\Box$

Once the application to the ``large sieve" is assumed (see below), a trivial example shows that $\pi$ is the best constant for $B_1$.

Note. The matrix $(\vert b_{m,n}\vert)$ does not define a bounded operator on $\ell_2$: to see this, take $x_n = 1/[(n^{1/2}(\log n +1)]$.

Some generalizations of the symmetric Hilbert inequalities

There is a massive literature on such generalizations, and we only mention a few selected results. Most of the literature relates to $\ell_p$ for general $p$, so in this subsection we drop the restriction $p = 2$. We denote by $p^*$ the conjugate index, defined by $1/p + 1/p^* = 1$.

Firstly, for $p > 1$, the norms of both $A_1$ and $A_2$, as operators from $\ell_p$ to $\ell_p$, are $\pi /\sin (\pi /p)$ (we denote this quantity by $C_p$). The statement for $A_2$ can be proved quite easily using a weighted version of Schur's lemma. This method fails for $A_1$; one alternative is to deduce the result from the continuous case. For both methods, see [HLP, chapter 9].

Some estimations have been given for the norm as an operator from $\ell_p$ to $\ell_q$, where $p < q$, (e.g. [Bon]), but exact values are not known.

[JL] gives the following generalization to the weighted $\ell_p$ space $\ell_p(w)$, with weighting sequence $w_n = 1/n^\alpha$. As an operator on this space, the norm of $A_2$ is $\pi /\sin[(1-\alpha )\pi /p]$. (The case of $A_1$ is seemingly intractable).

In [GY], the following strengthened version is proved: write $\varepsilon _p(n) = (1- \gamma )/n^{1/p}$ (where $\gamma$ is Euler's constant). Then for non-negative $x_m,\: y_n$.

$$\sum_{m=1}^\infty \sum_{n=1}^\infty a_{m.n} x_m y_n \leq \le... ...^\infty [C_p - \varepsilon _p^*(n)] y_n^{p^*} \right)^{1/p^*}.$$

A very different type of generalization is explored in [Benn]. The aim is to identify the set $hil(p)$ of all real sequences $x = (x_n)$ for which $A_1(\vert x\vert)$ is in $\ell_p$, so that

$$S(x) = : \sum_{m=1}^\infty \left( \sum_{n=1}^\infty a'_{m,n}\vert x_n\vert \right)^p$$

is convergent. It is clear that $hil(p)$ is a normed linear space, with norm defined by $\Vert x\Vert _{hil(p)} = S(x)^{1/p}$. By Hilbert's inequality, $hil(p)$ contains $\ell_p$ and $\Vert x\Vert _{hil(p)} \leq C_p \Vert x\Vert _p$. The following answer is given. Given a bounded sequence $x$, define $(Dx)(n) = \sup _{k \geq n}\vert x_k\vert$. Let $d(p^*)$ be the space of sequences $x$ for which $\Vert x\Vert _{d(p^*)} =: \Vert Dx\Vert _{p^*}$ is finite. Let $(X_p, N_p)$ be the Banach space dual of $d(p^*)$. Note that $X_p$ contains $\ell_p$ and $N_p(x) \leq \Vert x\Vert _p$. (There is an alternative description of $(X_p, N_p)$ in terms of factorizations $x = yz$, with stated norms for $y$ and $z$.) The result is that $hil(p) = X_p$, and the norms are equivalent:

$$(p-1)N_p(x) \leq \Vert x\Vert _{hil(p)} \leq C_p N_p(x).$$

The right-hand inequality is a strengthening of Hilbert's inequality, and the left-hand one represents new information. Remarkably, the same space $X_p$ is obtained if the process is applied to the averaging (alias Cesaro) operator $(Cx)(n) = \frac{1}{n}(x_1 + \cdots + x_n)$.

The Montgomery-Vaughan generalizations of the skew-symmetric inequalities

The following theorems of Montgomery and Vaughan have found important applications in analytic number theory.

THEOREM 5. Let $(\lambda_n)$ be a real sequence (finite or infinite, one-sided or two-sided) such that $\vert\lambda_n - \lambda_n\vert \geq \delta$ whenever $m \neq n$. Let G be the matrix defined by

$$g_{m,n} = \left\{ \begin{array}{ll} 1/(\lambda_m - \lambda... ...if } m \neq n, \\ 0 & \mbox{if } m = n. \end{array} \right.$$

Then $\Vert G\Vert \leq \pi /\delta$ as an operator on $\ell_2$.

THEOREM 6. Let $(\alpha_n)$ be a real sequence such that ${\rm dist}(\alpha_m - \alpha_n, \mathbb{Z}) \geq \delta$ whenever $m \neq n$, and let $H$ be the matrix defined by

$$h_{m,n} = \left\{ \begin{array}{ll} 1/\sin \pi (\alpha_m -... ...if } m \neq n, \\ 0 & \mbox{if } m = n. \end{array} \right.$$

Then $\Vert H\Vert \leq 1/\delta$.

Both theorems were first proved in [Montg&V], where Theorem 6 is proved first and Theorem 5 deduced from it. In [Montg1] and [Montg2], Theorem 5 is proved first and Theorem 6 is deduced. In fact, with careful formulation, the method really proves both theorems simultaneously. It works in the finite-dimensional context and starts with $G^*G$ (or $H^*H$) as in Schur's method, but cancellation no longer applies and essential use is made of the fact that a skew-hermitian operator attains its norm at an eigenvector.

Applications include the integral mean-value theorems for Dirichlet polynomials [Montg&V], and the best constant in the ``large sieve" inequality (however, without the best constant, this inequality is obtained more easily by Gallagher's method [Montg1]).

Stronger norm estimations in finite-dimensions

Here one must expect different results for the various operators considered. Some results are as follows. [Fra] shows that (in $N$ dimensions) $\Vert A_1\Vert \leq N \sin (\pi /N)$ ( $= \pi - O(N^{-2})$). For $A_2$, the weighted Schur method leads easily to $\pi - 2/(N+1)^{1/2}$, but with a deeper analysis [Wilf, Theorem 2.2] obtains the value

$$\Vert A_2\Vert = \pi - \frac{\pi^5}{2(\log N)^2} + O\left( \frac{\log \log N}{(\log N)^3} \right).$$

For the operator $B_1$, the Montgomery-Vaughan method gves

$$\Vert B_1\Vert^2 \leq 3 \sup_n \sum_m b_{m,n}^2,$$

which leads easily (in $N$ dimensions) to $\Vert B_1\Vert \leq \pi - 6/[\pi (N+1)]$. For the matrix $H$ in Theorem 6, with $\alpha_n = n/N$, [Matt] shows that the eigenvalues are $(2k -1 -N)i$ for $1 \leq k \leq N$, so that $\Vert H\Vert$ is exactly $(1 - 1/N)\pi$. This implies that $\Vert B_1\Vert \leq (1 - 1/N)\pi$. Montgomery has stated in private correspondence that $\pi - \Vert B_1\Vert$ is $O(\log N/N)$, but it would appear that no proof has ever been published.

References

[Benn] Grahame Bennett, Factorizing the Classical Inequalities, Mem. Amer.Math. Soc. 576 (1996).

[Bon] F.F. Bonsall, Inequalities with non-conjugate parameters, Quart. J. Math. Oxford (2) 2 (1951), 135-150.

[Fra] H. Frazer, Note on Hilbert's inequality, J. London Math. Soc. 21 (1946), 7-9.

[GY] Gao Mingzhe and Yang Bicheng, On the extended Hilbert's inequality, Proc. Amer. Math. Soc. 126 (1998), 751-759.

[HLP] G.H. Hardy, J. Littlewood and G. Polya, Inequalities, 2nd ed., Cambridge Univ. Press (1967).

[JL] G.J.O. Jameson and R. Lashkaripour, Norms of certain operators on weighted $\ell_p$ spaces and Lorentz sequence spaces, J. Ineq. Pure Appl. Math. (2002), to appear

[Matt] K.R. Matthews, Hilbert's inequalities, personal website www.maths.uq.edu.au/~krm .

[Montg1] H.L. Montgomery, The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), 547-567.

[Montg2] H.L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS no. 84, Amer. Math. Soc. (1990).

[Montg&V] H.L. Montgomery and R.C. Vaughan, Hilbert's inequality, J. London Math. Soc. (2) 8 (1974), 73-82.

[Wilf] H.S. Wilf, Finite sections of some classical inequalities, Ergebnisse der Mathematik vol. 52, Springer (1970).

[Young] N.J. Young, An Introduction to Hilbert Space, Cambridge Univ. Press (1988).