Nehari's theorem for convex domain Hankel and Toeplitz operators in several variables

We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows. Let $\Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2\Xi$, consider the Hankel operator $$\Gamma_f (g)(x)=\int_{\Xi} f(x+y) g(y) \, dy, \quad x \in\Xi.$$ Then $\Gamma_f$ extends to a bounded operator on $L^2(\Xi)$ if and only if there is a bounded function $b$ on $\mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2\Xi$. This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix. In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.

and consider a distribution f defined on . The associated general domain Hankel operator f = f , is the (densely defined) operator f : L 2 ( ) → L 2 ( ), given by where dy is the Lebesgue measure on R d .
The case = R + = (0, ∞) for d = 1 corresponds to the class of usual Hankel operators; when represented in the appropriate basis of L 2 (R + ), the operator f ,R + is realized as an infinite Hankel matrix {a n+m } ∞ n,m=0 [31,Ch. 1.8]. Nehari's theorem [25] characterizes the bounded Hankel matrices of this type, but it has an equivalent version for operators of the type f : L 2 (R + ) → L 2 (R + ), which reads as follows (we again refer to [31,Ch. 1.8], Theorem 8.1). For a function g on R d , we letĝ = Fg denote its Fourier transform,ĝ Theorem. Suppose that f is a distribution in R + , f ∈ D (R + ). Then f : L 2 (R + ) → For d > 1, the operators f ,R d + , = R d + , correspond to (small) Hankel operators on the product domain multi-variable Hardy space H 2 d . In this case, the analogue of Nehari's theorem remains true, apart from (1.1), but it is significantly more difficult to prove. It was established by Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2) [18,23]. A precise statement is given in Theorem 2.1.
The main purpose of this article is to prove Nehari's theorem when ⊂ R d is a simple convex polytope. When is convex note that + = 2 . to satisfy When d = 1, the only open connected sets ⊂ R are the intervals = I. In this case, Theorem 1.1 is due to Rochberg [35], who called the corresponding operators f ,I Hankel/Toeplitz operators on the Paley-Wiener space. They have also been called Wiener-Hopf operators on a finite interval [30]. These operators have inspired a wealth of theory in the single variable setting-see Section 2.5, where we shall interpret and in this case In the final part of the paper we shall give an application of Theorem can be found in [13].
The paper is laid out as follows. In Section 2 we will give a more formal background and introduce necessary notation. We will also discuss the relationship between f , , Paley-Wiener spaces, and co-invariant subspaces of the Hardy spaces.
In Section 3 we will prove approximation results for distribution symbols with respect to Hankel and Toeplitz operators, allowing us to reduce to smooth symbols. Section 4 brief ly outlines what we need to know about convex sets and polytopes. In Section 5 For the inverse transform we write F −1 (g) =ǧ. The product domain Hardy space H 2 d is the proper subspace of L 2 (R d ) of functions whose Fourier transforms are supported in We let P d : L 2 (R d ) → H 2 d denote the orthogonal projection and let J : It follows that the (possibly unbounded) operator f : is unitarily equivalent to the small Hankel operator Zf : Note that any b such thatb| R d + = f generates the same Hankel operator asf , Z b = Zf .
To justify the above computation easily we assumed that f ∈ L 2 (R d + ). An approximation argument is needed to consider general symbols f , which may only be distributions in R d + . We provide this later in Proposition 3.2. We can then read off the boundedness of f from the boundedness of the corresponding Hankel operator on H 2 d . When d = 1 and = = R + , the analogue of Theorem 1.1 is exactly the classical Nehari theorem. In higher dimensions the corresponding theorem is due to Ferguson-Lacey-Terwilleger [18,23]. In our notation, their results read as follows.
For d > 1 it is not possible to take c = 1 in (2.2), see for example [29]. This result, as stated in [18,23], requires that f ∈ L 2 (R d + ). The extension to the more general situation considered here is a technicality, but for completeness the details are provided in Section 3.

Hankel operators on bounded domains
We now discuss bounded domains , the setting of our main result. The only convex In the same article [35], it is posed as an open problem to characterize the bounded Hankel operators f , when is a disc in R 2 . We are not able to settle this question, but Theorem 1.1 does provide the answer when = (0, 1) d is a cube in R d .
As we will see, the Hankel operators f ,(0,1) d constitute a natural generalization of the unitarily equivalent to Toeplitz matrix operators on 2 (N). In this case the boundedness characterization is easy to both state and prove In Theorem 6.1 we extend (2.3) to Toeplitz operators f , for a class of "cone-like" domains ⊂ R d , for which = − = R d .

Truncated correlation operators
For open connected sets , ϒ ⊂ R d it is also convenient to introduce the more general "truncated correlation operators" f ,ϒ, : where f lives on = +ϒ. This class of operators includes both general domain Hankel and Toeplitz operators, by letting ϒ = and ϒ = − , respectively.
For our purposes, general truncated correlation operators will only appear in intermediate steps toward proving the main results, but they also carry independent interest. They were introduced in [1], where their finite rank structure was investigated.
In [2] it was shown that they have a fundamental connection with frequency estimation on general domains, motivating the practical need for understanding such operators, not only on domains of simple geometrical structure. In [3] it is explained how one may infer certain results for the integral operators f from their discretized matrix counterparts. We warn the reader that in naming the operators f , f , and f we have slightly departed from previous work, reserving the term (general domain) Hankel operator for truncated correlation operators of the form f , , .

Hankel operators on multi-variable Paley-Wiener spaces
Another viewpoint is offered through co-invariant subspaces of the Hardy spaces H 2 d . For a domain ⊂ R d , let PW denote the subspace of L 2 (R d ) of functions with Fourier transforms supported in , In the classical case = (0, 1) ⊂ R, note that Hence PW (0,1) is the ortho-complement (in H 2 1 ) of θ H 2 1 , the shift-invariant subspace of H 2 1 with inner factor θ . This space is usually denoted K θ , By a calculation similar to (2.1) we see that f ,(0,1) is unitarily equivalent to the compression of the Hankel operator Zf to PW (0,1) , 1) denotes the orthogonal projection onto PW (0,1) . Such truncated Toeplitz and Hankel operators are now very well studied on general K θspaces [6,7,9,10,14,20,27,30,36].
In the case of the cube the Hankel operator f , may, just as for d = 1, be understood as the compression of a Hankel operator to a co-invariant subspace of H 2 d . Namely, is an invariant subspace (under multiplication by bounded holomorphic functions), and as before we have that Finally, let us brief ly discuss the viewpoint of weak factorization. The Hardy As is well known, see for example [ with equivalence of norms. Here the projective tensor product norm on X X, X a Banach space of functions, is given by X X being defined as the completion of finite sums j G j H j in this norm.
The reason that Theorem 2.1 is equivalent to (2.4) is the following: by

yields a similar weak factorization theorem for Paley-Wiener
spaces. We postpone the proof to Section 5, but mention now that corresponding weak factorization for K θ spaces plays an important role in [6] and [9]. Corollary 5.3 might also be compared to the results in [37], where weak factorization for multivariate analytic polynomials is deduced as a consequence of Theorem 2.1.

Corollary 5.3.
Let be a simple convex polytope, and let = 2 . Then The norms of these Banach spaces are equivalent.
The 1st option leads to the "small" Hankel operators considered in Section 2.1, while the 2nd type of operator is commonly referred to as a "big" Hankel operator. In the notation of Section 2.4, a small Hankel operator is an operator f ,R d  [35], studying the case of a finite interval in one dimension. Furthermore, he posed as an open problem to understand the case when ⊂ R 2 is a disc. In this latter setting, L. Peng [32] characterized when f , belongs to the Schatten class S p , for 1 ≤ p ≤ 2, in terms of certain Besov spaces adapted to the disc. L. Peng also carried out a similar study [33] for the case of the multidimensional cube, = (−1, 1) d , describing membership in S p for all p, 0 < p < ∞, as well as giving a sufficient condition for boundedness.
Since then it seems that the field did not see progress until the results of Ferguson-Lacey-Terwilleger [18,23] settled the issue of boundedness of small Hankel operators.

Distribution symbols
Let , ϒ ⊂ R d be any open connected sets and let f ∈ D ( ) be a distribution on , = + ϒ. We follow the notation of [21] in our use of distributions. We then define Downloaded from https://academic.oup.com/imrn/advance-article-abstract/doi/10.1093/imrn/rnz193/5610521 by University of Reading user on 21 January 2020 Nehari's Theorem in Several Variables 11 the truncated correlation operator f as an operator f ,ϒ, : where (f , ϕ) denotes the action of f on ϕ and We reserve the notation f , ϕ for scalar products that are anti-linear in the 2nd entry.
Since T x ϕ is compactly supported in for x ∈ , it follows that f (ϕ) this is well defined and smooth in (see, e.g., [21,Theorem 4 gives rise to a densely defined operator on the latter space, which extends to a bounded operator f : . By slight abuse of notation, we write the action of f in this way even when f is not locally integrable. The central question in this paper is the following: for which domains ϒ and is the boundedness of f equivalent to the existence of a function b ∈ L ∞ (R d ) such thatb| = f ? Some care must be taken in interpreting this question. For example, the prototypical example of a bounded Hankel operator is the Carleman operator is in this case not a tempered distribution on R (so the meaning off is unclear)-it is, however, the restriction of the tempered distribution p. v. 1 x to R + . An example with a delta function makes it clear that it is not necessary for f to be locally integrable in either.
We first record the answer to our question in the trivial direction. is bounded and where M b is the operator of multiplication by b. The statement is obvious from here.
Next we establish two technical results on the approximation of distribution symbols by smooth compactly supported functions, Propositions 3.2 and 3.3. They will help us to overcome the technical issues mentioned earlier, in particular allowing us to deduce Theorem 2.1 from the corresponding statements in [18,23].
Let ψ ∈ C ∞ c (R d ) be a fixed non-negative function with compact support in the ball n=1 is an approximation of the identity. Since f ∈ D ( ) and supp ψ n ⊂ B(0, 1/2n), the convolution f * ψ n is well defined as a function in C ∞ ( 2n ). Let ρ n be a smooth cut-off function that is 1 in a neighborhood of n but zero in a neighborhood of c 2n , and note that ρ n ( f * ψ n ) then naturally defines a function in C ∞ (R n ). Finally, for a non-negative function and Let ω n (x) = ω(x/n). We introduce as an approximant of f , where the role of ω n is to enforce compact support in case is unbounded. By construction, f n ∈ C ∞ c ( ) and it is straightforward to check that f n → f in D ( ). As for f n ,ϒ n , , we have the following result. Proof. We can assume that f ,ϒ, < ∞, since otherwise there is nothing to prove.
This reduces our task to proving that the operators g n ,ϒ n , = ρ n (f * ψ n ),ϒ n , = f * ψ n ,ϒ n , are uniformly bounded in n. We have for ϕ ∈ C ∞ c (ϒ n ) and x ∈ that this completes the proof.
Since ϒ n = z n + R d + , z n = (2/n, . . . , 2/n), we have that , the computation that led to (2.1) is justified, and we conclude from [18,23] By Alaoglu's theorem it follows that there is a weak-star convergent subsequence It remains to prove that . However, this is clear from the construction; sinceφ ∈ L 1 we have that In Section 6 we will consider Toeplitz operators f , for which = − = R d .
In this case f * ψ n is a smooth function defined in all of R d , and there is no need to multiply with ρ n or to introduce the subdomains ϒ n . In this case we simply let Clearly, f n → f in D (R d ) and we have, with the exact same proof as for Proposition 3.2, the following approximation result.

On convex sets and polytopes
We recall some basic properties of convex sets. Given an unbounded convex set ⊂ R d which is either open or closed, its characteristic cone, also known as its recession cone, is the closed set The support function h : We refer to [21,Sec. 7.4] for the basic properties of h . The barrier cone of is the set The characteristic cone cc coincides with the polar cone of the barrier cone bc , that is, To give a complete reference for this claim, first note that for closed convex sets , cc coincides with the asymptotic cone of , giving (4. We next recall some standard terminology and facts of polytopes, referring to for example [12,. By an open half-space in R d we mean a set where ν ∈ R d is a non-zero vector and r ∈ R. A closed half-space is the closure of such a set. A finite intersection of half-spaces is called a polyhedral set. A convex polytope is a bounded polyhedral set. A closed convex polytope is the convex hull of a finite set of points. The minimal set of such points coincides with the extreme points of the polytope, that is, its vertices. If the minimal number of defining hyperspaces of a convex polytope is d + 1 (equivalently, if it has precisely d + 1 vertices), the polytope is called a simplex. For a non-closed polytope we define its vertices (and its edges and facets) as those of its closure.
The boundary of a polytope set is made up of a finite amount of facets (i.e. d − 1 dimensional faces), see Corollary 7.4 and Theorem 8.1 of [12]. For a polytope with vertex x j , we denote by ∂ far,x j the part of its boundary made up of all facets not containing x j .
A vertex of a polytope will be called simple if it is contained in precisely d of its edges. We say that a polytope is simple if all of its vertices are simple, which coincides with the standard terminology. Equivalently, this means that each vertex is contained in precisely d of its facets (cf. [12,Theorem 12.11]).
By an affine linear transformation we mean a map of the form where L is a linear map, and we call x 0 the origin of such a map. The following simple lemma gives a 3rd characterization of simple vertices.
where {e k } d k=1 denotes the standard basis of R d .
Proof. We may assume that x 1 = 0 is simple and that For the converse, simply note that the property of being a simple vertex is preserved under affine isomorphisms.
By compactness it is easy to construct a partition of unity adapted to the vertices of .

Lemma 4.2. Given a polytope with vertices {x
Proof. For ε > 0 and 1 ≤ j ≤ J, let Since every x ∈ is contained in some set V ε j , there is by compactness a fixed ε 0 > 0 such that Proof. The main fact to be proved is that Since C ∞ c ( ) is dense in L 2 ( ), it then follows that F −1 (C ∞ c ( )) is dense in the product PW PW .
We will actually show a little more than the claim. Namely, every g ∈ C ∞ c ( ) can be written in such a way that the corresponding map g → k s k L 2 ( ) t k L 2 ( ) is continuous from C ∞ c ( ), equipped with the usual test function topology, to R. By employing a partition of unity in which each member is compactly supported in a cube, it is sufficient to prove the claim when = (0, 1/2) d . For this we employ Fourier series. Let λ(t) = 1/2 − |t − 1/2|, t ∈ [0, 1], and let Note that λ is in the Wiener algebra A ([0, 1]), the space of functions on [0, 1] with absolutely convergent Fourier series, equipped with pointwise multiplication. Therefore  ([0, 1] d ), since is a tensor power of λ. Since g ∈ C ∞ c ((0, 1) d ) and is non-zero on compact subsets of (0, 1) d it follows by Wiener's lemma [19,Ch. 5] that g/ ∈ A([0, 1] d ). Expanding g/ in a Fourier series, Then a computation shows that An inspection of the argument shows that the operation g → g/ is continuous from Since μ is a bounded functional on PW PW we conclude that that is, f , is bounded, and in fact f = μ . Conversely, if f is a distribution on such that f , is bounded, it is clear that f induces a bounded functional μ on PW PW by (5.1). This proves that X is isometrically isomorphic to the Banach space (PW PW ) * , which also entails that X is closed, completing the proof.
In the remainder of this section we assume that is a convex polytope. Next we prove Theorem 1.1 under the additional assumption that f is supported around one simple vertex of . and let f ∈ D ( ) be such that supp f ∩ ∂ far,x = ∅. If f is bounded as an operator on Proof. As in Lemma 4.1, let A be an affine transformation with origin x such that A(R d + ) locally coincides with around x. It is straightforward to verify that it suffices to prove the proposition for f •A,A −1 ( ) . Since A −1 ( ) is also a convex polytope, we may hence assume that x = 0 and that locally coincides with R d + around 0. In particular, ⊂ R d + . Since supp f ⊂ and supp f ∩ ∂ far,0 = ∅, we can extend f to a distribution on all of R d + by letting it be zero outside . Our strategy is to show that the operator f ,R d + is bounded and to then apply Theorem 2.1.
For n ∈ N d let C n denote the cube (n 1 , By considering test functions g ∈ C ∞ c (R d + ) such that supp g ∩ rC m ⊂ rC m for every m, we give meaning to the equality Hence there are only finitely many non-zero terms in the decomposition. Since recalling the definition of f from Section 2, it therefore suffices to prove that f ,rC m ,rC n is bounded whenever (5.2) holds. If rC m , rC n ⊂ there is nothing to prove since f , is bounded by hypothesis. For the other terms, note that (5.2) and the choice of r implies that since 2 √ dr is the diameter of rC m + rC n . For any z ∈ R d , x ∈ rC n , and g ∈ C ∞ c (rC m ) we have that and hence f ,rC m ,rC n = f ,rC m +z,rC n −z .
In particular, for z = r(n − m)/2 we obtain that We have just demonstrated that f . This in particular implies thatb| = f when we return to the initial interpretation of f as a distribution on .
We are now ready to provide the proof of Theorem 1.1.
Proof. Assume that f is bounded. Let {x j } J j=1 be the vertices of , and let {μ j } J j=1 be partition of unity as in Lemma 4.2. For ϕ ∈ C ∞ c ( ) and x ∈ we have that where ϕ ξ (y) = e 2π iy·ξ ϕ(y). Hence, μ j f : L 2 ( ) → L 2 ( ) is bounded, Therefore, by Proposition 5.2 there are functions b j ∈ L ∞ such that μ j f =b j | . Thus We immediately obtain the corresponding result for Toeplitz operators, when is a simple convex polytope, which, possibly after a translation, is symmetric under Proof.
We also deduce the weak factorization result for PW 1 , see Section 2.5. According to Proposition 5.1, . Then, recalling that JG(x) = G(−x), we have that Since D is an involution, it follows that I * is onto.
In other words, I * : Then for ϕ ∈ C ∞ c ( ) and x = (x 1 , x 2 ) ∈ we have that where ϕ ξ (y) = e 2π iy 2 ξ ϕ(y), y = (y 1 , y 2 ) ∈ . Hence, as before we see that As in Proposition 5.2 and Theorem 1.1 it is sufficient to see that μ 1 f : 1)) define bounded operators, and by symmetry it is sufficient to consider the first of the two.
For n ∈ N, let S n denote the strip R + × (n, n + 1), and let r > 0 be such that We decompose μ 1 f : L 2 (R 2 + ) → L 2 (R 2 + ) according to strips instead of cubes, There are only a finite number of non-zero terms in this decomposition, and for any such term we have by our choice of r that rS m + rS n ⊂ . (5.6) For n, m corresponding to a non-zero term, we have that where z = (0, r(n − m)/2). Since rS m+n 2 ⊂ by (5.6) and μ 1 f : L 2 ( ) → L 2 ( ) is bounded by (5.5), we conclude that each non-zero term P rS n μ 1 f , R 2 + P rS m is bounded.
Hence μ 1 f : class of operators f , partially extends the class of generalized Toeplitz operators considered in [22], see Section 2.6. The next theorem can also be recovered by verifying the hypotheses of and keeping track of the constants in the proof of [22,Theorem 5.4].
However, for completeness we prefer to give our own concrete proof. Theorem 6.1. Let be a set as above. Then − = R d and, for f ∈ D (R d ), we have Proof. Fix z ∈ R d and set |z| = R. Pick a vector e ∈ int(cc˜ ) with distance greater than R to the complement of cc˜ , which is possible since cc˜ is a cone with non-empty interior. Then e + z ∈ cc˜ , so for any x ∈˜ we have that Similarly, x 1 + x + e ∈ . Since z is the difference of these two vectors, the 1st claim follows.
Suppose that we have proven the theorem for all f ∈ C ∞ c (R d ). If f is a general symbol for which f is bounded, consider the sequence of functions f n ∈ C ∞ c (R d ) from Proposition 3.3. Thenf n has, by Alaoglu's theorem, a subsequencef n k that converges weak star in L ∞ to some element g. Since f n converges to f in distribution, it must be that g =f . Hence f ∈ F −1 (L ∞ ) and, by Propositions 3.1 and 3.3, we have that This proves the theorem for general symbols.
Hence we assume that f ∈ C ∞ c (R d ). Fix ξ ∈ R d , pick any vector ν in int(bc˜ ), and consider for ε > 0 the function By [1, Lemma 9.5] this function is in L 2 (x 0 +˜ ) (The set bc˜ was denoted in [1].) and hence E ε ∈ L 2 ( ). We use E ε as a test function: uniformly on compacts in z. Since ξ is arbitrary this establishes that f ≥ f L ∞ and by Proposition 3.1 we then conclude that f = f L ∞ . Fix R > 0 and suppose that z ∈ R d with |z| < R. Again, pick a vector e ∈ int(cc˜ ) with distance greater than R to the complement of cc˜ . Then e + z ∈ cc˜ , and therefore With x 2 = x 1 +e−x 0 we have just shown that x 2 + ⊂ −z+ . It also holds that x 2 + ⊂ , by the last inclusion in the above chain and the fact that x 1 + e +˜ ⊂ x 1 +˜ ⊂ . This gives us that and hence that The desired equality (6.2) is now immediate, completing the proof. We present only the Toeplitz case. Such matrices appear in various applications, for example in multi-dimensional frequency estimation. Note in particular that Pisarenko's famous method for one-dimensional frequency estimation [34], which relies on the classical Carathéodory-Fejér theorem, was recently extended to the multi-variable case [39] (see also [3]).
When d = 1 our statement reduces to a well-known theorem on extending finite (ordinary) Toeplitz matrices, appearing previously for example in [5] and [26]. To describe it, recall that a finite N × N Toeplitz matrix is characterized by its constant diagonals, whose values we denote by a = (a −N+1 , . . . a N−1 ). As an operator T a on 2 ({0, . . . , N − 1}), its action is given by We can also consider the case when N = ∞, the definitions extending in the obvious way. The completion result then states that it is always possible to extend a to a biinfinite sequenceã such that the corresponding Toeplitz operator Tã : 2 (N) → 2 (N) satisfies Tã ≤ 3 T a .
It is an open problem whether the constant 3 is the best possible in this inequality. A discussion offering different approaches to the optimal constant can be found in [9]. See also [36]. For instance, a multilevel block Toeplitz matrix for d = 2 is an N × N Toeplitz matrix whose entries are N × N Toeplitz matrices. Again, we allow for the possibility that N = ∞. We now provide the multilevel block Toeplitz matrix analogue of the Toeplitz matrix completion theorem. In other words, with g r = {g(r + n)} n∈{0,...,N−1} d , we have that Integrating both sides over r ∈ (0, 1) d gives us that f (g) 2 ≤ T a 2 g 2 . In other words, Noting that the constant c in Corollary 1.2 is invariant under homotheties, we find that there exists a distributionf =b ∈ D (R d ), coinciding with f on (−N, N) where C d only depends on the dimension d. Of course, f ,R d : L 2 (R d ) → L 2 (R d ) is nothing but the operator of convolution withf . Now pick any function ϕ ∈ C ∞ c ((−1/2, 1/2) d ) with |ϕ| 2 dx = 1 and consider the isometry I : 2 (N d ) → L 2 (R d ) given by Then That is, I * f I = Tã. It is clear by construction thatã is an extension of a, a n = a n R d |ϕ(y)| 2 dy = a n , n ∈ {−N + 1, . . . , N − 1} d .
This finishes the proof.