Composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions

In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined on $\mathbf{R}^d$. We prove that only affine transforms can do so in a pretty large class of RKHS. Our result covers not only the Paley-Wiener space on the real line, studied in previous works, but also much more general RKHSs corresponding to analytic positive definite functions where existing methods do not work. Our method only relies on an intrinsic properties of the RKHSs, and we establish a connection between the behavior of composition operators and the asymptotic properties of the greatest zeros of orthogonal polynomials on a weighted $L^2$-spaces on the real line. We also investigate the compactness of the composition operators and show that any bounded composition operators cannot be compact in our situation.


Introduction
In this paper, we establish that the composition operator generated by a map in the Euclidean space R d enjoys the boundedness property in a reproducing kernel Hilbert space (RKHS for short) only if the map is affine when the reproducing kernel is an analytic positive definite function with some conditions. In addition, we will characterize affine maps which induce bounded composition operators. We summarize several basic notation at the end of this section.
Recall the definition of composition operators. Let f : E → E ′ be a map from a set E to a set E ′ , and let V and W be function spaces on E and E ′ , respectively. The composition operator C f : f → h • f is the linear operator (C f , D(C f )) from W to V whose domain is D( Since a composition operator is always a closed operator, it is worth remarking that the preserving property, that is, D(C f ) = W implies the boundedness of C f if V and W are sufficiently good topological linear spaces, for example, Banach spaces.
We also recall the notion of RKHSs. A function k defined on the cross product of E × E, where E is a set, is said to be positive definite if for any arbitrary function X : E → C and for any finite subset F of E, ∑ p,q∈F X (p)X (q)k(p, q) ≥ 0.
A fundamental theorem in the theory of RKHSs is that such a function k generates a unique reproducing kernel Hilbert space H k . See [20], for example. Now, let us state our main result. We will adopt the following definition of the Fourier transform: Let 0 = w ∈ L 1 ∩ L ∞ (⊂ L 2 ) and assume w ≥ 0 almost everywhere. Then, w is a positive definite function, namely, becomes a positive definite kernel, and thus k determine a RKHS H k (see Section 2 for more details). The RKHS H k is realized in the space of continuous and square integrable functions on R d whose Fourier transform vanishes almost everywhere on {w = 0}, and its norm is given by It is worth considering the case of d = 1 and w = 1 [−1/2,1 /2] . In this case, the RKHS constructed by w is called the Paley-Wiener space. This space is composed of functions whose fourier transforms are supported on [−1/2, 1/2]. As warping operators, many researchers have been studying the condition for which the D(C f ) is equal to H k , which is equivalent to the boundedness of C f , in the field of signal processing, a brunch of engineering [1,3,[5][6][7]23]. In [17], a general dimensional case is treated although their setting slightly differs from ours. We also note that composition operators have recently attracted researchers in a field of data science and machine learning [13,14,16], and mathematical properties of composition operators are quite important to provide their theoretical guarantee.
For z ∈ C d , we denote by m z the pointwise multiplication operator on L 2 (w): where ⊤ stands for the transpose of vectors or matrices. For each n ∈ N, the space P n ⊂ C[ξ 1 , ξ 2 , . . ., ξ d ] stands for the linear space of all polynomials of (total) degree at most n. In the case of |ξ | 2n w(ξ ) ∈ L 1 , we regard P n ⊂ L 2 (w), and for any y ∈ R d , we define We define a.e. ξ for some λ > 0 .
We will check that G (w) coincides with the linear maps inducing a bounded composition operator on H k (see Proposition 9). Under certain assumptions on w, the boundedness of composition operators force the original maps to be affine as our main result below shows: Theorem 1. Let w ∈ L 1 ∩ L ∞ \ {0} be non-negative almost everywhere, and let k(x, y) := w(x − y).
We impose the following three assumptions on w: (A) for any a > 0, there exists c a > 0 such that w(ξ ) ≤ c a e −a|ξ | for almost all ξ ∈ R n , (B) there exists B > 0 such that for any y ∈ R d , lim sup n→∞ E − n (y; w), lim sup n→∞ E + n (y; w) < B, We will prove Theorem 1 as a corollary of Theorems 3 and 4. Our method is based on an intrinsic structure of L 2 (w), and thus can treat a quite general class of w on higher dimensional Euclidean space R d (d ≥ 1). For some special cases, we have a more concrete result as follows: Theorem 2. Let w ∈ L 1 ∩ L ∞ \ {0} be a nonnegative spherical function. Assume that there exists a locally L 1 -function Q : [0, ∞) → R such that w(ξ ) = e −Q(|ξ |) . We further assume that there exists c ≥ 0 such that Q(t) + ct is nondecreasing for sufficiently large t ≥ 0 and that , where α ≥ 0 and β : [0, ∞) → R is an non-decreasing function going to ∞, satisfy the conditions in Theorem 2. We actually obtain a more general result (Theorem 6), and Theorem 2 above is a corollary (see Corollary 9) of Theorem 6. We also obtain a similar result in the case where w is a tensor product of even functions on R (see Section 4.4 for details).
Our results cover the previous works [6,7], namely, we see that Assumptions (A)-(C) in Theorem 1 hold if d = 1 and w is compactly supported, since m z is a bounded linear isomorphism on L 2 (w) and 1 ∈ G (w). Needless to say, 1 [−π,π] satisfies Assumptions (A)-(C) in Theorem 1. Thus, only affine maps can induce bounded linear operators on the Paley-Wiener space in this case.
Furthermore, our result also provides a non-trivial improvement even in a one-dimensional case. In fact, in [6,7], their proof is based on finiteness of the order of the entire function w. In their situation, the RKHS is composed of entire functions of order at most 1, and they directly use Pólya's theorem [19] with some careful analysis, in other words, finiteness of the order of w is crucial in their proof. In contrast, our method does not need finiteness of the order, and by means of Theorem 2, we actually find an example of RKHS containing entire functions of infinite order, but only affine maps can induce bounded composition operators (see Section 4.5 for details) as in the following corollary: Then, if a map induces bounded composition operators on the RKHS H k associated to k(x, y) = w(x − y), then the original map is affine, but the RKHS H k contains entire functions of infinite order.
In addition, in previous works, they always impose some good properties, such as entireness, on the original map. On the other hand, we do not need to assume the map f is entire in advance as we prove the boundedness of C f automatically induces the holomorphy and continuity of f (Theorem 4). The compactness of composition operators is an important problem as well. However, unfortunately, compactness fails as our next theorem shows.
Corollary 2. Using the same notation as Theorem 1, let w satisfy Assumptions (A)-(C). Then, no composition operators C f can be compact.
We can rephrase Corollary 2 in Proposition 10 on the basis of Theorem 1, namely, thanks to Theorem 1, it suffices to prove that affine maps cannot induce compact composition operators. See Section 3.1 for the details.
Under Assumption (A), H k is composed of entire functions on C d , thus our study is also located as a study of composition operators on entire functions. This topic has been extensively studied, for example, as in [2,8,9,21] and so on. However, the function space H k in our study is quite different from those in these previous literature. As a result, the behavior of composition operators dramatically changes compared with them, for example, composition operators therein can become compact linear operators, but does not in our situation as in the above corollary.
Let us explain the outline of the proof of Theorem 1 and 2. Regarding Theorem 1, the "if" part is not so hard, and we prove it in Section 3.1. The harder part of the proof is the "only if" part, and is obtained as a corollary of Theorems 3 and 4 in Section 3. Theorem 3 states that the "only if" part of Theorem 1 holds under Assumptions (A), (B), and (C), and assuming the existence of the holomorphic map F : C d → C d with F| U = f . Since there exists a natural isomorphism Ψ w : H k ∼ = L 2 (w) (see Corollary 4), the study of composition operators is reduces to that of the corresponding operators on L 2 (w). Thus, by considering the action of the corresponding operators on spaces of polynomials of various degrees in L 2 (w), the boundedness of the composition operators enables us to control the derivative of the holomorphic map F. Theorem 4 deduces that under Assumption (A) and D(C f ) = H k , there exists a holomorphic map F : C d → C d with F| U = f . The proof involves an explicit construction of the analytic continuation of f in terms of the boundedness of C f . Here, we emphasize that f is originally a mere map defined on an open subset U ⊂ R d , not the whole space, but we prove that the boundedness of C f shows the map f is a restriction of holomorphic map F defined on C d . As for Theorem 2, the hardest part in the proof is to verify Assumption (B). We show that the left hand side in Assumption (B) is closely related to an asymptotic behavior of greatest zeros of orthogonal polynomials in a weighted L 2 -space on R. We improve Freud's methods in his works [11,12] on asymptotic behaviors of orthogonal polynomials, and check Assumption (B). We include the details in Section 4.
Notation: In this paper, we always work in the d-dimensional Euclidean space. For r ≥ 0, we denote by C r the space of C r functions on R d . For a non-negative measurable function w on R d , and p ∈ [1, ∞), we denote by L p (w) the space composed of the equivalent classes of measurable functions h vanishing on In the case of w ≡ 1, we abbreviate L p (1) to L p . We also denote by L ∞ the space of essentially bounded measurable functions on R d . For any measurable set A in an Euclidean space, we denote by 1 A the characteristic function supported on A, and by |A| the volume of A with respect to the Lebesgue measure. We denote by M d (R) (resp. GL d (R)) the set of square real matrices of size d (resp. regular real matrices of size d).

Preliminaries
In this section, we review the notion of reproducing kernel Hilbert spaces associated with positive definite functions and composition operators and then show some of their basic properties.

Reproducing kernel Hilbert spaces with positive definite functions
Let E be an arbitrary abstract (non-void) set and k : E × E → C be a map. Denote by C E the linear space of all maps from E to C. A reproducing kernel Hilbert space (RKHS for short) with respect to k is a Hilbert space H k ⊂ C E satisfying the following two conditions: 1. For any x ∈ E, the map k x := k(x, ·) is an element of H k . 2. For any x ∈ E and h ∈ H k , we have h, k x H k = h(x).
If such H k exists, H k is unique as a set and we call k a positive definite kernel. We note that if k is a positive definite function in the sense described in Introduction, there exists a unique Hilbert space H k ⊂ C E satisfying the above two conditions (see, [20] for more detail). The second condition is known as the reproducing property of the RKHS H k . Here, we define the feature map by For any subset F ⊂ E, we define a closed subspace of H k by which is the closure of the linear subspace generated by the set φ k (F). Accordingly, we see that H k,F is isomorphic to H k| F 2 : Proposition 1. Let k be a positive definite kernel on E, and let F ⊂ E. Then, the restriction map C E → C F induces an isomorphism r k,F : Proof. For any x ∈ F, the restriction map allocates φ k| F 2 (x) to φ k (x); thus, it induces the isomorphism r k,F between the Hilbert spaces.
We shall now prove a proposition for the feature map for an RKHS associated with a positive definite function u:  y)), the continuity of u implies that of φ k . We will prove the injectivity of φ k . Suppose to the contrary that there exists a, b ∈ R d with a = b such that φ k (a) = φ k (b). Then, for any x ∈ R d , which contradicts the assumption that u vanishes at infinity. Thus, the feature map is injective. Now, we prove the continuity of φ −1 for some a ∈ R d but that {x n } n≥0 does not converge to a. Since φ k is injective and continuous, any convergent subsequence of {x n } n≥0 converges to a; thus, we may assume |x n | → ∞ as n → ∞. For any x ∈ R d , Since u is not a constant function, this is a contradiction. Thus, φ −1 k is continuous.
By the Riemann-Lebesgue theorem, we have the following corollary: Corollary 3. Assume H k is an RKHS associated with a positive definite function in the form of w for some non-negative w ∈ L 1 ∩ L ∞ \ {0}. Then the feature map φ k is a homeomorphism from R d onto φ k (R d ).
In this paper, we will only consider the RKHSs associated with a positive definite function in the form of w for some w ∈ L 1 ∩ L ∞ \ {0}.
We have an explicit description of the RKHS H k above: and its inner product is given by g, . By direct computation, we see that it suffices to show that the map h is continuous. In fact, it is a consequence of the following inequality: This inequality can be deduced from Schwartz's inequality. Thus, we may re- Then, we immediately obtain the following corollary: . Then, Ψ w is an isomorphism from H K to L 2 (w).

Properties of RKHSs for positive definite functions with a certain decay condition
Let w ∈ L 1 ∩ L ∞ \ {0} be a non-negative measurable function. We define the following decay condition: For a positive integer n > 0 and positive real number a ≥ 0, we define a function u that satisfies (DC) d n,a if for any ε > 0, there exists L ε > 0 such that For a > 0, we define We also define By virtue of Proposition 4, if u satisfies (DC) d n,a for some a > 0 (resp. a = 0), then any element of H k is holomorphic on X d a (resp. C n on X d 0 ). As a result, we have the following proposition: Proof. It suffices to prove that H ⊥ k,U = {0}. Take an arbitrary g ∈ H ⊥ k,U . Then, we see that for any x ∈ U , Since g is an analytic function on R d , we have g = 0. Therefore, Under the condition (DC) d n,a , for each z ∈ X d a , we define e z ∈ L 2 (w) by and we define the map, We should remark that in the case of a > 0, for is defined as the evaluation of the analytic continuation of u at x − z (note that u is originally defined on R d ). Accordingly, we have the following proposition: n,a for some integer n > 0 and a > 0 (resp. a = 0). Then the map ϕ is holomorphic (resp. differentiable) in X d a in the sense that for any Proof. For any positive number ε > 0, any non-negative integer n ≥ 0, j = 1, . . ., d, and any function ψ : Moreover, for n n n = (n 1 , . . ., n d ), we define It suffices to show that for any n n n if a > 0, or |n n n| ≤ n if a = 0. Here, we denote |n n n| := ∑ j n j . By direct computation, we have (left hand side of (2)) − (right hand side of (2)) 2 Thus, from the definition of (DC) d n,a , we see that the last integral converges to 0.

Composition operators on RKHS
We give a definition of composition operators for our setting:

Definition 2. Let k and ℓ be positive definite kernels on sets E and F, respectively. For any map f
Since we see that a composition operators is closed, by the closed graph theorem, we have the following proposition: Accordingly, the adjoint of C f has the following property: Proposition 8. Let k and ℓ be a positive definite kernel on E and F, respectively, and let f : Proof. The proof entails a straightforward computation.
Consequently, we have the following corollary: Since the right-hand side is continuous, so is f .
At the end of this section, we define another linear operator K f under the condition (DC) d n,a for some a > 0, keeping in mind that H k = H k,U according to Proposition 5.
Here, r k,U is the restriction map defined in Proposition 1.

Main results
Here, we prove the main results. We establish the criterion of the boundedness of composition operators in the case that the map f is affine.

Boundedness and compactness of composition operators for affine maps
Recall that we defined As the following proposition shows, G (u) is a natural class.
Proof. First, we prove the "if" part. Since any element of H k vanishes at ∞ and C f preserves H k , the matrix A has to be regular. Let h be an arbitrary nonnegative smooth function with compact support and vanishing in an open set including {w = 0}. We define g : Since by Proposition 4, we see that Since h is arbitrary, we have for almost all ξ . As in the same computation as above, for We also observe that the composition operators induced by affine maps cannot be compact: Since u(x) converges to 0 as |x| → ∞ by the Riemann-Lebesgue theorem, by taking the limit m, n → ∞ with m = n, we find that u(0) = 0. Since u is a positive definite function, |u(x)| = | k x , k 0 H k | ≤ u(0) by the Cauchy-Schwarz inequality. Thus, we have u = 0. This is a contradiction.

Affineness of holomorphic maps with bounded composition operators
In this section, we prove that maps are affine if they admit an analytic continuation and the domains of their composition operators are the whole space H k , namely, the following theorem: be a non-negative function, and let k(x, y) = w(x − y). We impose the following three assumptions on w: We always regard the space of d-variable polynomials C[ξ 1 , . . . , ξ d ] as a subspace of L 2 (w) as functions of (ξ 1 , . . . , ξ d ). We also fix an open set U ⊂ R d and a map f :

Then, for any open set U ⊂ R d and any map f
The following simple proposition shows that the information of F is included in K f although K f is defined by the map f initially defined only on U : Proposition 11. Assume Assumption (A) in Theorem 3. For any z ∈ C d , we have K f (ϕ(z)) = ϕ(F(z)).
where r ∈ where r z is a polynomial of degree smaller than k and its coefficients are entire functions with respect to the variable z. We prove (5) by induction on k. In the case of k = 1, since K f is continuous, we see that The left hand side is equal to K f [−2πiξ i 1 e z ], and the right hand side is equal to . Thus, we have (5). Let k > 1 and for ε > 0, put where e i k is the i k -th elementary vector in C d . Then, by induction hypothesis, we have where s z ∈ C[ξ 1 , . . ., ξ d ] is a polynomial of degree smaller than k − 1 and their coefficients are entire with respect to z. Then, if we take ε to 0, since K f is continuous, we see that there exists r z ∈ C[ξ 1 , . . ., ξ d ] of degree smaller than k whose coefficients are entire with respect to z such that Thus, by induction, we prove (5).
For each n > 0, we denote the space of homogeneous polynomials of degree n by C[ξ 1 , . . ., ξ d ] n . Then, we have the following corollary: The situation is the same as that in Lemma 1. Write Then, the following diagram is commutative: proj.
Q n,z /Q n−1,z Here, proj. is the natural surjection to the quotient, [·] means the natural morphism induced by (·), and we define S n [iJ F (z)/2π] to be the restriction of Proof. This is clear from Lemma 1.
Regarding Corollary 6, we have the following lemma: The situation is the same as in Corollary 6. Then, we have the following inequality on the norm of the operators: Here, the topologies of the quotients space the above operators act on are induced from L 2 (w).
Proof. This lemma immediately follows the fact that [ * ] is the same as the norms of the compositions of an inclusion and a projection for subspaces of L 2 (w) with * .
Next, we have the following key lemma: Proof. For each n > 0, we denote by · n the norm on C[ξ 1 , . . . , ξ d ] n induced from P n /P n−1 via the isomorphism (see Corollary 6). Here, the norm of P n is the restriction of that of H k , and the norm of P n /P n−1 is the quotient norm. Let α z be an arbitrary eigenvalue of J F (z) that acts on C[ξ 1 , . . . , ξ d ] 1 . Also, let v ∈ C[ξ 1 , . . . , ξ d ] 1 be its eigenvector. Then, we have By Corollary 6 and Lemma 2 (we use the notation in this corollary,) we have If we take the n-th root and then "lim sup n→∞ ", by combining this limit and Assumption (B), there exists B > 0 independent of z such that Thus, any eigenvalue of J F (z) is bounded by a constant independent of z. In particular, the holomorphic function trJ F is bounded on C d , and hence, trJ F is constant by Liouville's theorem.

Now we prove Theorem 3.
Proof. The "only if" part immediately follows from Proposition 9. The "if" part is proved as follows: If f is such a map, then, by Lemma 3, trJ A• f (z) = tr(AJ F (z)) is constant for any A ∈ G (w) as C A • C f = C A• f is bounded. By Assumption (C) in Theorem 3: G (w) R = M d (R), it follows that J F itself is independent of z. Thus, f is an affine map.

Analytic continuation
In this section, we prove that any map inducing bounded composition operators in H k has an analytic continuation: Proof. Put u = w. By Lemma 4, we can find vectors a 1 , a 2 , . . ., a d such that {∇u(a j )} d j=1 spans C d . Fix b ∈ U arbitrarily. It suffices to show that f is C 1 at a neighborhood of b in U . Define Then, , is also a C 1 -function defined on R d . Furthermore, Consequently, . Therefore, f is a C 1 -function on a neighborhood of b.

Now let us prove Theorem 4.
Proof. First, we claim that we may replace w withw(ξ ) := (w(ξ )+w(−ξ ))/2. In particular, this allows us to assume that w is an even function, and thus, −1 ∈ G (w). Let v := w, and we define ℓ(x, y) := v(x − y). We show the claim as follows: in fact, it is obvious that v satisfies (DC) d n,a . We prove that the composition operator from H ℓ to H ℓ| U 2 is defined everywhere and bounded. We define a densely defined linear map K f : L 2 (w) → L 2 (w) with domain D( K f ) = {e x } x∈U C by allocating e f (x) to e x (Here, we may define such a linear map since e x 's are linearly independent). Let u := w, and let h = r ∑ j=1 a j e x j ∈ D( K f ) (a j ∈ C d and x j ∈ U for j = 1, . . . , r). Then Thus, we see that K f is bounded and we can uniquely extend K f as a bounded linear operator on L 2 (w). We define . Therefore C f is simply the composition operator from H ℓ to H ℓ| U 2 , which is defined everywhere and bounded. By the above claim, we may assume that w is an even function and −1 ∈ G (u). Fix y = (y 1 , . . ., y d ) ∈ U , and define the holomorphic map F : Let m : L 2 (w) ⊗ L 2 (w) → L 1 (w) be the natural multiplication map, and let ι : Cξ i ⊂ L 1 (w) be the linear isomorphism defined by allocating ξ j to the vector e j := (0, . . . , 0, j 1, 0, . . . , 0). Then, we have the following proposition: Proposition 12. Under the above notation, let U 0 be the connected component Moreover, for any y ∈ U 0 , we have Proof. Since m and (K f ⊗ K − f ) are bounded linear operators and F is obviously holomorphic, the composition m • (K f ⊗ K − f ) • F is also holomorphic. Let x ∈ U 0 . Thanks to Proposition 6 and Lemma 1, we see that Now, we complete the proof. Since Lemma 6 below implies Cξ i , a and gives the analytic continuation of f .

Lemma 6. Let U ⊂ R d be an open set, and let X ⊂ C d be a connected open set
containing U . Also, let V be a locally convex space over C, and let γ : Proof. Suppose that γ(z 0 ) / ∈ V 0 for some point z 0 ∈ X . Then, the Hahn-Banach theorem guarantees that there is a continuous linear functional λ : V → C such that λ (γ(z 0 )) = 1 and λ (V 0 ) = {0}. Therefore, λ • γ : X → C is a holomorphic function vanishing at U ; thus, λ • γ ≡ 0 on X . This contradicts λ • γ(z 0 ) = 1.

Boundedness for special positive definite functions
In this section, we investigate the boundedness of composition operators on RKHSs associated to a spherical positive definite function and a convolution of real positive definite functions on R. Unless w is compactly supported, Assumption (B), that is the existence of B > 0 such that for any y ∈ R d , is the hardest condition to verify in our main theorem (Theorem 1). In the case where w is spherical or convolution of real positive definite functions, we may relate the left-hand-side to an asymptotic behavior of greatest zeros of certain orthogonal polynomials. On the other hand, the asymptotic properties of greatest zeros of orthogonal polynomials are extensively studied in previous works (see, for example, [18,Part 2]), and we can utilize various techniques developed there. Analysis illustrated in this section also provides a non-trivial consequence even in a one-dimensional case. Indeed, our result is still valid even if an RKHS contains an entire function of infinite order where previous frame work does not work.

Orthogonal polynomials
First, we briefly review basic properties of orthogonal polynomials. For more details, see [18,22]. Let µ be a Borel measure on R. Assume, for any integer Then, we define (normalized) orthogonal polynomials by polynomials such that p n (t; µ) is of degree n and R p m (t; µ)p n (t; µ)dµ(t) = δ m,n for any m, n ≥ 0. We note that each p n (t; µ) is uniquely determined up to sign. We denote by X n (µ) the greatest zero of p n (t; µ). We denote by γ n (µ) the leading coefficient of p n (t; µ), and we define the Christoffel function λ n (s; µ) by We recall the following somewhat known properties: Proof. We invoke the following formulas (see, for example, [

Asymptotic estimation of greatest zeros of orthogonal polynomials
We denote by L p (R) the usual L p -space with respect to the Lebesgue measure on the real line R for p ∈ (0, ∞]. We denote by · p the L p -norm of L p (R). Let Before getting into the main body of this section, we introduce some notation: For any L ∈ R and for any measurable function Q : [0, ∞) → R, we define We also define the "monotonic" part and "oscillated" part for Q as follows: We note that Q o is non-negative, and Q m is non-decreasing and upper semicontinuous (thus right continuous: lim For simplicity, we abbreviate (Q L ) m and (Q L ) o to Q m L and Q o L . We will prove the following theorem: Theorem 5. Let W be as above. Suppose that there exists a function Q : [0, ∞) → R such that W (t) = e −Q(|t|) . Let L > 0 be a positive number and assume t n e Lt W (t) ∈ L 1 (R) for all n ≥ 0. For σ ≤ L, we define Assume in addition the following two conditions: and there exists B > 0 such that for any sufficiently large t > 0, Then, X n (µ σ )n −1 uniformly converges to 0 over σ ≤ L.
The proof of Theorem 5 is based on some auxiliary estimates. The following lemma is essential: Lemma 7. Under the same notation as Theorem 5, for any ρ > 0, we define Ξ ρ,σ := | · | ρ W σ ∞ , ξ ρ,σ := sup ε>0 ess inf t ≥ 0 : t ρ W σ (t) > Ξ ρ,σ − ε Then, for any σ ≤ L, we have In order to prove Lemma 7, we improve results in [10]. Before proving Lemma 7, we provide several fundamental inequalities. For ξ ≥ 0 and a function W as in the beginning of this section, as Freud did in [10], we define and we denote µ ξ := W ξ (t)dt. First, we prove an elementary inequality for holomorphic functions. Lemma 8 below will substitute for the technique used in [10], where Freud used the Hardy space H 2 over the unit disc.
Proof. Let α 1 , . . . , α n ∈ D be the zeros of f contained in D. Then, we may assume that f takes the form: where g is a holomorphic function in D and m 1 , . . . , m n > 0. Thus, it suffices to prove the case of f = e g or f = z − α for some α ∈ D \ {0}. In the case of f = e g , it immediately follows from the mean value theorem for harmonic function. In the case of f (z) = z − α, we employ the following well-known fact for |α| ≤ 1: 1 2π 2π 0 log e iθ − α dθ = 0.
In this case, the inequality (15) is equivalent to the validity of the inequality for |α| ≤ 1.
Then, we have an estimation of the Christoffel function as a corollary of this lemma: Corollary 7. For any ξ > 0 and s ∈ R such that |s| > ξ > 0, we have Proof. The proof is completely the same as [10, Lemma 2], where Freud factorized p n (z; µ ξ ) and used the fact that any zeros of p n (t; µ ξ ) are contained in [−ξ , ξ ] and that ξ 2 ≥ ξ 2 − a 2 for all ξ , a ∈ R with |ξ | ≥ |a|.
As a corollary of this lemma, we have the following estimation: Corollary 8. Under the same notation as Lemma 10. For any ρ > 0, we define Proof. Let A = 2 and ξ = ξ 4n in Lemma 10. Since ∞ 2ξ 4n we have the desired estimation.
We refer back to the proof of Lemma 7.
First, we claim that In fact, we first prove Ξ ρ ≥ sup t≥0 t ρ V (t). Since for any t ≥ 0 and δ > 0, by definition, the set S := {s ≥ t : W L (s) > V (t) − δ } has positive measure, we see that Ξ ρ ≥ s ρ W L (s) ≥ t ρ V (t) − t ρ δ for almost all s ∈ S. It implies the desired inequality. As sup and define η δ := ess infΩ δ . Take arbitrary η 1 > η δ . Then, we see that (η δ , η 1 )∩ Ω δ has positive volume and is contained in Since η 1 is arbitrary and V is right continuous, we see that Ξ ρ − δ ≤ η ρ δ V (η δ ) for sufficiently small δ > 0. Since, η δ ր ξ ρ as δ → 0, we have lim Let us assume ξ ρ ≥ 1. By (22), we have for all 0 ≤ t ≤ ξ ρ , or equivalently, Let q > 0 be an arbitrarily positive number. By multiplying the both sides by qt q−1 and integrating them from 0 to ξ ρ , we have Since we see that the second term of (23) goes to Q m L (0) when q → 0 as Q m L is right continuous and bounded on [0, 1]. Thus, by taking limit q → 0 on (23), we have By (11), we see that the left hand side is +∞, thus so is the right hand side.

Boundedness for spherical positive definite functions
Let w be a spherical function on R d , and let L > 0. Take W : [0, ∞) → R be a function such that w(ξ ) = W (|ξ |). Assume |ξ | n e 2L|ξ | w(ξ ) ∈ L 1 for all n ≥ 0. For y ∈ R d with |y| ≤ L, we introduce E n (y; w) := n sup P∈P n \{0} e |y||·| P L 2 (w) First, we show a relation between the zeros of orthogonal polynomials and E + y (w). Lemma 12. Let w, W and L be as above. For 0 ≤ σ ≤ L, we define the measure ν σ on R by ν σ = |t| d−1 e 2σ |t| W (|t|)dt.
Then, for y ∈ R d with |y| ≤ L, we have Proof. The first inequality is obvious in terms of the Cauchy-Schwarz inequality. We prove the second inequality. For any real-coefficients one-variable poly- By the Cauchy-Schwarz inequality, we have By Theorem 1 of [12], we have Therefore, by integrating both sides of (24), we have On the other hand, by a direct computation, we have Since h is a positive polynomial of degree at most 2n, it is well known that there exists a finite collection of polynomials h 1 , . . . , h K of degree at most n such that h = h 2 1 + · · · + h 2 K (for example, see [4]). Thus we have Therefore, by (25), we have the desired inequality.
We prove the following theorem: . Assume there exists a measurable function Q : [0, ∞) → R such that w(ξ ) = e −Q(|ξ |) and for all L > 0, there exists B > 0 such that for any sufficiently large t > 0, Then, the function w further satisfies Assumptions (B) and (C).
Proof. Since w is a spherical function, the set G (w) contains all the orthogonal matrices, thus, G (u) generates the space of matrices, namely, (C) holds. Let us prove Assumption (B). First, we prove lim sup n→∞ E n (y; w) ≤ 1.
Thanks to Lemma 7, it suffices to prove that R L (t) := Q(t) − (d − 1) logt − Lt satisfies the conditions (11) and (26) for all L > 0. The condition (11) is immediate since w satisfies Assumption (A). Regarding condition (26), since Since the second term is constant for sufficiently large t, we see that R L satisfies condition (26) as Q L+d−1 satisfies (26). Thus, lim sup n→∞ E + n (y; w) ≤ 1 by the left inequality of Lemma 12. We prove lim sup n E − n (y; w) ≤ 1. Let w 0 (ξ ) := e −2|y|·|ξ | w(ξ ). Let P ∈ P n . By the Cauchy-Schwarz inequality, we have Then, we have Thus, we may use Lemma 12 and Theorem 5 as in the above argument, and we obtain lim sup n→∞ E − n (y; w) ≤ lim sup n→∞ E n (y; w 0 ) ≤ 1.
As a result, we obtain a simple sufficient condition for w so as to satisfy Assumption (B): be a nonnegative spherical function. Assume that there exists a locally L 1 -function Q : [0, ∞) → R such that w(ξ ) = e −Q(|ξ |) . We further assume that there exists c ≥ 0 such that Q(t) + ct is nondecreasing for sufficiently large t ≥ 0 and that Q(t + R) − Q(t) → ∞ as t → ∞ for some R > 0. Then, the function w satisfies Assumptions (A), (B), and (C).
Proof. We deduce from the assumtions of the corollary that Q is bounded any interval [a, b] as long as b > a ≫ 1. Assumption (A) immediately follows from the condition that Q(t + R) − Q(t) → ∞ as t → ∞. We will prove that for any L > 0, Q o L (t) is bounded for any sufficiently large t > 0. We easily see that the boundedness of Q o L implies the condition (26) in Theorem 5, and thus, w satisfies the condition (B) and (C) by Theorem 6.
First, we claim that we may assume Q is non-decreasing. In fact, Let R ′ > 0 be a positive number such that Q −c (t) = Q(t) + ct is non-decreasing for t ≥ R ′ . We define a non-decreasing functionQ bỹ Note that we immediately see thatQ(t + R) −Q(t) → ∞ as t → ∞. Then, we have Thus, we have Since the first term does not affect the condition (26), we may replace Q(t) with the non-decreasing functionQ to prove the condition (26). Now, we assume Q is non-decreasing. Fix an arbitrary sufficiently large number s ≥ 0 satisfying Q(t + R) − Q(t) > LR for any t ≥ s. Since for any t ∈ [s, s + R), we have Q L (t + nR) − Q L (t) > 0 for all positive integer n ≥ 1. Thus, for any arbitrary ε > 0 there exists t ε ∈ [s, s + R) such that Q m L (s) + ε > Q L (t ε ). Then, by definition of Q o L and by the fact that Q is non-decreasing, we see that Since ε is arbitrary, we have Q o L (s) ≤ LR, which means that for any sufficently large s > 0, Q o L (s) bounded.

Boundedness for positive definite functions of tensor products of even functions
In this subsection, we discuss the case where the non-negative function w is a tensor product of even functions on R. First, we prove the following lemma: Lemma 13. Let w ∈ L 1 ∩ L ∞ \ {0} be a nonnegative measurable function. Assume there exists w 1 , . . . , w d : R → R such that each w i satisfies Assumption (A) and w(ξ 1 , . . . , ξ d ) = w 1 (ξ 1 ) · · · w d (ξ d ). If each w i satisfies Assumption (B), namely, there exists B i > 0 such that for all y ∈ R, lim sup n→∞ E ± n (y; w i ) < B i , then w also satisfies Assumption (B).
Proof. Fix y = (y i ) d i=1 ∈ R. Then, for any P ∈ P n , we see that m y P 2 where P i is a one-variable polynomial of degree at most 2n.
Therefore, we see that lim sup n E + n (y, w) ≤ B 1 · · · B d .
We also obtain lim sup n E − n (y, w) ≤ B 1 · · · B d .
in the same manner as above.
Then, we obtain a similar theorem to Theorem 6: for all i, j ∈ {1, . . . , d}, |Q i − Q j | is bounded, and for each i = 1, . . ., d and any L > 0, there exists B i > 0 such that for any sufficiently large t > 0, Then, the function w further satisfies Assumptions (B) and (C).
Proof. By Lemma 13 with Theorem 6, it suffices to show that w satisfies Assumption (C). The boundedness of |Q i − Q j | implies G (w) contains all the symmetric group S d ⊂ GL d (R d ). Since each Q i is an even function, G (w) also contains C d 2 ⊂ GL d (R d ), the group of diagonal matrices A satisfying A 2 = I. Since the group generated by S d and C d 2 generates M d (R) as a linear space over R, we obtain (C).
We also have a similar corollary to Corollary 9: for all i, j ∈ {1, . . ., d}, |Q i − Q j | is bounded. We further assume that for each i = 1, . . ., d, there exists c i such that Q i (t)+c i t is non-decreasing for sufficiently large t ≥ 0 and Q i (t + R i ) − Q(t) → ∞ as t → ∞ for some R i > 0. Then, the function w satisfies Assumption (A), (B), and (C).
Proof. By Lemma 13 with Corollary 9, it suffices to show that w satisfies Assumption (C), but its proof is the same as in that of Theorem 7.

An example of entire functions of infinite order
Even in one dimensional case, our result contains an essentially new contribution. In the case of d = 1 and w = 1 [−1/2,1/2] , there are several works treated the boundedness of composition operators in RKHS [6,7]. Their method based on finiteness of the order of the entire function w. As the RKHS associated to the positive definite function w is composed of entire functions of order 1, they [6,7] directly apply Pólya's theorem [19] with some careful analysis, and deduce affiness of original maps inducing bounded composition operators. We may apply this method without using ours discribed in this paper if the RKHS is composed of entire functions of finite order. However, there exists an example of RKHSs containing entire functions of infinite order, but only affine maps can induce bounded composition operators on the RKHS based on our framework. Let us explain the example. We define Then, we easily see that = e e 2πiz + e e −2πiz − 1 · sin(πz) πz The entire function w(z) is of infinite order since | w(iy)| = O(e e 2πy ). Let Q := − log w. Then, we immediately see that Q is non-decreasing and Q(t + 1) − Q(t) → ∞ as t → ∞ since Q(t + 1) − Q(t) = log(n + 1) for t ∈ [−1/2 + n, 1/2 + n). Thus, thanks to Corollary 9 and Theorem 1, if a composition operator on the RKHS associated with the positive definite function w is bounded, then the original map is an affine map.