Superoscillating sequences and supershifts for families of generalized functions

We construct in this paper a large class of superoscillating sequences, more generally of $\mathscr F$-supershifts, where $\mathscr F$ is a family of smooth functions (resp. distributions, hyperfunctions) indexed by a real parameter $\lambda\in \R$. The key model we introduce in order to generate such families is the evolution through a Schr\"odinger equation $(i\partial/\partial t - \mathscr H(x))(\psi)=0$ with a suitable hamiltonian $\mathscr H$, in particular a suitable potential $V$ when $\mathscr H(x) = -(\partial^2/\partial x^2)/2 + V(x)$. The family $\mathscr F$ is in this case $\mathscr F= \{(t,x) \mapsto \varphi_\lambda(t,x)\,;\, \lambda \in \R\}$, where $\varphi_\lambda$ is evolved from the initial datum $x\mapsto e^{i\lambda x}$. Then $\mathscr F$-supershifts will be of the form $\{\sum_{j=0}^N C_j(N,a) \varphi_{1-2j/N}\}_{N\geq 1}$ for $a\in \R\setminus [-1,1]$, taking $C_j(N,a) =\binom{N}{j}(1+a)^{N-j}(1-a)^j/2^N$. We prove the locally uniform convergence of derivatives of the supershift towards corresponding derivatives of its limit. We analyse in particular the case of the quantum harmonic oscillator, which forces us, in order to take into account singularities of the evolved datum, to enlarge the notion of supershifts for families of functions to a similar notion for families of hyperfunctions, thus beyond the frame of distributions.


Introduction
The Aharonov-Berry superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in the study of Aharonov weak measurements, [3,10,11,23]. The literature related to superoscillations is very large; without claiming completeness, we mention [17,18,19,21,22,33,34,36,39]. Quite recently, this class of functions has been investigated from the mathematical point of view, see [4,5,6,7,8,12,13,16,25,27,28] and the monograph [9]. Their theory is now very well developed, even though there are still open problems associated with superoscillatory functions, in particular as it concerns their longevity, when evolved according to a wide class of partial differential equations.
Let a > 1 be a real number. The archetypical superoscillatory sequence is the sequence of complex valued functions {x → F N (x, a)} N ≥1 defined on R by and N j denotes the binomial coefficient. The first thing one notices is that if we fix x ∈ R, and we let N go to infinity, we immediately obtain that lim N →∞ This new representation, together with the computation of the limit of x → F N (x, a) when N goes to infinity, explains why such a sequence {x → F N (x, a)} N ≥1 is called superoscillatory. Even though, for every N , the frequencies (1 − 2j/N ) that appear in the Fourier representation of F N are bounded by one, the limit function is x → e iax , where a can be an arbitrarily large real number. If one considers the map λ ∈ R → ϕ λ , where ϕ λ : x ∈ R → e iλx , one says that λ → {x → F N (x, λ)} N ≥1 realizes a supershift for λ → ϕ λ , or also that λ → ϕ λ admits λ → {x → F N (x, λ)} N ≥1 as a supershift. Such a notion will be made precise in Definition 5.1.
Let t ∈ R or t ∈ R + be a real parameter and P = γ 0 + γ 1 X + · · · + γ d X d ∈ C[X] with γ d = 0. For any λ ∈ R and N ∈ N * , let In the particular case where P is even with real coefficients, namely P = d ′ κ ′ =0 γ 2κ ′ X 2κ ′ with γ 2κ ′ ∈ R for κ ′ = 0, ..., d ′ , γ 2d ′ = 0 andP = d ′ κ ′ =0 (−1) κ ′ +1 γ 2κ ′ X 2κ ′ , the function (t, x) ∈ R 2 t,x −→ ψ P,N (a, t, x) is the global solution of the Schrödinger type partial differential equation i∂/∂t −P (∂/∂x) (ψ) ≡ 0 in R 2 t,x evolved from the initial datum x → F N (x, λ) on the line {t = 0} in R 2 t,x . Let D x := ∂/∂x and ⊙ denote the composition law between differential operators in the variable x with coefficients depending on the parameter t. Observe then that one can rewrite formally for any N ∈ N * ψ P,N (t, x, λ) = We will justify such a rewriting in section 3 and exploit it in order to prove (theorem 3.3) that for any (µ, ν) ∈ N 2 ∂ µ+ν ∂t µ ∂x ν (ψ P,N (t, x, λ)) N −→∞ −→ ∂ µ+ν ∂t µ ∂x ν (e itP (λ) e iλx ) Let a ∈ R \ [−1, 1]. We deduce in this way from the original "superoscillating" sequence {x → F N (x, a)} N ≥1 a large class of superoscillating sequences which are of the form {x → ψ P,N (t, x, a)} N ≥1 where t ∈ R is interpreted as a real parameter, the superoscillating convergence being uniform with respect to the parameter t ; moreover the superoscillating convergence property propagates through any differential operator in t, x, the convergence being uniform on any compact subset of R 2 t,x . As we already mentioned, the class of superoscillating sequences {x → ψ P,N (t, x, a)} N ≥1 introduced previously includes examples where (t, x) → ψ P,N (t, x, a) is realized through the evolution in t (from the initial value t = 0) of the solution of a Cauchy problem (with initial datum on {t = 0}) attached to a Schrödinger operator i∂/∂t −P (∂/∂x), where P is a real even differential operator with constant coefficients (the case whereP (∂/∂x) = −∂ 2 /∂x 2 corresponds for example to the classical case of Schrödinger equation for a free particle). Indeed if, for λ ∈ R, one denotes the response (at the time t) to the input datum x → e iλx (when t = 0) with the function (t, x) −→ ϕ λ (t, x), then for any a ∈ R the function (t, x) → ψ P,N (t, x, a) can be expressed as Since, when λ > 1 is arbitrarily large and N ∈ N * , the function (t, x) → ψ P,N (t, x, λ) arises from shifted functions (t, x) → ϕ λ (t, x) with |λ| ≤ 1, one can still say that λ ∈ R → ϕ λ (considered from R into the space of functions of the variables (t, x) in the phase domain, here R 2 t,x ) admits λ → (t, x) → ψ P,N (t, x, λ) N ≥1 as a supershift.
Given a Schrödinger operator i ∂ ∂t + 1 2 ∂ 2 ∂x 2 + V (x) with a suitable real potential V and Green function G V : (t, x, x ′ ) → G V (t, x, 0, x ′ ) such that for λ ∈ R and (t, x) in the phase domain (as a regularized integral on R or R + in a sense that will be precised in section 4), we will settle sufficient conditions that ensure in particular that the integral operator locally uniformly in some open subset U of the phase space (in R 2 t,x ) on which V is smooth and which is entirely determined by the explicit expression of the Green function G V (theorem 5.1). In such a situation the map λ ∈ R → (ϕ λ ) |U admits then the sequence (where a ∈ R \ [−1, 1]) as a supershift. Moreover, for any λ ∈ R, ϕ λ ∈ C ∞ (U , C) and for any (µ, ν) ∈ N 2 , locally uniformly in U .
Interesting situations occur when λ ∈ R → ϕ λ makes sense as a continuous map from R into D ′ (U ′ , C) for some open subset U ′ U in the phase space. Such is the case in the example of the quantum harmonic oscillator, where V (x) = x 2 /2, the phase space is R + * × R and In such case, given k ′ ∈ N and (when a ∈ R \ [−1, 1]) as a supershift for λ −→ (ϕ λ ) |about ((2k ′ +1)π/2,x 0 ) (all maps being considered here as distribution-valued about ((2k ′ + 1)π/2, x 0 )), while it is possible to do so about a point (k ′′ π, x 0 ), where k ′′ ∈ N * . In order to interpret (1.3) as a supershift for λ → (ϕ λ ) about ((2k ′ +1)π/2,x 0 ) , one needs to consider (ϕ λ ) |about ((2k ′ +1)π/2,x 0 ) as a hyperfunction (in t) times a distribution (in x) instead of distribution in (t, x). We will discuss such questions in section 6. The plan of the paper is the following: the paper contains five sections, besides this introduction. In section 2 we introduce the spaces A p (C), A p,0 (C), and we define some infinite order differential operators with nonconstant coefficients which will play a crucial role to prove our main results. In section 3 we recall the definition of generalized Fourier sequence and (complex) superoscillating sequence in one and several variables together with some examples; we then study two Cauchy-Kowalevski problems (one of which of Schrödinger type) and we show that superoscillations persist in time. In section 4 we address the problem of explaining the process of regularization of formal Fresnel-type integrals which is a necessary step to obtain further results in the paper. Fresnel-type integrals are shown to be continuous on A 1 (C) in section 5, in which we also treat a Cauchy problem for the Schrödinger equation with centrifugal potential and also for the quantum harmonic oscillator. Finally, in section 6, we investigate the evolution of superoscillating initial data with respect to the notion of supershift for the quantum harmonic oscillator, and we focus on singularities. It is interesting to note that in this case one needs to extend the concept of supershift in the case of hyperfunctions.
Notations. We use the notations with capital letters Z, d/dZ, W, d/dW,Ž in the expressions of formal differential operators, besides the usual notation z for the complex variable and t (time) x, x ′ (space) real variables.

On continuity of some convolution operators
Let f be a non-constant entire function of a complex variable z. We define for r ≥ 0.
The non-negative real number ρ defined by is called the order of f . If ρ is finite then f is said to be of finite order and if ρ = ∞ the function f is said to be of infinite order.
In the case f is of finite order we introduce the non-negative real number and call it the type of f . If σ ∈ (0, ∞) we say f is of normal type, while we say it is of minimal type if σ = 0 and of maximal type if σ = ∞. Constant entire functions are considered of minimal type and order zero. In the sequel we will extensively make use of weighted spaces A p (C) or A p,0 (C) of entire functions whose definition follows ; such spaces are classical, see e.g. [15,41].
Definition 2.1. Let p be a strictly positive number. We define the space A p (C) as the C-algebra of entire functions such that there exists B > 0 such that The space A p,0 (C) consists of those entire functions such that To define a topology in these spaces we follow [15, Section 2.1]. For p > 0, B > 0 and for any entire function f , we set

is a Banach space and the natural inclusion mapping
. For any sequence {B n } n≥1 of positive numbers, strictly increasing to infinity, we can introduce an LF-topology on A p (C) given by the inductive limit Since this topology is stronger than the topology of the pointwise convergence, it is independent of the choice of the sequence {B n } n≥1 . Thus, in this inductive limit topology, given f and a sequence {f N } N ≥1 in A p (C), we say that f N → f in A p (C) if and only if there exists n ∈ N * such that f, f N ∈ A Bn p (C) for all N ∈ N * , and f N − f Bn → 0 for N → ∞. The topology on A p,0 (C) is given as the projective limit where {ε n } n≥1 is a strictly decreasing sequence of positive numbers converging to 0. It can be proved, see [15,Section 6.1], that A p (C) and A p,0 (C) are respectively a DFS space and an FS space. When p > 1, A p,0 (C) is the strong dual of A p ′ (C) (where 1/p + 1/p ′ = 1), the duality being realized as In the extreme case p = 1, A 1 (C) (also denoted as Exp(C)) is isomorphic to the space H(C) of analytic functionals, the duality being realized as Here H(C) is equipped with its usual topology of uniform convergence on any compact subset.
The following result is an immediate consequence of the definition of the topology in the spaces A p (C) for p > 0.
be a sequence of elements in A p (C). The two following assertions are equivalent: • the sequence f converges towards 0 in A p (C) ; • the sequence f converges towards 0 in H(C) and there exists Proof. The first assertion means that there exists B > 0 with lim N →∞ f N B = 0 (in particular f N B ≤ 1 for N ≥ N 1 ), which implies that the sequence f converges to 0 in H(C) and that |f N (z)| ≤ Ae B|z| p with B and A = sup(Ã 1 , ...,Ã N 1 , 1) independent of N (Ã j = sup C (|f j (z)| e −B|z| p ) for j = 1, ..., N 1 ). Conversely, assume that the second assertion holds and take B > B f , so that, given ε > 0, there exists R ε > 0 such that On the other hand, since f converges to 0 uniformly on any compact subset of C, in particular on D(0, R ε ), there exists N ε ∈ N * such that Therefore sup N ≥Nε f N B < ε and the sequence f converges to 0 in A p (C).
To prove our main results we need an important lemma that characterizes entire functions in A p (C) in terms of the behaviour of their Taylor development, see Lemma 2.2 in [12].
Lemma 2.1. The entire function f : z → ∞ j=0 f j z j belongs to A p (C) if and only if there exists . (2. 2) The following lemmas are refinements of results previously stated in [12], except that we need here some extra dependency with respect to auxiliary parameters. They will be of crucial importance in order to prove the main results in the next sections.
Lemma 2.2. Let T be a set of parameters and τ ∈ T → D(τ ) be a differential operator-valued map for some p ≥ 1 and B ≥ 0. Then D(τ ) acts as a continuous operator from A 1 (C) into itself uniformly with respect to the parameter τ ∈ T .
Proof. It follows from Lemma 2.1 that the coefficient functions τ −→ b j (τ ) satisfy then uniform estimates for some positive constants C = C(D) and b = b(D) depending only on the finite quantity A in (2.3) and B. Let f : W → ∞ ℓ=0 a ℓ W ℓ ∈ A 1 (C). There are then (see again Lemma 2.1) positive constants γ and β such that |a ℓ | ≤ (γ/ℓ!) β ℓ for any ℓ ∈ N. Consider the action of D on such f . One has (for the moment formally) Therefore the formal identity (2.4) is in fact a true one for any W ∈ C, which shows that Let f = {f N } N ≥1 be a sequence converging towards 0 in A 1 (C) which is equivalent to say that sup(b f N + C f N ) < +∞ and that f converges towards 0 in H(C), see Proposition 2.1. Then the sequence for some positive constants A f and B f depending only on D and f. Let B > B f and ε > 0. Let R = R ε large enough such that for some constants C f and b f independent on τ ∈ T and on N (see (2.5)) and the sequence f converges to 0 in H(C), one can find N = N ε such that Hence the sequence D(τ )(f) converges towards 0 in A 1 (C), uniformly with respect to the parameter τ .
Since the next lemma involves as set of parameters T the set which is now given as split in the form T = T × C Z , where C Z is already a copy of the complex plane, one needs to duplicate C Z into an extra copy of C denoted as C W . Lemma 2.3. Let T be a set of parameters and t ∈ T → D(t, Z) be a differential operator-valued map for somep > 1, p ≥ 1 and B ≥ 0. Then D(t, Z) acts as a continuous operator from A 1 (C) into Ap ,0 (C) uniformly with respect to the parameter t ∈ T.
Proof. The function is well defined and depends as an entire function of two variables of the variables Z and W (which also justifies in (2.7) the application of Fubini theorem). Cauchy formulae in C × C show that for any t ∈ T, for any j, κ ∈ N, for each η > 0, with constants C η ,b and b independent on the parameter t. Let now f = {f N } N ≥1 be a sequence of elements in A 1 (C) which converges to 0 in A 1 (C). All differential operators act continuously on A 1 (C), as seen in Lemma 2.2. Moreover, one has (plugging in (2.5) the estimates (2.8)) that is the entire (with order 1/p and type 1) Mittag-Leffler function. One has therefore for such and (taking now W = Z) Since the Mittag-Leffler function E 1/p,1 has orderp > 1, the estimates (2.10) (uniform in the parameter t as well as on the function f ∈ A 1 γ,β (C)) show that the differential operator acts continously from A 1 (C) into Ap ,0 (C), uniformly with respect to the parameter t ∈ T. One just needs to repeat here the end of the proof of Lemma 2.2.
We conclude this section by proving a quantitative lemma which reveals to be essential in the sequel. It is a refinement of Lemma 1 in [28].
Lemma 2.4. Let a ∈ C with α := max(1, |a|) and, for any z ∈ C, . For any N ∈ N * and any z ∈ C, one has be the sinus cardinal function; it satisfies |sinc(z)| ≤ e |Im(z)| for any z ∈ C. One has then the upper uniform estimates which is the first chain of inequalities in (2.11). For any N ∈ N * , one has also 14) It follows from the identity (2.14) and (2.12), that for any N ∈ N * and z ∈ C, The second inequality in (2.11) is thus proved.
One can now state as a consequence of Proposition 2.1 and Lemma 2.4 the following theorem.
Proof. It follows from estimates (2.12) that the sequence f = {z → F N (z, a)} N ≥1 satisfies the estimates (2.1) with p = 1, B f = |a| + 1 and C f = 1. Lemma 2.4 implies on the other hand that the sequence f converges towards z → e iaz in H(C). The result is then a consequence of Proposition 2.1.

Uniform convergence of superoscillating sequences
Let m ∈ N * and (F (R m , C)) N * be the family of all sequences Y = {x ∈ R m → Y N (x)} N ≥1 of complex valued functions defined on R m . We first recall in this section the notions of (complex) generalized Fourier sequence (CGFS) and (complex) superoscillating sequence (CSOscS) in (F (R m , C)) N * . We start first with the case m = 1.
where C j (N ) ∈ C and k j (N ) ∈ R for any N ∈ N * and j ∈ N.
, is any subsequence of the Fourier (resp. Fourier- then it realizes, after re-indexation, an archetypical example of a complex generalized Fourier sequence in (F (R, C)) N * This fact justifies the terminology.
2. When m = 1 and a ∈ R, the sequence {x → F N (x, a)} N ≥1 is also an example of a complex generalized Fourier sequence in (F (R, C)) N * . In this case, note that C j (N ) = C j (N, a) ∈ R for any j ∈ N.
3. Let P = κ∈Z * γ κ X κ ∈ C[X, X −1 ] be a Laurent polynomial and L(P ) the diameter of its support.
is after re-indexation a complex generalized Fourier sequence in (F (R, C)) N * .
converges uniformly on any compact subset of U sosc to the restriction to U sosc of a trigonometric polynomial function where P ∞ ∈ C[X, X −1 ] is a Laurent polynomial with no constant term and k(∞) ∈ R \ [−1, 1].
Remark 3.1. If Y is a superoscillating sequence in the sense of Definition 3.2, it is Y ∞superoscillating in the sense of Definition 1.1 in [27], with superoscillation set any segment [a, b] such that b − a > 0 is included in the superoscillation domain U sosc .
1. Any subsequence of the Fourier (resp. Fourier- This follows from Lemma 2.4 (namely from the inequalities (2.11) for a ∈ R and x ∈ R). This is the model that inspired us originally and that we will generalize in this paper.
Inspired by physical considerations which we will discuss later on, we extend as follows Definition 3.1 and Definition 3.2 to the higher dimensional setting where m > 1. The model we will use in order to extend Definition 3.1 will be the one in Example 3.1 (3).
is a complex generalized Fourier sequence in the two real variables t, x, the polynomial P ∈ C[T, X] being here P (T, X) = T X.
Definition 3.2 extends to the multivariate case as follows.
converges uniformly on any compact subset of U sosc to the restriction to U sosc of a trigonometric polynomial function In order to illustrate Definition 3.4 with an example which is derived from Example 3.2 (2), consider, for p ∈ N and a ∈ R \ [−1, 1], the complex generalized Fourier sequence in two real variables t, x ). An immediate computation shows that for any (t, x) ∈ R 2 , One can extend analytically ψ p,N (·, ·, a) as a function from R×C to C, such that one has formally One can prove here the following result.
Proof. The first assertion follows from Lemma 2.2 with R t = T as set of parameters and p ≥ 1 as order of the symbol of the differential operator D p (t) as a differential operator in W . Since {z → F N (z, a)} N ≥1 converges to z → e iaz in A 1 (C) (see theorem 2.1), the sequence {z → D p (t) (F N (·, a))(z)} N ≥1 converges towards z → D p (t)(e ia(·) )(z) locally uniformly with respect to t ∈ R. One can check that D p (t)(e ia(·) )(z) = e ia p t e iaz thanks to an immediate computation. Since (1 − 2j/N ) p and (1 − 2j/N ) lie in [−1, 1] for any j ∈ {0, ..., N } and a ∈ R \ [−1, 1], the generalized Fourier sequence (3.3) is superoscillating with P ∞ (T, X) = T X, k 1 (∞) = a p and k 2 (∞) = a, the superoscillation domain being here R 2 . The expressions of the partial derivatives in t in terms of the partial derivatives in x in (3.6) follow from the fact that ψ p,N (·, ·, a) satisfies the partial differential equation in the Cauchy-Kowalevski problem (3.4). The last assertion in the theorem results from the continuity of the differentiation d/dz as an operator from A 1 (C) into itself.
Let now P ∈ R[X] be an even polynomial P (X) = γ 0 + γ 1 X 2 + · · · + γ 2d ′ X 2d ′ and a ∈ R \ [−1, 1]. Consider in this case the generalized Fourier sequence . (3.7) As in the previous case, an easy computation shows that the function ψ P,N (·, ·, a) is the unique global solution (on the whole space R 2 ) of the Cauchy-Kowalevski problem and the partial differential operator is here of Schrödinger type. Let us introduce the differential operator D P (t) defined as with symbol in A 2d ′ (C W ) (the set of parameters T being again T = R t ).
Theorem 3.2. Let P ∈ R[X] be an even polynomial with degree 2d ′ . For any λ ∈ R, the Cauchy-Kowalevski problem (of Schrödinger type) admits as unique global solution in R 2 the function (t, x) → ϕ λ (t, x) = e itP (λ) e iλx . One has ψ P,N (·, ·, λ) = N j=0 C j (N, λ) ϕ 1−2j/N and the sequence {(t, x) → ψ P,N (t, x, λ)} N ≥1 converges uniformly on any compact set in R 2 to (t, x) → e itP (λ) e iλx . For any (µ, ν) ∈ N 2 , the sequence of functions converges uniformly on any compact in R 2 to the function Proof. One has and ϕ λ (0, x) = e iλx for all x ∈ R. It follows from Lemma 2.2 that the operator D P (t) acts continuously on A 1 (C), locally uniformly with respect to the parameter t ∈ R. Since the sequence {z ∈ C → F N (·, λ)} N ≥1 converges to z → e iλz in A 1 (C), the sequence {z ∈ C → D P (t)(F N (·, λ))(z)} N ≥1 converges to z → D P (t)(e iλ(·) )(z) = e itP (λ) e iλz in A 1 (C) locally uniformly with respect to the parameter t ∈ R. The first equality in (3.10) follows from the fact that ψ P,N (·, ·, λ) is solution of the Cauchy-Kowalevski problem (3.8). The final assertion follows from the continuity of d/dz : One can even drop the hypothesis about P and take P = d κ=0 γ κ X κ as polynomial of degree d in C[X] with associate polynomialP = d κ=0 (−i) κ+1 γ κ X κ . The Cauchy-Kowalevski problem (3.8) is not anymore of the Schrödinger type (sinceP / ∈ R[X] in general), which makes the only difference with the case previously studied. Nevertheless, one can state exactly the same result, with this time is superoscillating as a generalized Fourier sequence in two variables (t, x), with R 2 as domain of superoscillation.
Proof. The proof follows that one of Theorem 3.2. The fact that the sequence {x → ψ B,N (t, x, a)} N ≥1 is superoscillating for any t ∈ R follows from the fact that it converges on any compact of R x (locally uniformly in t) to x → e itP (a) e iax . As for the last assertion, to define Y ∞ one takes P ∞ (T, X) = T X, κ(∞) = P (a) and k(∞) = a in Definition 3.4.
] be a power series with radius of convergence ρ ∈]0, +∞], together with the convolution operator Since F and ∞ κ=0 i 1−κ γ κ X κ share the same radius of convergence ρ > 0, F E (t) realizes, for each t ∈ R) an holomorphic function in D(0, ρ) ⊂ C W (with Taylor series about 0 depending on t ∈ R). More precisely, one has where, for R > 0, the radius of convergence of the power series j≥0 κ≥0 |β j,κ |R κ X j is at least equal to ρ.
For any λ ∈] − ρ, ρ[ and z ∈ C, one has formally One requires the following lemma in order to justify the formal relations (3.11).
Lemma 3.1. When ρ = +∞, the convolution operator D E (t) acts continuously locally uniformly with respect to t ∈ R from A 1 (C) into itself. When ρ ∈]0, +∞[ it acts continuously locally uniformly with respect to t ∈ R from the space (where {B ρ,n } n≥1 is a strictly increasing sequence converging to ρ) into itself.
Proof. Suppose first that ρ = +∞. Let R > 0 and K ⊂ [−R, R] ⊂ R t be a compact set. One recalls here that the radius of convergence of the power series j≥0 κ≥0 |b j,κ |R κ X j equals +∞. Let γ > 0, β > 0 and f = ℓ≥0 f ℓ W ℓ ∈ A γ,β 1 (C). One can check as in the proof of Lemma 2.2 (compare to (2.5)) that, for any t ∈ K and j ∈ N, This is indeed enough to conclude as in the proof of Lemma 2.2 that D E (t) acts continuously locally uniformly in t from A 1 (C) into itself. Consider now the case where ρ ∈]0, +∞[. For any R > 0, the radius of convergence of the power series j≥0 κ≥0 |b j,κ | R κ X j is now at least equal to ρ. Repeating the preceeding argument (but taking now β ≤ ρ − ε for some ε > 0 arbitrary small), one concludes that D E (t) acts continuously locally uniformly in t from lim We can now state the last result of this section. is superoscillating. Moreover, for any such a and (µ, ν) ∈ N 2 , the sequence of functions converges then uniformly on any compact in R 2 t,x to the fonction Proof. The fact thatĚ acts continuously from lim ←− A Bρ,n 1 (C W ) into itself follows from Lemma 3.1, considering justĚ (independent of the parameter t) instead of D E (t). For any λ ∈ R with |λ| < ρ, the operatorĚ then acts on e iλ(·) and it is immediate to check that for any t ∈ R Therefore, for any a ∈ R and N ∈ N * , one has by linearity (since ρ > 1) Lemma (3.1) (applied this time with D E (t)), combined with Theorem 2.1 and the estimates in the first line of (2.11) in Lemma 2.4, imply that as soon as one has |a| < ρ − 1 the sequence {z ∈ C → D E (t)(F (·, a))(z)} N ≥1 converges (locally uniformly with respect to the parameter t) to z → e iaz in A 1 (C). The last assertion in the particular case µ = ν = 0 follows. The first equality in (3.12) comes from the identity (3.14), while the second one comes from (3.11) (as justified by Lemma 3.1). The last assertion of the theorem when µ, ν are arbitrary is then a consequence of the continuity of d/dz from ←− is also superoscillating, this time according to Definition 3.4 (with P ∞ (T, X) = T X, κ(∞) = E(a) and k(∞) = a).

Regularization of formal Fresnel-type integrals
In order to settle from the mathematical point of view the approach to non-absolutely convergent integrals on the half-line R + * or the whole real line R through the so-called principle of regularization that we will invoke in the remaining sections 5 and 6 (with respect to supershift considerations related to Schrödinger equations with specific potentials), we need to explain what regularization of formal Fresnel-type integrals on R + * or R means.
Suppose that T is a set of parameters. Let G : (t, Z) ∈ t × C −→ G(t, Z) be a function which is entire as a function of Z for each t ∈ T fixed. Let also φ be a non-vanishing real function on T that will play the role of a phase function. Let finally χ be a real number such that χ > −1.
Proof. The absolute convergence follows from the estimates (4.4), together with the fact that if u = e iθ , Re((tu) 2 ) = t 2 cos(2θ) > 0 for t > 0. The fact that the integrals do not depend of u follows from residue theorem (applied on the oriented boundary of conic sectors with apex at the origin).
In view of this lemma, the regularization of an integral of the Fresnel-type such as (4.1) consists in the successive two operations : 1. first transform the formal expression (4.1) into one of the representations (4.2) or (4.3) according to sign(φ(t)) ; 2. then invoke Lemma 4.1 (provided the required hypothesis are satisfied) and consider then the regularization of (4.1) as and proceed as above for the two formal expressions involved into this formal decomposition.
It is immediate to compare this approach to regularization to the alternative following one.
exists and coincides with the integral regularized under the approach described above.
Proof. It is enough to prove the result when ̟ = ±1 since one reduces to one of these two cases up to a homothety on the real half line. One has where F + (Z) = e −i(1+χ)π/4 F (e −iπ/4 Z) and F − (Z) = e i(1+χ)π/4 F (e iπ/4 Z). Let ρ ε = √ 1 + ε 2 , and ξ ε = arg [0,π/2[ √ 1 + iε. One has then (4.7) In the two integrals on the right-hand side of the equalities (4.7), the integration contour can be replaced by the half-line R + * as a consequence of Lemma 4.1. It is then possible to take the limit when ε tends to 0. Lebesgue's domination theorem then applies and since ρ ε tends to 1 and ξ ε to 0, one gets This concludes the proof of the Proposition.

Continuity on A 1 (C) of Fresnel-type integral operators
Let T be a set of parameters and t ∈ T → D(t, Z) (as in the statement of Lemma 2.3) be a differential operator-valued map for somep ∈]1, 2], p ≥ 1 and B ≥ 0. Let also φ be a non-vanishing real function on T and χ > −1. It follows from the estimates (5.1), together with Lemma 4.1, that the regularization approach described in section 4 allows to define the operator One needs to consider for the moment these operators as acting on entire functions of the complex variable Z. For α ∈ C, let also H α be the dilation operator H α : f → f (α(·)) acting on such functions. The symbol ⊙ still stands for the composition of operators. The discussion is with respect to the sign of φ(t).
Theorem 5.1. Suppose that the parameter space T is a topological space and that φ is continuous. Consider functions B j : T × C × C → C (j ∈ N) which are entire in the two complex entries and such that for some p ≥ 1,p ∈]1, 2], and B ≥ 0. Then the operator (understood through the process of regularization as described above) acts continuously locally uniformly in t from A 1 (C) into Ap(C).
Proof. It is enough to consider T as a neighborhood of a point t 0 in which φ(t) ≥ ε 0 > 0 (since φ is continuous). Let f = {Z → f N (Z)} N ≥1 be a sequence of elements in A 1 (C) that converges towards 0 in A 1 (C), which means (see Proposition 2.1) that all f N belong to some A C,b 1 (C) for some constants C, b > 0 independent on N (namely f N = ℓ a N,ℓ Z ℓ with |a N,ℓ | ≤ C b ℓ /ℓ! for any ℓ ∈ N). It is clear that the operator is such that for each ε > 0, there exists A (ε) ≥ 0 (depending on T, A (ε) , the B j , b and C, but not on the N ) such that Take in particular ε < ε 0 . Then the functioň y χ e −ε 0 y 2 e εyp dy e B (ε) |Ž|p ∀Ž ∈ C (remember thatp ∈]1, 2]). It remains to show that the sequence converges to 0 in Ap ,0 (C). It is enough (see Proposition 2.1) to prove that it converges to 0 uniformly on any closed disk D(0, r) in C. Fix ε < ε 0 and η > 0. Choose then R η >> 1 such that y χ e −ε 0 y 2 e εyp dy e B (ε) |Ž|p ≤ η e −B (ε) rp e B (ε) |Ž|p ≤ η.
Note that our estimates show that the convergence towards 0 in Ap ,0 (C) thus obtained is uniform in t ∈ T.

Superoscillations and supershifts
Consider the Schrödinger equation where H denotes the Hamiltonian operator attached to the physical system which is under is a Cauchy-Kowalevski problem (assuming that x lies in some open set U ⊂ R where the Hamiltonian operator is regular), any entry x ∈ U → Y N (x) evolves in a unique way from t = 0 towards t > 0 as (t, x) → ψ N (t, x). Assume in addition that x lies in the maximal superoscillation domain U suposc max ; the limit function x ∈ U ∩ U sosc max → Y ∞ (x) then also evolves from U ∩ U sosc max into some function (t, x) → ψ ∞ (t, x). A natural question then occurs. As long as the evolution persists (let say for t ∈ [0, T ]), is it true that the sequence {x ∈ U → ψ N (t, x)} N ≥1 is such that its restriction to U ∩ U sosc max converges (uniformly on any compact subset of U ∩ U sosc max ) to x → ψ ∞ (t, x)? If this is the case, one will say that the superoscillating character of the sequence Y persists in time through the Schrödinger evolution operator ∂/∂t − H which is here considered.
In order to formulate such question in a different way, let us now consider the (t, x) domain [0, T ]×(U ∩U sosc max ) = T×(U ∩U sosc max ) = T as a parameter set and focus on the map λ ∈ R −→ ϕ λ , where ϕ λ : T → R is evolved to [0, T ] × U (through the Schrödinger operator) from the initial datum x ∈ U → e iλx , then restricted to the parameter set T . Previous considerations lead to the following definition, which is inspired by Definition 3.2.

The Schrödinger Cauchy problem with centrifugal potential
We will consider in this subsection the case where U = {x ∈ R ; x > 0} and the hamiltonian in (5.6) is x ∈ U → H (x) = −(∂ 2 /∂x 2 )/2 + u/(2x 2 ), where u denotes a real strictly positive physical constant. The corresponding Cauchy-Kowalevski problem (with [0, +∞[×U as phase space) is the Schrödinger Cauchy problem with centrifugal potential, see [26] for more references. For this Cauchy-Kowalevski problem, the analysis of the evolution t → ψ(t, ·) of the solution (t, x) ∈ [0, ∞[×U → ψ(t, x) from an initial datum x ∈ U → ψ(0, x) can be carried through thanks to the explicit form of the Green function (t, x, x ′ ) → G(t, x, t ′ = 0, x ′ ).
with F -supershift domain equal to T . Moreover, for any (µ, ν) ∈ N 2 , the sequence of functions converges uniformly on any compact K ⊂⊂ T to the function Proof. Let λ ∈ R. The evolution of the initial datum x ∈ U → e iλx through the Schrödinger equation (5.6) is explicited (for the moment formally) thanks to the expression (5.8) of the Green function as For any M ∈ N such that 2M > ν − 1/2 and any y > 0, one has [43, pp. 207-209]). It follows from such developments, together with Proposition 4.1, that the integral in (5.9) exists for any (t, x) ∈ T as a semi-convergent integral (of the Fresnel-type), whose value coincides with the regularized integral described in section 4. Set now , χ := ν + 1 2 , in order to fit with the setting described in Theorem 5.1. Since E ν ∈ A 1 (C) and the operator with order 0 given as t → B 0 (t, Z,Ž) (d/dZ) 0 satisfies the hypothesis of this theorem with p = 1 andp = 2. Then the operator acts continuously locally uniformly in t ∈]0, +∞[ from A 1 (C) into A 2 (C). For any λ ∈ R and t > 0, the function x ∈]0, +∞[ → ϕ λ (t, x) is C ∞ because of its expression (5.9). Moreover, when a ∈ R \ [−1, 1], it follows from Theorem 2.1 that the sequence converges in A 2 (C) (locally uniformly with respect to t > 0) to z → D(t)(e ia(·) ). One concludes then to the second assertion in the statement of the theorem. As for the last assertion, it follows from the fact that the action of i∂/∂t and H (x) coincide on solutions of (5.6), together with the continuity of the operator d/dz from A 2 (C) into itself.
Proof. Consider the two (for the moment formal) operators (see Remark 4.1). Set now ) in order to fit with the setting described in Theorem 5.1. As in the proof of Proposition 5.1, this theorem applies here and the two operators (5.11) act continuously from A 1 (C) to A 2 (C) (locally uniformly with respect to the parameter t ∈ T). Note again that the Fresneltype integrals (5.11), where Z → e iλZ (λ ∈ R) is taken inside the bracket andŽ ∈ R, are semi-convergent and their values as semi-convergent integrals coincide with the values that are obtained by regularization as in section 4. In fact, in the case whereŽ = x ∈ R and t ∈ T, the value of 1 2iπ sin t e i cotan t 2Ž . Since (t, x) ∈]0, +∞[×R −→ (cos t) −1/2 e −ix 2 tan(t) e −iλ 2 tan(t) is a locally integrable function, the initial datum x ∈ R → e iλx evolves through the Schrödinger equation (5.6) as a distribution ϕ λ (in fact defined by a locally integrable function). Let D(t) the differential operator Since (cos t) −1/2 e −iŽ 2 tan(t) 2 e −iλ 2 tan(t) 2 ⊙ H 1/ cos t (e iλ(·) )(Ž) = (cos t) −1/2 e −iŽ 2 tan(t) 2 D(t)(e iλ(·) )(Ž), and D acts continuously locally uniformly in t from A 1 (C) to A 2 (C) thanks to Lemma 2.2, the sequence is, for any a ∈ R \ [−1, 1], a supershift for the family F = {(ϕ λ ) |U ; λ ∈ R} (with F -supershift domain U ). The last assertion follows from the same argument than that used for the last assertion in Proposition 5.1.

Singularities in the quantum harmonic oscillator evolution
This section is the natural continuation of subsection 5.4. We continue to investigate with respect to the notion of supershift the evolution of initial data x ∈ R → e iλx , when λ ∈ R, through the Cauchy-Schrödinger problem for the quantum harmonic oscillator and focus now on singularities. In this section we keep the same notations as in Proposition 5.2 and fix a point (t 0 , x 0 ) in T \ U . We will just consider the case t 0 = π/2 since the situation is essentially identical at any point ((2k + 1)π/2, x 0 ) with k ∈ N and x 0 ∈ R.
Proof. The coefficients of A κ as a polynomial in d/dZ satisfy κ,j∈N,Ž∈C |a κ,j (Ž)| ≤ C 0 b j 0 Γ(j/p) + 1 eB |Ž|p for some absolute constants C 0 and b 0 (Lemma 2.1). As in the proof of Lemma 2.2, one concludes that for any f ∈ A C,b 1 (C) and any κ ∈ N, one has uniform estimates |A κ (Ž, d/dZ)(f )| ≤ C A (b) exp(b 0 b|Z| +B|Ž|p) for some positive constant A (b) . One gets the required estimates when evaluating at Z = 0.
Proof. Let θ ∈ D(R 2 t,x , C) with support a small neighborhood V of the point (π/2, x 0 ) (x 0 ∈ R) and θ the test-function with support V − (π/2, 0) ∋ (0, x 0 ) that corresponds to it through the successive transformations explicited previously. One has for any λ ∈ R, with p =p = 2. These two operators act then continuously (locally uniformly with respect to the parameterŽ) from A 1 (C) into the space of infinite order differential operators in d/dv (depending on the parameterŽ ∈ C). Such differential operators can be considered as hyperfunctions on R v (elements of H(R v )). Since v ) and depend continuously (locally uniformly with respect toŽ) on the entry f in A 1 (C). Proposition 6.1 follows then from Theorem 2.1 and from the expression (6.4) (together with its formal reformulation (6.5)) for the evaluations ϕ λ , θ when λ ∈ R and ϕ λ is considered as an element in D ′ (T , C) (acting on θ ∈ D(T , C)) which can be also interpreted an a hyperfunction on T .