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In [5] was proved that E2(Q) A2(Q), ⊂R⊂ where E2(Q) is the Hardy-Smirnov space on Q and A2(Q) is the (unweighted) Bergman space on Q. Thus, in this work well-known spaces of analytic functions with some structure appeared for the first time.
In [8] the author has proved that the null reachable space is the sum of two Bergman spaces on sectors (whose intersection is Q), i.e. R = A2(∆) + A2(1 ∆), R − where ∆ := z C : arg z < π/4 . { ∈ | | } As the author has remarked, given any function f hol(Q), how can we write f as a sum of two functions in those different Bergman∈ spaces? Notice that A2(∆) and A2(1 ∆) are RKHSs on different domains. − In this work our main result (Theorem 4) shows that the null reachable space is the sum of two RKHSs on the same domain Q, therefore is a RKHS on Q (byR properties of the RKHSs). Corollary 2.1 in [9, page 98] givesR a necessary and sufficient condition so that a function f hol(Q) can be in . ∈ R
2. Statement of the results To get the characterization of as a RKHS on Q, we proceed in several steps. R We characterize the subspaces in corresponding to the cases either ur = 0 or u =0 or u = u or u = u as RKHSsR of analytic functions on a square. ℓ r − ℓ r ℓ In [2] the authors ask for a characterizacion of the null reachable space, at time T > 0, with just one Dirichlet boundary control. So we consider the null reachable space, at time T > 0,
ℓ 2 := w( ,T ): w is solution of system (1) with u LC(0,T ),u 0 RT { · ℓ ∈ r ≡ } with one Dirichlet boundary control on the left. As before, we can see that the space ℓ does not depend on T > 0, so ℓ will denote this space. RT R Motivated by the idea in [5] of writing the solution of system (1) in terms of integral operators having well known heat kernels (see 6), and by using the charac- terization of the image of a linear mapping as a RKHS (see [9, page 134]) we have obtained the characterization of ℓ as a RKHS on a square. R First, we introduce some notation and definitions. Consider the square D := R2 ℓ (x, y) : y < x, y < 2 x , the open set D∞ := n∈Z(2n + D) and the following∈ positive| | definite| | function− on the sector ∆, S zw z2+w2 1 1 − 4T 0(z, w; T ) := e 2 + 2 , z,w ∆. K π (z2 + w )2 4T z2 + w ! ∈ THE REACHABLE SPACE OF THE HEAT EQUATION 3
Theorem 1. For each T > 0 fixed, we have that ℓ = f hol(Dℓ ): f(z +2)= f(z)= f( z),f ℓ (D) R { ∈ ∞ − − |D ∈ HT } where ℓ (D) is the RKHS of analytic functions on D with reproducing kernel HT (2) ℓ(z, w; T ) := 0(z +2n, w +2m; T ), z,w D. K Z K ∈ m,nX∈ The space ℓ (D) is endowed with the norm given in (14). HT Notice that the properties of 2-periodicity, oddness and analyticity domain of f ℓ are inherited from those of the analytic extension of the heat kernel ∈ R (∂xθ)(x, t), see Remarks 10, 11.
We also have the corresponding result for the null reachable space with one Dirichlet boundary control on the right, defined as follows r 2 := w( ,T ): w is solution of system (1) with u 0,u LC(0,T ) . RT { · ℓ ≡ r ∈ } Theorem 2. Let Dr = 1+ Dℓ . For each T > 0 fixed, we have that ∞ − ∞ r = f hol(Dr ): f(z +2)= f(z)= f( z),f r ( 1+ D) R { ∈ ∞ − − |−1+D ∈ HT − } r where T ( 1+ D) is the RKHS of analytic functions on 1+ D with reproducing kernel H − −
(3) r(z, w; T )= 0(z +2n +1, w +2m + 1; T ), z,w 1+ D. K Z K ∈− m,nX∈ The space r ( 1+ D) is endowed with the norm given in (15). HT − By (6) and Theorem 1 we also have r = f hol(Dr ): f(z +2)= f(z),f( z)= f(z),f( 1) T (D) . R { ∈ ∞ − − ·− |D ∈ Hℓ }
Our approach also provides the characterization of the following subspaces in , R + 2 := w( ,T ): w is solution of system (1) with u LC(0,T ),u = u , RT { · ℓ ∈ r − ℓ} − 2 and := w( ,T ): w is solution of system (1) with u LC(0,T ),u = u . RT { · ℓ ∈ r ℓ}
Once again, (2.14), (2.15) and Theorem 3.3 in [4] imply the null controllability of 2 2 system (1) with initial datum w(x, 0) L (0, 1) and the controls uℓ,ur LC(0,T ) ∈ + − ∈ satisfying either ur = uℓ or ur = uℓ. Thus, the spaces T , T do not depend on T > 0. − R R
Theorem 3. Let Q∞ := n∈Z(n + Q). For each T > 0 fixed we have that (1) + = f hol(Q ): f(z +1) = f(z) = f( z),f +(Q) where R { ∈ S∞ − − |Q ∈ HT } +(Q) is the RKHS on Q with reproducing kernel HT +(z, w; T ) := 0(z + n, w + m; T ) K Z K m,nX∈ = (z, w; T )+ (z +1, w + 1; T )+ (z +1, w; T )+ (z, w + 1; T ), Kℓ Kℓ Kℓ Kℓ for z, w Q. The space +(Q) is endowed with the norm given in (16). ∈ HT 4 MARCOS LOPEZ-GARC´ ´IA
(2) − = f hol(Q ): f(z +1) = f(z) = f( z),f −(Q) where R { ∈ ∞ − − − |Q ∈ HT } −(Q) is the RKHS on Q with reproducing kernel HT (z, w; T ) := (z, w; T )+ (z +1, w + 1; T ) (z +1, w; T ) (z, w + 1; T ), K− Kℓ Kℓ − Kℓ − Kℓ for z, w Q. The space −(Q) is endowed with the norm given in (17). ∈ HT As a consequence, we get a description of the null reachable space . R Theorem 4. We have that = + + − and + − = 0 . R R R R ∩ R { } Moreover, := f : f = +(Q)+ −(Q) with reproducing kernel R|Q { |Q ∈ R} HT HT (z, w; T )+ (z +1, w + 1; T ), z,w Q. Kℓ Kℓ ∈ The space is endowed with the norm given in (18). R|Q Clearly, the condition + − = 0 follows from the functional equations in the last result. R ∩ R { }
Next we consider the case with Neumann boundary data. ∂ v ∂ v =0, 0
In some situations, the null reachable space at time T > 0 of a certain heat equation can be described in terms of well known analytic functions spaces. For instance, consider the heat equation for the half line, ∂ v ∂ v =0, 0
This paper is organized as follows. In the next section we introduce notation, give some results about RKHSs, and make the computations needed to prove the results. In Section 4 we provide the proofs.
3. Preliminaries In this section we use some results about the one dimensional heat equation that can be found in [1]. First, consider the heat kernel on the upper half plane R2 := (x, t) R2 : t> 0 given as follows + { ∈ } 2 1 − x 2 K(x, t) := e 4t , (x, t) R . √4πt ∈ + In order to describe the solution w(x, t) of system (1) we introduce the so-called theta function R2 θ(x, t) := K(x +2n,t), (x, t) +, Z ∈ nX∈ so we have the system (1) admits a unique solution w C([0, ), W −1,2(0, 1)) given by (see [1, Theorem 6.3.1], [5]) ∈ ∞ t t (6) w(x, t)= 2 (∂ θ)(x, t τ)u (τ)dτ +2 (∂ θ)(x 1,t τ)u (τ)dτ. − x − ℓ x − − r Z0 Z0 For an open set Ω C, the (unweighted) Bergman space on Ω is the vector space ⊂ A2(Ω) := f :Ω C f analytic on Ω and f L2(Ω) { → | ∈ } endowed with the inner product 1 f,g 2 := f(z)g(z)dxdy. h iA (Ω) π ZΩ We also consider the Hardy space on the half space C+, ∞ H2(C+) := f : C+ C f analytic on C+ and sup f(x + iy) 2dy < → | | | ∞ x>0 Z−∞ endowed with the inner product ∞ f,g 2 C+ := sup f(x + iy)g(x + iy)dy. h iH ( ) x>0 Z−∞ Consider the following positive definite functions on ∆ 4zw 1 1(z, w) := , 2(z, w) := , z,w ∆, K (z2 + w2)2 K z2 + w2 ∈ and the biholomorphism ϕ(z)= z2 from ∆ onto C+.
′ ′ Remark 7. Notice that 1(z, w) = B(ϕ(z), ϕ(w))ϕ (z)ϕ (w) where B (z, w) is K K 2 + K the reproducing kernel for the Bergman space A (C ), so that 1(z, w) is the re- producing kernel for the Bergman space A2(∆) (see [3, page 12]).K
The following result shows that 2(z, w) is the reproducing kernel for the pull- back space, induced by the functionKϕ, of the Hardy space H2(C+).
Lemma 8. 2(z, w) is the reproducing kernel for the RKHS ϕ(∆) := f ϕ : f H2(C+) K. H { ◦ ∈ } 6 MARCOS LOPEZ-GARC´ ´IA
2 + 2 + Proof. Here H stand for the reproducing kernel for H (C ). Let evz : H (C ) C be the functionalK evaluation at z C+. If g ker(ev ) then g →0, ∈ ∈ p∈∆ ϕ(p) ≡ so Theorem 2.9 in [9, page 81] implies that the RKHS with reproducing kernel T (ϕ(z), ϕ(w)) = (z, w) is the space (∆) equipped with the inner product KH K2 Hϕ (7) f ϕ, g ϕ = f,g 2 C+ . h ◦ ◦ iHϕ(∆) h iH ( )
For each t> 0 fixed, consider the entire function 2 z − z (∂ K)(z,t) := e 4t , z C, x −4√πt3/2 ∈ which is the analytic extension of the function (∂ K)(x, t), x R. x ∈ Lemma 9. i) For each t> 0 fixed, the function
(8) (∂xθ)(z,t) := (∂xK)(z +2n,t) Z nX∈ is holomorphic on D and continuous on D. ii) For each compact set D and t > 0, there exists a constant CF,t > 0 such that F ⊂ t 1/2 2 (9) (∂xK)(z +2n,t τ) dτ CF,t for all z . Z 0 | − | ≤ ∈F nX∈ Z Proof. For n 2 we have that n2 2nx if 0 x 1, and 3n2 4 4x 4nx if 1 x 2; therefore≤− ≥− ≤ ≤ ≥ − − ≤ ≤ 2 − n 4ne t , n 1, z D, ℜ(z2) ≥ ∈ 2 − (z+2n) 2e 4t , n =0, z D, − 4t (10) z +2n e ℜ − 2 ∈ | | ≤ − ((z 2) ) 2e 4t , n = 1, z D, 2 − ∈ − n 4 n e 4t , n 2, z D. | | ≤− ∈ −s −σ Since e Cσs for all s,σ > 0, together the Weierstrass M-test imply the series in (8) converges≤ absolutely and uniformly on D, and the result i) follows.
For n Z 1, 0 , z D we have ∈ \{− } ∈ t ∞ 2 4 4 1 n2 −ρ − 2t (∂xK)(z +2n,t τ) dτ 2 ρe dρ 1+ e . | − | ≤ πn 2 ≤ π t Z0 Zn /(2t) Let z be such that (z2) = min (z2), therefore 0 ∈F ℜ 0 z∈F ℜ t ∞ 2 ℜ 2 2 1 1 (z ) (z0 ) −ρ 0 − 2t (∂xK)(z,t τ) dτ 2 2 ρe dρ = 2 2 1+ ℜ e 0 | − | ≤ π( (z )) ℜ(z2)/(2t) π( (z )) 2t Z ℜ 0 Z 0 ℜ 0 for all z . ∈F 2 2 In a similar way, let z1 be such that ((z1 2) ) = minz∈F ((z 2) ), therefore ∈ F ℜ − ℜ −
t 2 ℜ − 2 2 1 ((z1 2) ) − ((z1 2) ) (∂ K)(z 2,t τ) dτ 1+ ℜ − e 2t | x − − | ≤ π( ((z 2)2))2 2t Z0 ℜ 1 − for all z . ∈F THE REACHABLE SPACE OF THE HEAT EQUATION 7
Remark 10. In fact, by making an easy modification in the last proof we get that (∂xθ)( ,t) hol(D∞). Clearly, for each t > 0 fixed the function (∂xθ)( ,t) fulfills the following· ∈ functional equations · (11) f(z +2)= f(z)= f( z), z D . − − ∈ ∞ 2 Remark 11. Let t > 0 fixed and u LC(0,t). Lemma 9 together Morera and Fubini’s theorems imply that the continuous∈ function (by (10) and the dominated convergence theorem) t z u(τ)(∂ θ)(z,t τ)dτ 7→ x − Z0 is holomorphic on D∞ and satisfies the functional equations in (11). We write D for the closure of the set D. Proposition 12. Let t> 0 fixed. The series introduced in (2) converges absolutely and uniformly on D D (or D D). Thus ℓ( , w; t) is an analytic function on D, and (z, ; t) is an× anti-analytic× functionK on D· . Kℓ · Proof. For n , m 3 and z, w D we have that | | | |≥ ∈ (z +2n)2 + (w +2m)2 4(n2 + m2 2 n 2 m 2) 16. ≥ − | |− | |− ≥ By using (10) and the last inequality we have the series defining ( , ; t) converges Kℓ · · absolutely and uniformly on D D whenever we sum over all the indexes satisfying n , m 3. × | | | |≥ Now suppose that there exist z D, w D such that (z +2n)2 + (w +2m)2 = 0. Then z +2n = i(w +2m), thus ∈2n + ∈z = w 1, so n = 0 or n = 1. By simmetry, we also± have m =0 or m| = ℜ1.| If n|ℑ= m| ≤= 0 then z = iw, which− is a contradiction because D ( iD)= .− In any case, we get a contradiction± because ∩ ± ∅ (D 2) ( i(D 2)) = , (D 2) ( iD) = and D ( i(D 2)) = . This completes− ∩ the± proof.− ∅ − ∩ ± ∅ ∩ ± − ∅
Let (E) be the vector space consisting of all complex-valued functions on a set F E, and let ( , , H) be a Hilbert space. For a mapping h : E , consider the induced linearH mappingh· ·i L : (E) defined by → H H→F Lf(p)= f, h(p) . h iH The vector space (L) := Lf : f is endowed with the norm R { ∈ H} f L = inf f : f ,f = L(f) . k kR( ) {k kH ∈ H } A fundamental problem about the linear mapping L is to characterize the vector space (L). The following result summarizes Theorems 2.36, 2.37 in [9, pages 135–137]R and provides an answer to the last question.
Theorem 13. (1) ( (L), L ) is a RKHS with reproducing kernel R k·kR( ) K(p, q)= h(q), h(p) , p,q E. h iH ∈ (2) The mapping L : (L) is a surjective bounded operator with operator norm less than 1.H → R 8 MARCOS LOPEZ-GARC´ ´IA
4. Proofs of the results
Proof of Theorem 1. By (6) and Remark 11 we have that w( ,T ) hol(D∞) and fulfills the functional equations in (11). · ∈
2 Lemma 9 ii) implies that the function h : D LC(0,T ) given by →
(12) h (t)= (∂ θ)(z,T t), t (0,T ), z x − ∈ ℓ 2 makes sense, and Remark 11 implies that the linear operator T : LC(0,T ) hol(D) given by L → ℓ ( u)(z)= u,h 2 , z D, LT h ziLC(0,T ) ∈ is well defined. By (6) we have ℓ 2 w(z,T )= 2( u )(z), z D, u LC(0,T ). − LT ℓ ∈ ℓ ∈ ℓ ℓ Theorem 13 implies that T (D) := ( T ) is a RKHS on D with reproducing kernel H R L ∗ (z, w; T )= h ,h 2 . K h w ziLC(0,T ) The inequality in (9), the dominated convergence theorem and Proposition 12 allow us to compute T (z +2n)(w +2m) 1 (z+2n)2 (w+2m)2 ∗ − 4(T −t) − 4(T −t) (z, w; T ) = lim 3 e dt K N,M→∞ 16π 0 (T t) |nX|≤N |mX|≤M Z − (13) = lim 0(z +2n, w +2m; T )= ℓ(z, w; T ). N,M→∞ K K |nX|≤N |mX|≤M We also have ℓ 2 (14) w( ,T ) Hℓ (D) = inf u L2(0,T ) : w( ,T )= 2 T u,u LC(0,T ) . k · k T k k C · − L ∈ n o
2 Proof of Theorem 2. We only give a sketch. Let h : 1+ D LC(0,T ) given by − → h (t)= (∂ θ)(z +1,T t), t (0,T ), z x − e ∈ r 2 and the linear operator : LC(0,T ) hol( 1+ D) given by e LT → − r ( u)(z)= u, h 2 , z 1+ D. LT h ziLC(0,T ) ∈− By (6) and Remark 10 we have e r 2 w(z,T )=2( u )(z), z 1+ D, u LC(0,T ). LT r ∈− r ∈ r r Theorem 13 implies that T ( 1 + D) := ( T ) is a RKHS on 1 + D with reproducing kernel H − R L −
h , h 2 = (z, w; T ). h w ziLC(0,T ) Kr We also have e e r 2 (15) w( ,T ) Hr (−1+D) = inf u 2 : w( ,T )=2 u,u LC(0,T ) . k · k T k kLC(0,T ) · LT ∈ n o THE REACHABLE SPACE OF THE HEAT EQUATION 9
ℓ Remark 14. 1) Since ℓ( , w; T ) T (D) for all w D (see [9, Proposition 2.1, K · ∈ H ∈ ℓ page 71]), we get that the function ℓ( ,y;1) : (0, 2) R is in for all y (0, 2). r K · → R ∈ 2) Since r( , w; T ) T (D) for all w D, we get that the function r( ,y; 1) : (0, 2) RKis· in r for∈ H all y (0, 2). ∈ K · → R ∈ Proof of Theorem 3. (1) We set u = u in (6) to get r − ℓ T w(x, T ) = 2 [(∂ θ)(x, T τ) + (∂ θ)(x 1,T τ)]u (τ)dτ − x − x − − ℓ Z0 T = 2 (∂ θ)(x, T τ)u (τ)dτ − x − ℓ Z0 where e R2 θ(x, t)= K(x + n,t), (x, t) +. Z ∈ nX∈ e For t> 0 fixed, clearly the function (∂xθ)(z,t) has similar properties to the analytic theta function (∂xθ)(z,t) in Lemma 9, and also satisfies the following functional equations, e f(z +1)= f(z)= f( z), z Q . − − ∈ ∞ Therefore, w( ,T ) hol(Q ) and fulfills the last functional equations. · ∈ ∞ Now we proceed as in the proof of Theorem 1: consider the function h+ : Q 2 → LC(0,T ) given by
h+(t) = (∂ θ˜)(z,T t) z x − = (∂ θ)(z,T τ)+ (∂ θ)(z +1,T τ), t (0,T ), x − x − ∈ + 2 and the linear operator : LC(0,T ) hol(Q) given by LT → + + ( u)(z)= u,h 2 , z Q. LT h z iLC(0,T ) ∈ By (6) we have
+ 2 w(z,T )= 2( u )(z), z Q, u LC(0,T ). − LT ℓ ∈ ℓ ∈ + + Theorem 13 implies that T (D) := ( T ) is a RKHS on Q with reproducing kernel (the computation is similarH to (13))R L
+ + + h ,h 2 = (z, w; T ). h w z iLC(0,T ) K By the other hand
+ + h ,h 2 = h ,h 2 + h ,h 2 + h ,h 2 + h ,h 2 h w z iLC(0,T ) h w ziLC(0,T ) h w+1 z+1iLC(0,T ) h w+1 ziLC(0,T ) h w z+1iLC(0,T ) = (z, w; T )+ (z +1, w + 1; T )+ (z, w + 1; T )+ (z +1, w; T ), Kℓ Kℓ Kℓ Kℓ for z, w Q, where h is the function in (12). ∈ We also have
+ 2 + 2 (16) w( ,T ) H (D) = inf u LC(0,T ) : w( ,T )= 2 T u,u LC(0,T ) . k · k T k k · − L ∈ n o 10 MARCOS LOPEZ-GARC´ ´IA
(2) We set ur = uℓ in (6) to get T w(x, T ) = 2 [(∂ θ)(x, T τ) (∂ θ)(x 1,T τ)]u (τ)dτ − x − − x − − ℓ Z0 T = 2 [(∂ θ)(x, T τ) (∂ θ)(x +1,T τ)]u (τ)dτ − x − − x − ℓ Z0 By Lemma 9 and Remark 10 we have (∂ θ)( ,t) (∂ θ)( +1,t) hol(Q ), for all t> 0, x · − x · ∈ ∞ and satisfies the functional equations f(z +1)= f(z)= f( z), z Q . − − − ∈ ∞ Therefore, w( ,T ) hol(Q ) and fulfills the last functional equations. · ∈ ∞ − 2 Consider the function h : Q LC(0,T ) given by → h−(t)= (∂ θ)(z,T t) (∂ θ)(z +1,T t), t (0,T ), z x − − x − ∈ − 2 and the linear operator : LC(0,T ) H(Q) given by LT → − − ( u)(z)= u,h 2 , z Q. LT h z iLC(0,T ) ∈ − − Theorem 13 implies that T (D) := ( T ) is a RKHS on Q with reproducing kernel H R L − − h ,h 2 = h ,h 2 + h ,h 2 h ,h 2 h ,h 2 h w z iLC(0,T ) h w ziLC(0,T ) h w+1 z+1iLC(0,T ) − h w+1 ziLC(0,T ) − h w z+1iLC(0,T ) = (z, w; T )+ (z +1, w + 1; T ) (z, w + 1; T ) (z +1, w; T ), Kℓ Kℓ − Kℓ − Kℓ for z, w Q, where h is the function in (12). ∈ We also have − 2 − 2 (17) w( ,T ) H (D) = inf u LC(0,T ) : w( ,T )= 2 T u,u LC(0,T ) . k · k T k k · − L ∈ n o
+ Remark 15. 1) Since +( , w; T ) T (Q) for all w Q, we get that the function K · + ∈ H ∈ +( ,y;1) : (0, 1) R is in for all y (0, 1). K · → − R ∈ 2) Since −( , w; T ) T (D) for all w Q, we get that the function −( ,y; 1) : (0, 1) RK is· in − for∈ H all y (0, 1). ∈ K · → R ∈ 2 Proof of Theorem 4. Let u ,u LC(0,T ). By (6) we have ℓ r ∈ w(z,T )= +[u u ](z)+ −[u + u ](z), z Q. − LT ℓ − r LT ℓ r ∈ Therefore w( ,T ) +(D)+ −(D). Since +(D) and −(D) are RKHS on Q · ∈ HT HT HT HT with reproducing kernels +(z, w; T ) and −(z, w; T ) respectively, it follows that KT KT +(D)+ −(D) is a RKHS with reproducing kernel HT HT +(z, w; T )+ −(z, w; T )=2 (z, w; T )+2 (z +1, w + 1; T ), KT KT Kℓ Kℓ z, w Q, and is equipped with the norm (see [9, page 93]) ∈ 2 2 2 + − − (18) f = min f1 H+(D)+ f2 H (D) : f = f1+f2,f1 T (D),f2 T (D) . k k {k k T k k T ∈ H ∈ H } THE REACHABLE SPACE OF THE HEAT EQUATION 11
2 Proof of Corollary 5. If v C([0,T ]; LC(0, 1) is solution of system (4), with uℓ,ur 2 ∈ d ∈ LC(0,T ), then (∂xv)(x, t) is solution of system (1), therefore (∂xv)(z,t)= dz (v(z,t)) , 0
2 2 − z − w 4zw e 4 e 4 , and (z2 + w2)2
z2 z2 − 4 − 4 e f,e g −z2/4 2 = f,g 2 . h ie A (∆) h iA (∆) −z2/4 By Lemma 8 and (7) we have that ze ϕ(∆) is a RKHS with reproducing kernel H 2 2 − z − w zw e 4 e 4 , and z2 + w2 for all f,g H2(C+) we have ∈ z2 z2 − 4 − 4 ze f ϕ,ze g ϕ −z2/4 = f ϕ, g ϕ H (∆) = f,g H2(C+). h ◦ ◦ ize Hϕ(∆) h ◦ ◦ i ϕ h i 12 MARCOS LOPEZ-GARC´ ´IA
−z2/4 2 −z2/4 q Therefore e A (∆)+ze ϕ(∆) is the RHKS with reproducing kernel 4π (z, w; 1), and with norm H K (20) 2 2 2 2 2 z 2 −1 z 2 − z 2 − z f = min e 4 f1 2 + z e 4 f2 : f = f1+f2,f1 e 4 A (∆),f2 ze 4 ϕ(∆) . k k∗ {k kA (∆) k kHϕ(∆) ∈ ∈ H } Finally, consider the biholomorphism ψ : ∆ ∆ given by ψ(z) = T −1/2z. Hence q(z, w; T )= T q(ψ(z), ψ(w);1) is the reproducing→ kernel of the space (see Theo- remK 2.9 in [9, pageK 81])
2 2 f ψ : f e−z /4A2(∆) + ze−z /4 (∆) = ( q ), { ◦ ∈ Hϕ } R LT with norm f ψ = f . k ◦ k∗∗ k k∗ q z2 z2 − 4 2 − 4 Thus ( ( T ), ∗∗) is isometrically isomorphic to (e A (∆) + ze ϕ(∆), R L k·k q z2 z2 H k · − 4 2 − 4 ∗) for all T > 0. As a vector space ( T ) = e A (∆) + ze ϕ(∆) for all kT > 0. R L H
Remark 16. The integral operator in (19) is the operator ΦT in [5, 8].
Conclusion e It is well known that any RKHS is determined by its reproducing kernel, unfortu- naly in our cases we cannot write the reproducing kernels in terms of “elementary functions”. Here we have a fundamental problem: is it posible to found a more ℓ + manageable description for the RKHSs ( T (D), Hℓ (D)), ( T (D), H+(D)) H k·k T H k·k T − − and ( T (D), H (D))? The proof of Moore’s theorem (see [9, pages 68-71]) gives H k·k T an equivalent description of the latter spaces, but it is hard to handle too. We would like to have a description as in Theorem 6.
Our work shows that there is no unique description for the reachable space of the heat equation for a finite rod with two Dirichlet boundary controls. It wouldR be interesting to establish a connection between our characterization and the ones obtained in [6, 8].
References
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