A Generalization of the Fock Space

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1 A Generalization of the Fock Space Southeastern Analysis Meeting Joel Rosenfeld, Ben Russo* March 2017

2 Definition The Fock space F 2 is a RKHS consisting of entire functions where f 2 = f (z) 2 1 π e z 2 dz < C

3 Definition The Fock space F 2 is a RKHS consisting of entire functions where f 2 = f (z) 2 1 π e z 2 dz < C Equivalently, { } F 2 = f (z) = a n z n : a n 2 n! <. n=0 n=0

4 Definition The Fock space F 2 is a RKHS consisting of entire functions where f 2 = f (z) 2 1 π e z 2 dz < C Equivalently, { } F 2 = f (z) = a n z n : a n 2 n! <. n=0 n=0 The reproducing kernels for F 2 are given by: K(z, w) = e z w

5 Definition The one parameter Mittag-Leffler function is defined as E q (z) = n=0 z n Γ(qn + 1) q R +.

6 Definition The one parameter Mittag-Leffler function is defined as E q (z) = n=0 z n Γ(qn + 1) q R +. When q = 1: When q = 1/2: E 1 (z) = e z E 1/2 (z) = e z erfc(z 1/2 )

7 We define the following generalization of the Fock Space: Definition { ML 2 (C; q) = f (z) = a n z n : n=0 } a n 2 Γ(qn + 1) < n=0 is a RKHS with kernel functions given by K q (w, z) = E q ( wz)

8 We define the following generalization of the Fock Space: Definition { ML 2 (C; q) = f (z) = a n z n : n=0 } a n 2 Γ(qn + 1) < n=0 is a RKHS with kernel functions given by K q (w, z) = E q ( wz) Equivalently, we can consider ML 2 (C; q) to consist of entire functions for which f 2 = f (z) 2 1 qπ z 2 q 2 e z 2 q dz < C

9 Proposition If f F 2, then f ML 2 (C; q) for all 0 < q 1. Moreover, lim f q 1 ML 2 (q) = f F 2.

10 Proposition If f F 2, then f ML 2 (C; q) for all 0 < q 1. Moreover, lim f q 1 ML 2 (q) = f F 2. To see this we note that { z n } { } and z n n! n=0 Γ(qn+1) n=0 are orthonormal basis for F 2 and ML 2 (q) respectively.

11 Proposition If f F 2, then f ML 2 (C; q) for all 0 < q 1. Moreover, lim f q 1 ML 2 (q) = f F 2. To see this we note that { z n } { } and z n n! n=0 Γ(qn+1) n=0 are orthonormal basis for F 2 and ML 2 (q) respectively. f 2 ML 2 (q) = ( ) Γ(qn + 1) a n 2 n! and f 2 F 2 = a n 2

12 By a simple application of Cauchy-Schwarz f (z) 2 E q ( z 2 ) f ML 2

13 By a simple application of Cauchy-Schwarz Proposition f (z) 2 E q ( z 2 ) f ML 2 If f is in the Mittag-Leffler space ML 2 (C; q) for 0 < q < 2 then f is of order less than or equal to 2/q.

14 By a simple application of Cauchy-Schwarz Proposition f (z) 2 E q ( z 2 ) f ML 2 If f is in the Mittag-Leffler space ML 2 (C; q) for 0 < q < 2 then f is of order less than or equal to 2/q. Proposition Let {z n } be the zero sequence, repeated according to multiplicity and arranged so that 0 < z 1 z 2..., of a function f ML 2 (C, q) such that f (0) 0. If 0 < q < 2 then there exists a positive constant c such that z n cn q 2.

15 Originally, a real version of the Mittag-Leffler space was developed for an approximation scheme for Caputo fractional derivatives.

16 Originally, a real version of the Mittag-Leffler space was developed for an approximation scheme for Caputo fractional derivatives. Definition The Riemann-Liouville fractional integral for q R + is defined as (J q f )(t) = 1 Γ(q) t 0 (t τ) q 1 f (τ)dτ

17 Originally, a real version of the Mittag-Leffler space was developed for an approximation scheme for Caputo fractional derivatives. Definition The Riemann-Liouville fractional integral for q R + is defined as (J q f )(t) = 1 Γ(q) t 0 (t τ) q 1 f (τ)dτ Definition Let n N. For an n-times differentiable function f : R + R, the Caputo fractional derivative of order q, where n 1 < q n, is given by D q f (t) = J n q d n dt n f (t) 1 Γ(n q) = f (n) (t) t 0 f (n) (τ) dτ n 1 < q < n, n N (t τ) q+1 n q = n N

18 The Caputo derivative has a pointwise interpolation property. lim q n Dq f (t) = f (n) (t)

19 On monomials we have that Γ(n+1) D q t n Γ(n q+1) tn q m 1 < q < m, n > m 1, n R = 0 m 1 < q < m, n m 1, n N

20 On monomials we have that D q t n = Γ(n+1) Γ(n q+1) tn q m 1 < q < m, n > m 1, n R 0 m 1 < q < m, n m 1, n N The Mittag-Leffler function of order q is an eigenfunction for Caputo differentiation. D q E q (λt q ) = λe q (λt q )

21 The real Mittag-Leffler space is defined as follows: Definition For q > 0, the (real) Mittag-Leffler space of order q is the (real valued) RKHS corresponding to the kernel function K q (t, λ) = E q (λ q t q ). We denote this space by ML 2 (R + ; q) Hence, { ML 2 (R + ; q) = f (t) = a n t qn : n=0 } a n 2 Γ(qn + 1) < n=0

22 The real Mittag-Leffler space is defined as follows: Definition For q > 0, the (real) Mittag-Leffler space of order q is the (real valued) RKHS corresponding to the kernel function K q (t, λ) = E q (λ q t q ). We denote this space by ML 2 (R + ; q) Hence, { ML 2 (R + ; q) = f (t) = a n t qn : n=0 } a n 2 Γ(qn + 1) < n=0 For functions in the real valued Mittag-Leffler space we have that D q f (t) = n=0 Γ(q(n + 1) + 1) a n+1 t qn Γ(qn + 1)

23 To extend the Caputo derivative to functions of a complex variable we define the following: Definition ML 2 (C \ R ; q) = {f : C \ R C : f (z 1/q ) ML 2 (C; q)}

24 To extend the Caputo derivative to functions of a complex variable we define the following: Definition ML 2 (C \ R ; q) = {f : C \ R C : f (z 1/q ) ML 2 (C; q)} { ML 2 (C \ R ; q) = f (z) = a n z qn : n=0 } a n 2 Γ(qn + 1) < n=0

25 To extend the Caputo derivative to functions of a complex variable we define the following: Definition ML 2 (C \ R ; q) = {f : C \ R C : f (z 1/q ) ML 2 (C; q)} { ML 2 (C \ R ; q) = f (z) = a n z qn : n=0 } a n 2 Γ(qn + 1) < n=0 We will define f (z), g(z) ML 2 (C\R ;q) f = (z 1/q ), g(z 1/q ) ML 2 (C;q).

26 To extend the Caputo derivative to functions of a complex variable we define the following: Definition ML 2 (C \ R ; q) = {f : C \ R C : f (z 1/q ) ML 2 (C; q)} { ML 2 (C \ R ; q) = f (z) = a n z qn : n=0 } a n 2 Γ(qn + 1) < n=0 We will define f (z), g(z) ML 2 (C\R ;q) f = (z 1/q ), g(z 1/q ) ML 2 (C;q). We can view ML 2 (C \ R ; q) as the pullback of ML 2 (C; q) using the mapping ϕ : C \ R C; ϕ(z) = z q.

27 Definition Let f (z) ML 2 (C \ R; q), then f (z) = a n z qn n=0 where an 2 < Define the Caputo derivative of f (z) as follows: D q f (z) = n=1 a n Γ(qn + 1) Γ(q(n 1) + 1) zq(n 1)

28 Definition Let f (z) ML 2 (C \ R; q), then f (z) = a n z qn n=0 where an 2 < Define the Caputo derivative of f (z) as follows: D q f (z) = n=1 a n Γ(qn + 1) Γ(q(n 1) + 1) zq(n 1) Since this definition agrees with the definition of the Caputo fractional derivative for functions in the real valued Mittag-Leffler space on the real line, by an application of the identity theorem this is an appropriate generalization to C \ R

29 Consider the space L 2 (R, dx) and the densely defined unbounded operators and X : Dom(X ) L 2 (R) L 2 (R); D : Dom(D) L 2 (R) L 2 (R); X (f ) = xf (x) D(f ) = i d dx f (x)

30 Consider the space L 2 (R, dx) and the densely defined unbounded operators and X : Dom(X ) L 2 (R) L 2 (R); D : Dom(D) L 2 (R) L 2 (R); X (f ) = xf (x) D(f ) = i d dx f (x) Define the annihilation and creation operators on L 2 (R, dx) by respectively. W = 1 ( X + i ) 2 D, W = 1 ( X i ) 2 D

31 Let {h n (x)} be the hermite functions defined by: Definition and h n (x) = H n (x) = ( 1) n e x2 d n dx n e x2 ( ) π 2 n n! e x2 H n ( 2x).

32 Let {h n (x)} be the hermite functions defined by: Definition and h n (x) = H n (x) = ( 1) n e x2 d n dx n e x2 ( ) π 2 n n! e x2 H n ( 2x). The hermite functions are an orthonormal basis for L 2 (R, dx) that lie in the domains of both W and W.

33 Let {h n (x)} be the hermite functions defined by: Definition and h n (x) = H n (x) = ( 1) n e x2 d n dx n e x2 ( ) π 2 n n! e x2 H n ( 2x). The hermite functions are an orthonormal basis for L 2 (R, dx) that lie in the domains of both W and W. Moreover, Wh n = n h n 1 and W h n = n + 1 h n+1.

34 In a similar fashion, consider F 2 (C) with orthonormal basis {e n } = { } z n n! given by the scaled monomials.

35 In a similar fashion, consider F 2 (C) with orthonormal basis {e n } = { } z n n! given by the scaled monomials. Let A and A be the unbounded densely defined operators given by Af (z) = d dz f (z), A f (z) = zf (z).

36 In a similar fashion, consider F 2 (C) with orthonormal basis {e n } = { } z n n! given by the scaled monomials. Let A and A be the unbounded densely defined operators given by Af (z) = d dz f (z), A f (z) = zf (z). Again we note that Ae n = n e n 1 and A e n = n + 1 e n+1.

37 The Bargmann transform, B : L 2 (R, dx) F 2 (C) is given by for z C. [Bf ](z) = ( ) 2 1/4 f (x)e 2xz x2 1 2 z2 dx π R

38 The Bargmann transform, B : L 2 (R, dx) F 2 (C) is given by [Bf ](z) = ( ) 2 1/4 f (x)e 2xz x2 1 2 z2 dx π R for z C. The inverse transform is given by [ B 1 f ] ( ) 2 1/4 (x) = f (z)e 2x z x2 1 2 z2 e z 2 dz. π C

39 The Bargmann transform, B : L 2 (R, dx) F 2 (C) is given by [Bf ](z) = ( ) 2 1/4 f (x)e 2xz x2 1 2 z2 dx π R for z C. The inverse transform is given by Where, [ B 1 f ] ( ) 2 1/4 (x) = f (z)e 2x z x2 1 2 z2 e z 2 dz. π C ( ) 2 1/4 e 2xz x2 1 2 z2 = π n=0 h n (x) zn n!.

40 B : a n h n (x) n=0 n=0 a n z n n!. Hence Theorem (Bargmann 61) W = B 1 AB W = B 1 A B We wish to leverage the above result into a result about fractional differentiation and multiplication by z q on the Mittag-Leffler space on the slitted plane.

41 Definition Let (q, p) R 2+, call H p,q (z) = n=0 z n Γ(pn + 1) Γ(qn + 1) the modified Mittag-Leffler function for (p, q).

42 Definition Let (q, p) R 2+, call H p,q (z) = n=0 z n Γ(pn + 1) Γ(qn + 1) the modified Mittag-Leffler function for (p, q). Theorem The operator given by U q : F 2 ML 2 (C \ R ; q) U q f = 1 f (w)h 1,q (wz q )e w 2 dw π C is an isometry between F 2 and ML 2 (C \ R ; q) such that z n n! z qn Γ(qn + 1)

43 Theorem There exists an isometry from L 2 (R, dx) onto ML 2 (C \ R ; q) given by B q = U q B : L 2 (R, dx) ML 2 (C \ R ; q).

44 Definition Define the operators Z q and Y q on ML 2 (C \ R; q) as follows: Z q : Dom(Z q ) ML 2 (C \ R; q) (Z q f )(z) = z q f (z) and Y q : Dom(Y q ) ML 2 (C \ R; q) (Y q f )(z) = D q f (z).

45 Definition Define the operators Z q and Y q on ML 2 (C \ R; q) as follows: Z q : Dom(Z q ) ML 2 (C \ R; q) (Z q f )(z) = z q f (z) and Y q : Dom(Y q ) ML 2 (C \ R; q) (Y q f )(z) = D q f (z). Proposition The operators Z q and Y q are closed, Z q = Y q, and Y q = Z q.

46 Definition Define the operators Z q and Y q on ML 2 (C \ R; q) as follows: Z q : Dom(Z q ) ML 2 (C \ R; q) (Z q f )(z) = z q f (z) and Y q : Dom(Y q ) ML 2 (C \ R; q) (Y q f )(z) = D q f (z). Proposition The operators Z q and Y q are closed, Z q = Y q, and Y q = Z q. For convenience we will apply the following relabeling: A q = Y q and A q = Z q

47 We note that on our orthonormal basis for ML 2 (C \ R ; q) given by { } z qn {g n } = Γ(qn + 1) we have that

48 We note that on our orthonormal basis for ML 2 (C \ R ; q) given by { } z qn {g n } = Γ(qn + 1) we have that A q g n = D q Γ(qn + 1) g n = Γ(q(n 1) + 1) g n 1 and A qg n = z q g n = Γ(q(n + 1) + 1) g n+1. Γ(qn + 1)

49 We note that on our orthonormal basis for ML 2 (C \ R ; q) given by { } z qn {g n } = Γ(qn + 1) we have that A q g n = D q Γ(qn + 1) g n = Γ(q(n 1) + 1) g n 1 and A qg n = z q g n = Γ(q(n + 1) + 1) g n+1. Γ(qn + 1) This tells us that the g n (z) lie in the domains of both operators.

50 Proposition Let S q : L 2 (R, dx) L 2 (R, dx) be the operator given by Γ(q(n + 1) + 1) S q h n = (n + 1)Γ(qn + 1) h n. S q is a compact self-adjoint operator.

51 Proposition Let S q : L 2 (R, dx) L 2 (R, dx) be the operator given by Γ(q(n + 1) + 1) S q h n = (n + 1)Γ(qn + 1) h n. S q is a compact self-adjoint operator. We define the following operators: Definition W q : Dom(W ) L 2 (R, dx); W q : Dom(W ) L 2 (R, dx); W q = S q W W q = W S q Note that S q (Dom(W )) Dom(W )

52 Given the orthonormal basis of the hermite functions we note: Γ(qn + 1) W q h n = S q Wh n = Γ(q(n 1) + 1) h n 1 W q h n = W S q h n = Γ(q(n + 1) + 1) h n+1. Γ(qn + 1)

53 Given the orthonormal basis of the hermite functions we note: Γ(qn + 1) W q h n = S q Wh n = Γ(q(n 1) + 1) h n 1 W q h n = W S q h n = Γ(q(n + 1) + 1) h n+1. Γ(qn + 1) Hence Proposition For 0 < q 1 and Bq 1 A q B q = W q B 1 q A qb q = W q.

54 Proposition For 0 < q 1 the function w(z) = 1 qπ z 2 q 2 e z q 2 is the unique, continuous, radial weight such that for polynomials p(z) = a n z qn a 0 and s(z) = b m z qm +... b 0 under the inner product p(z), s(z) = p(z 1/q )r(z 1/q )w(z)dz C the operations of multiplication by z q and Caputo differentiation are adjoint, and the function f (z) = 1 has norm 1.

55 Thanks!

Introduction to The Dirichlet Space

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