Uncertainty Principle and Prolate Spheroidal Wave functions

Uncertainty Principle and Prolate Spheroidal Wave functions

Aline Bonami, Universit´e d’Orl´eans

Joint work with Abderrazek Karoui Hammamet, 23 mars 2011

Heisenberg Uncertainty Principle

Heisenberg Inequality (in Dimension one) is

∆x∆p ≥~/2 ~=h/2π.

Mathematically, this is equivalent to the inequality Z

R

x²|f(x)|²dx 1/2

× Z

R

ξ²|ˆf(ξ)|²dξ 1/2

≥ Z

R

|f(x)|²dx, with equality for Gaussian functions.

fˆ(ξ) := 1

√2π Z +∞

−∞

e^−ixξf(x)dx.

Heisenberg Uncertainty Principle

Heisenberg Inequality (in Dimension one) is

∆x∆p ≥~/2 ~=h/2π.

Mathematically, this is equivalent to the inequality Z

R

x²|f(x)|²dx 1/2

× Z

R

ξ²|ˆf(ξ)|²dξ 1/2

≥ Z

R

|f(x)|²dx, with equality for Gaussian functions.

fˆ(ξ) := 1

√2π Z +∞

−∞

e^−ixξf(x)dx.

The spectral gap of the Harmonic oscillator

Z

R

x²|f(x)|²dx 1/2

× Z

R

ξ²|ˆf(ξ)²dξ 1/2

≥ Z

R

|f(x)|²dx

leads to Z

R

x²|f(x)|²dx+ Z

R

ξ²|fˆ(ξ)|²dξ≥ 1 2

Z

R

|f(x)|²dx, or, which is equivalent,

* x²f −

d dx

2

f,f +

≥ 1 2kfk²₂. 1/2 is the smallest eigenvalue of the harmonic oscillator H:=x²− _dx^d 2

ande^−x2/2 is the corresponding eigenfunction.

The spectral gap of the Harmonic oscillator

Z

R

x²|f(x)|²dx 1/2

× Z

R

ξ²|ˆf(ξ)²dξ 1/2

≥ Z

R

|f(x)|²dx

leads to Z

R

x²|f(x)|²dx+ Z

R

ξ²|fˆ(ξ)|²dξ≥ 1 2

Z

R

|f(x)|²dx, or, which is equivalent,

* x²f −

d dx

2

f,f +

≥ 1 2kfk²₂.

1/2 is the smallest eigenvalue of the harmonic oscillator H:=x²− _dx^d 2

ande^−x2/2 is the corresponding eigenfunction.

The spectral gap of the Harmonic oscillator

Z

R

x²|f(x)|²dx 1/2

× Z

R

ξ²|ˆf(ξ)²dξ 1/2

≥ Z

R

|f(x)|²dx

leads to Z

R

x²|f(x)|²dx+ Z

R

ξ²|fˆ(ξ)|²dξ≥ 1 2

Z

R

|f(x)|²dx, or, which is equivalent,

* x²f −

d dx

2

f,f +

≥ 1 2kfk²₂. 1/2 is the smallest eigenvalue of the harmonic oscillator H:=x²− _dx^d 2

ande^−x2/2 is the corresponding eigenfunction.

The eigenvalues of the Harmonic oscillator

n+¹₂ is thenth eigenvalue of the harmonic oscillator H:=x²− _dx^d 2

andH(x)e^−x2/2 is the corresponding eigenfunction.

Moreover,n+ ¹₂ is equal to

fmax1,···fn

min{hHf,fi; f ∈(f1,· · ·,fn)^⊥,kfk₂ = 1}. Extrema are given by Hermite functions.

The eigenvalues of the Harmonic oscillator

n+¹₂ is thenth eigenvalue of the harmonic oscillator H:=x²− _dx^d 2

andH(x)e^−x2/2 is the corresponding eigenfunction.

Moreover,n+ ¹₂ is equal to

fmax1,···fn

min{hHf,fi; f ∈(f1,· · ·,fn)^⊥,kfk₂ = 1}.

Extrema are given by Hermite functions.

Slepian Extremal problem

How large on [−1,+1] are functions with spectrum in [−c,+c] ?

I Then

λ0(c) = max Z +1

−1

|f|²dx; kfk₂ = 1,Supp(bf)⊂[−c,+c]

.

I

λ_n(c) = min

f1,···fn

max{ Z +1

−1

|f|²dx ; f ∈(f₁,· · ·,f_n)^⊥,kfk₂ = 1}. Consider the operator onL²([−1,+1])

F_c(f)(x) := c

2π

1/2Z +1

−1

e^icxξf(ξ)dξ. Thenλn(c) are the eigenvalues of F_c^∗F_c.

Key point :F_c^∗F_c(f) =f ∗χ\_[−c,+c] is compact.

Slepian Extremal problem

How large on [−1,+1] are functions with spectrum in [−c,+c] ?

I Then

λ0(c) = max Z +1

−1

|f|²dx; kfk₂ = 1,Supp(bf)⊂[−c,+c]

.

I

λ_n(c) = min

f1,···fn

max{ Z +1

−1

|f|²dx ; f ∈(f₁,· · ·,f_n)^⊥,kfk₂ = 1}. Consider the operator onL²([−1,+1])

F_c(f)(x) := c

2π

1/2Z +1

−1

e^icxξf(ξ)dξ. Thenλn(c) are the eigenvalues of F_c^∗F_c.

Key point :F_c^∗F_c(f) =f ∗χ\_[−c,+c] is compact.

Slepian Extremal problem

How large on [−1,+1] are functions with spectrum in [−c,+c] ?

I Then

λ0(c) = max Z +1

−1

|f|²dx; kfk₂ = 1,Supp(bf)⊂[−c,+c]

.

I

λ_n(c) = min

f1,···fn

max{

Z +1

−1

|f|²dx ; f ∈(f₁,· · ·,f_n)^⊥,kfk₂ = 1}.

Consider the operator onL²([−1,+1]) F_c(f)(x) :=

c 2π

1/2Z +1

−1

e^icxξf(ξ)dξ. Thenλn(c) are the eigenvalues of F_c^∗F_c.

Key point :F_c^∗F_c(f) =f ∗χ\_[−c,+c] is compact.

Slepian Extremal problem

How large on [−1,+1] are functions with spectrum in [−c,+c] ?

I Then

λ0(c) = max Z +1

−1

|f|²dx; kfk₂ = 1,Supp(bf)⊂[−c,+c]

.

I

λ_n(c) = min

f1,···fn

max{

Z +1

−1

|f|²dx ; f ∈(f₁,· · ·,f_n)^⊥,kfk₂ = 1}.

Consider the operator onL²([−1,+1]) F_c(f)(x) :=

c 2π

1/2Z +1

−1

e^icxξf(ξ)dξ.

Thenλn(c) are the eigenvalues ofF_c^∗F_c. Key point :F_c^∗F_c(f) =f ∗χ\_[−c,+c] is compact.

The asymptotic behavior of λ

(c ).

(Landau and Widom 1980)

Asymptotically forc tending to ∞, the sequence λn(c) stays close to 1 forn ≤ _π²c, then decreases exponentially rapidly.

Letψ_n,c be the eigenfunction corresponding toλ_n,c. The main tool : (Slepian) The eigenfunctionsλn are also

eigenfunctions of an explicit Sturm-Liouville operator on (−1,+1) L_cφ:=−d

dx

(1−x²)φ⁰

+c²x²φ.

The asymptotic behavior of λ

(c ).

(Landau and Widom 1980)

Asymptotically forc tending to ∞, the sequence λn(c) stays close to 1 forn ≤ _π²c, then decreases exponentially rapidly.

Letψ_n,c be the eigenfunction corresponding toλ_n,c. The main tool : (Slepian) The eigenfunctionsλn are also

eigenfunctions of an explicit Sturm-Liouville operator on (−1,+1) L_cφ:=−d

dx

(1−x²)φ⁰

+c²x²φ.

Why Prolate Spheroidal Wave Functions ?

L_cφ:=−d dx

(1−x²)φ⁰

+c²x²φ.

Arises in finding particular “radial” solutions of the Helmotz equation ∆Φ +k²Φ = 0 in three dimensions in the “ prolate spheroidal coordinates”. Explains the name : Prolate Spheroidal Wave Functions PSWF.

C. Niven :On the Conduction of Heat in Ellipsoids of Revolution,Philosophical Trans. R. Soc. Lond. 1880 171, 117-151.In this remarkable piece of work, Niven has developed a detailed computational and asymptotic methods for the PSWFs and the eigenvaluesχ_n(c).

Why Prolate Spheroidal Wave Functions ?

L_cφ:=−d dx

(1−x²)φ⁰

+c²x²φ.

Arises in finding particular “radial” solutions of the Helmotz equation ∆Φ +k²Φ = 0 in three dimensions in the “ prolate spheroidal coordinates”. Explains the name : Prolate Spheroidal Wave Functions PSWF.

C. Niven :On the Conduction of Heat in Ellipsoids of Revolution,Philosophical Trans. R. Soc. Lond. 1880 171, 117-151.In this remarkable piece of work, Niven has developed a detailed computational and asymptotic methods for the PSWFs and the eigenvaluesχ_n(c).

Why Prolate Spheroidal Wave Functions ?

L_cφ:=−d dx

(1−x²)φ⁰

+c²x²φ.

Arises in finding particular “radial” solutions of the Helmotz equation ∆Φ +k²Φ = 0 in three dimensions in the “ prolate spheroidal coordinates”. Explains the name : Prolate Spheroidal Wave Functions PSWF.

C. Niven :On the Conduction of Heat in Ellipsoids of Revolution,Philosophical Trans. R. Soc. Lond. 1880 171, 117-151.In this remarkable piece of work, Niven has developed a detailed computational and asymptotic methods for the PSWFs and the eigenvaluesχ_n(c).

What is known on PSWF ?

I The ψ_n,c are the bounded eigenfunctions on (−1,+1) of L_cφ:=− d

dx

(1−x²)φ⁰

+c²x²φ related to the eigenvaluesχn(c).

I ψ_n,c is of the same parity as n.

I They are also eigenfunctions of F_c : for all x, ψn,c(x) :=µn

Z +1

−1

e^icxξf(ξ)dξ.

I They are entire functions.

They are used for numerics (for Helmotz Equation, in Signal Processing, as an orthonormal basis for the representation of functions on (−1,+1)). Work of Rokhlin Xiao et al., Chen Gottlieb and Hesthaven, Wang, Boyd, Karoui, Moumni, etc.

What is known on PSWF ?

I The ψ_n,c are the bounded eigenfunctions on (−1,+1) of L_cφ:=− d

dx

(1−x²)φ⁰

+c²x²φ related to the eigenvaluesχn(c).

I ψ_n,c is of the same parity as n.

I They are also eigenfunctions of F_c : for all x, ψn,c(x) :=µn

Z +1

−1

e^icxξf(ξ)dξ.

I They are entire functions.

They are used for numerics (for Helmotz Equation, in Signal Processing, as an orthonormal basis for the representation of functions on (−1,+1)). Work of Rokhlin Xiao et al., Chen Gottlieb and Hesthaven, Wang, Boyd, Karoui, Moumni, etc.

The WKB method for L

.

ψ_n,c, related to the eigenvalue χ_n:=χ_n(c), is the bounded solution of

d dx

(1−x²)ψ⁰(x)

+χn(1−qx²)ψ(x) = 0, x ∈(−1,1). (1) Here we assume thatq =c²/χn<1.

Remark.χn is not known. Heuristically (and asymptotically) equivalent toc < ^π₂n.

S(x) :=S_q(x) = Z 1

x

s

1−qt²

1−t² dt, x ∈[0,1). (2) We look forψ under the form

ψ(x) = (1−x²)^−1/4(1−qx²)^−1/4U(S(x)). (3)

The WKB method for L

.

ψ_n,c, related to the eigenvalue χ_n:=χ_n(c), is the bounded solution of

d dx

(1−x²)ψ⁰(x)

+χn(1−qx²)ψ(x) = 0, x ∈(−1,1). (1) Here we assume thatq =c²/χn<1.

Remark.χ_n is not known. Heuristically (and asymptotically) equivalent toc < ^π₂n.

S(x) :=S_q(x) = Z 1

x

s

1−qt²

1−t² dt, x ∈[0,1). (2) We look forψ under the form

ψ(x) = (1−x²)^−1/4(1−qx²)^−1/4U(S(x)). (3)

The WKB method for L

.

ψ_n,c, related to the eigenvalue χ_n:=χ_n(c), is the bounded solution of

d dx

(1−x²)ψ⁰(x)

+χn(1−qx²)ψ(x) = 0, x ∈(−1,1). (1) Here we assume thatq =c²/χn<1.

Remark.χ_n is not known. Heuristically (and asymptotically) equivalent toc < ^π₂n.

S(x) :=S_q(x) = Z 1

x

s

1−qt²

1−t² dt, x ∈[0,1). (2) We look forψ under the form

ψ(x) = (1−x²)^−1/4(1−qx²)^−1/4U(S(x)). (3)

The WKB method for L

Lemma.Forq <1,there exists a functionF :=F_q that is continuous on [0,S(0)],satisfies the inequality

|F(s)| ≤ 3

(1−q)³, (4)

and such thatU is a solution of the equation U⁰⁰(s) +

χn+ 1 4s²

U(s) =F(s)U(s), s ∈[0,S(0)]. (5)

The WKB method for L

.

The homogeneous equation U⁰⁰(s) +

χ_n+ 1 4s²

U(s) = 0 has the two independent solutions

U1(s) =χ^1/4n

√sJ0(√

χns), U2(s) =χ^1/4n

√sY0(√ χns).

The functionU is given by

U =AU₁+R with√

χ_nR a bounded function.

The WKB method for L

.

The homogeneous equation U⁰⁰(s) +

χ_n+ 1 4s²

U(s) = 0 has the two independent solutions

U1(s) =χ^1/4n

√sJ0(√

χns), U2(s) =χ^1/4n

√sY0(√ χns).

The functionU is given by

U =AU₁+R with√

χ_nR a bounded function.

Uniform approximation of PSWF

Theorem.[A.B., A.K. (2010)] For q =c²/χn(c)<1 and 0≤x≤1,

ψ_n,c(x) =Aχn(c)^1/4p

Sq(x)J0(p

χn(c)Sq(x))

(1−x²)^1/4(1−qx²)^1/4 +R_n,c(x) (6) with

sup

x∈[0,1]

|R_n,c(x)| ≤C_qχ_n(c)^−1/2. (7)

Close approximations were known, but for fixedx.

Uniform approximation of PSWF

Theorem.[A.B., A.K. (2010)] For q =c²/χn(c)<1 and 0≤x≤1,

ψ_n,c(x) =Aχn(c)^1/4p

Sq(x)J0(p

χn(c)Sq(x))

(1−x²)^1/4(1−qx²)^1/4 +R_n,c(x) (6) with

sup

x∈[0,1]

|R_n,c(x)| ≤C_qχ_n(c)^−1/2. (7) Close approximations were known, but for fixedx.

Uniform bounds of the PSWFs

sup

|x|≤1

|ψ_n(x)| ≤Cqχ^1/4n , with the maximum obtained at 1.

sup

|x|≤1

(1−x²)^1/4|ψ_n(x)| ≤Cq.

sup

x∈[−1,1]

|ψ⁰_n(x)| ≤Cχn5/4, sup

x∈[−1,1]

(1−x²)|ψ⁰_n(x)| ≤C√ χ_n.

The method does not give the derivatives of higher order.

Uniform bounds of the PSWFs

sup

|x|≤1

|ψ_n(x)| ≤Cqχ^1/4n , with the maximum obtained at 1.

sup

|x|≤1

(1−x²)^1/4|ψ_n(x)| ≤Cq.

sup

x∈[−1,1]

|ψ_n⁰(x)| ≤Cχn5/4, sup

x∈[−1,1]

(1−x²)|ψ⁰_n(x)| ≤C√ χ_n.

The method does not give the derivatives of higher order.

The main difficulty : estimate the constantA, using the fact that ψn has norm 1.

For the derivative, use of the equation : (1−x²)ψ_n⁰(x) =χ_n

Z 1

x

(1−qt²)ψ_n(t)dt.

The main difficulty : estimate the constantA, using the fact that ψn has norm 1.

For the derivative, use of the equation : (1−x²)ψ_n⁰(x) =χ_n

Z 1

x

(1−qt²)ψ_n(t)dt.

The exponential decay of λ

(c ).

Well-known equality [Slepian 1964, Rokhlin et al. (2007)] : that for any positive integern,we haveλ_n(c) =λ⁰×λ⁰⁰, with

λ⁰ : = c²ⁿ⁺¹(n!)⁴

2((2n)!)²(Γ(n+ 3/2))² (8)

λ⁰⁰ : = exp

2 Z c

0

(ψn,τ(1))²−(n+ 1/2)

τ dτ

. (9)

Numerical evidence, see [Rokhlin et al. 2007], indicates that (ψn,τ(1))²−(n+ 1/2)≤0,∀t ≥0.If we accept this assertion, then we observe that the sequenceλ_n(c) decays faster than c

2 ec

4n 2n

so that the exponential decay has started at [ec/4].

To compare to [Landau and Widom 1980] :∀c >0,∀0< α <1, N(α) = #{λ_i(c);λ_i(c)> α} is given by

N(α) = 2c

π +

1

π² log

1−α

α

log(c) +o(log(c)).

The exponential decay of λ

(c ).

Well-known equality [Slepian 1964, Rokhlin et al. (2007)] : that for any positive integern,we haveλ_n(c) =λ⁰×λ⁰⁰, with

λ⁰ : = c²ⁿ⁺¹(n!)⁴

2((2n)!)²(Γ(n+ 3/2))² (8)

λ⁰⁰ : = exp

2 Z c

0

(ψn,τ(1))²−(n+ 1/2)

τ dτ

. (9)

Numerical evidence, see [Rokhlin et al. 2007], indicates that (ψn,τ(1))²−(n+ 1/2)≤0,∀t ≥0.If we accept this assertion, then we observe that the sequenceλ_n(c) decays faster than c

2 ec

4n 2n

so that the exponential decay has started at [ec/4].

To compare to [Landau and Widom 1980] :∀c >0,∀0< α <1, N(α) = #{λ_i(c);λ_i(c)> α} is given by

N(α) = 2c

π +

1

π² log

1−α

α

log(c) +o(log(c)).

The exponential decay of λ

(c ).

(A.B., A.K. (2010)) : Letδ >0. There existsN and κ such that, for allc ≥0 and n≥max(N, κc),

λn(c)≤e^{−δ(n−κc)}.

Work in progress :

I Bounds below for the λ_n(c).

I Sharp estimates for the χ_n(c).

I Qualitative properties of the PSWFs.

The exponential decay of λ

(c ).

(A.B., A.K. (2010)) : Letδ >0. There existsN and κ such that, for allc ≥0 and n≥max(N, κc),

λn(c)≤e^{−δ(n−κc)}.

Work in progress :

I Bounds below for the λ_n(c).

I Sharp estimates for the χ_n(c).

I Qualitative properties of the PSWFs.

Almost band-limited functions.

LetT and Ω de two measurable sets. A function pair (f,bf) is said to beT−concentrated inT andΩ−concentrated in Ω if

Z

T^c

|f(t)|²dt ≤²_T, Z

Ω^c

|bf(ω)|²dω≤²_Ω.

Approximation of almost band-limited functions.

Theorem

If f is an L² normalized function that is_T−concentrated in T = [−1,+1]andΩ−band concentrated in Ω = [−c,+c],then for any positive integer N,we have

Z +1

−1

|f −SNf|²dt 1/2

≤Ω+p

λN(c) (10) and, as a consequence,

kf −χ_[−T,+T]S_Nfk₂ ≤_T+_Ω+p

λ_N(c). (11)

Key point : forf whose Fourier transform is supported on [−c,+c], f =X

n

a_nψ_n, f −S_Nf = X

n≥N

a_nψ_n.

So

Z +1

−1

|f −SNf|²dt = X

n≥N

|a_n|²

≤ λ_N(c)X

n≥N

|a_n|² λ_n(c)

≤ λ_N(c)kfk²₂.

Key point : forf whose Fourier transform is supported on [−c,+c], f =X

n

a_nψ_n, f −S_Nf = X

n≥N

a_nψ_n.

So

Z +1

−1

|f −SNf|²dt = X

n≥N

|a_n|²

≤ λ_N(c)X

n≥N

|a_n|² λ_n(c)

≤ λ_N(c)kfk²₂.

Approximation in Sobolev spaces.

Theorem

Let be a positive real number. Assume that f ∈H^s([−1,+1]), for some positive real number s>0. Then for any c ≥0 and any integer N≥1,we have

kf −SNfk₂ ≤K(1 +c²)^−s/2kfk_H^s+Kp

λN(c)kfk₂. (12) Here, the constant K depends only on s. Moreover it can be taken equal to1 when f belongs to the space H₀^s([−1,+1]).