# Fourier transform, null variety, and Laplacian's eigenvalues specgeom/ transform, null variety, and...

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Fourier transform, null variety, and Laplacianseigenvalues

Michael Levitin

Reading University

Spectral Geometry Conference, 19 July 2010

joint work with Rafael Benguria (PUC Santiago) and Leonid Parnovski (UCL)

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 1 / 22

Objects of study in BenguriaL.Parnovski (2009)

Rd simply connected bounded domain with connectedboundary ;

(x) =

{1, if x ,0 if x 6

the characteristic function of ;

() = F []() :=

eix dx its Fourier transform;

NC() := { Cd : () = 0} its complex null variety, or nullset;

C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

cos( x) dx = 0} and () := dist(N (), 0);(0

Objects of study in BenguriaL.Parnovski (2009)

Rd simply connected bounded domain with connectedboundary ;

(x) =

{1, if x ,0 if x 6

the characteristic function of ;

() = F []() :=

eix dx its Fourier transform;

NC() := { Cd : () = 0} its complex null variety, or nullset;

C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

cos( x) dx = 0} and () := dist(N (), 0);(0

Objects of study in BenguriaL.Parnovski (2009)

Rd simply connected bounded domain with connectedboundary ;

(x) =

{1, if x ,0 if x 6

the characteristic function of ;

() = F []() :=

eix dx its Fourier transform;

NC() := { Cd : () = 0} its complex null variety, or nullset;

C() := dist(NC(), 0) = min{|| : NC()};Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

cos( x) dx = 0} and () := dist(N (), 0);(0

Objects of study in BenguriaL.Parnovski (2009)

Rd simply connected bounded domain with connectedboundary ;

(x) =

{1, if x ,0 if x 6

the characteristic function of ;

() = F []() :=

eix dx its Fourier transform;

NC() := { Cd : () = 0} its complex null variety, or nullset;

cos( x) dx = 0} and () := dist(N (), 0);(0

Objects of study in BenguriaL.Parnovski (2009)

Rd simply connected bounded domain with connectedboundary ;

(x) =

{1, if x ,0 if x 6

the characteristic function of ;

() = F []() :=

eix dx its Fourier transform;

NC() := { Cd : () = 0} its complex null variety, or nullset;

C() := dist(NC(), 0) = min{|| : NC()};

Also, in particular for balanced (e.g. centrally symmetric domains) welook at N () := NC() Rd = { Rd : () = 0} = { Rd :

cos( x) dx = 0} and () := dist(N (), 0);(0

Objects of study in BenguriaL.Parnovski (2009)

Rd simply connected bounded domain with connectedboundary ;

(x) =

{1, if x ,0 if x 6

the characteristic function of ;

() = F []() :=

eix dx its Fourier transform;

NC() := { Cd : () = 0} its complex null variety, or nullset;

cos( x) dx = 0} and () := dist(N (), 0);

(0

Objects of study in BenguriaL.Parnovski (2009)

Rd simply connected bounded domain with connectedboundary ;

(x) =

{1, if x ,0 if x 6

the characteristic function of ;

() = F []() :=

eix dx its Fourier transform;

NC() := { Cd : () = 0} its complex null variety, or nullset;

cos( x) dx = 0} and () := dist(N (), 0);(0

What does N () look like?

Far field zero intensity diffraction pattern from the aperture ! Physicalmotivation.Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

What does N () look like?

Far field zero intensity diffraction pattern from the aperture ! Physicalmotivation.Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

What does N () look like?

Farfield zero intensity diffraction pattern from the aperture ! Physicalmotivation.Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

What does N () look like?

Far field zero intensity diffraction pattern from the aperture ! Physicalmotivation.

Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

What does N () look like?

Far field zero intensity diffraction pattern from the aperture ! Physicalmotivation.Also, it is of importance for inverse problems and image recognition. It isknown that the structure of N () far from the origin determines theshape of a convex set .

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 3 / 22

Questions and Motivation

Main question: study () and its relations to the eigenvalues.

Known links between N () and spectral theory: two unsolved problems

Pompeius Problem

Let M(d) be a group of rigidmotions of Rd , and be abounded simply connected domainwith piecewise smooth connectedboundary. Prove that the existenceof a non-zero continuous functionf : Rd Rd such thatm() f (x) dx = 0 for all

m M(d) implies that is a ball.

Schiffers conjecture

The existence of an eigenfunction v(corresponding to a non-zeroeigenvalue ) of a NeumannLaplacian on a domain such thatv const along the boundary (or, in other words, the existence ofa non-constant solution v to theover-determined problemv = v , v/n| = 0,v | = 1) implies that is a ball.

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 4 / 22

Questions and Motivation

Main question: study () and its relations to the eigenvalues.Known links between N () and spectral theory: two unsolved problems

Pompeius Problem

Let M(d) be a group of rigidmotions of Rd , and be abounded simply connected domainwith piecewise smooth connectedboundary. Prove that the existenceof a non-zero continuous functionf : Rd Rd such thatm() f (x) dx = 0 for all

m M(d) implies that is a ball.

Schiffers conjecture

The existence of an eigenfunction v(corresponding to a non-zeroeigenvalue ) of a NeumannLaplacian on a domain such thatv const along the boundary (or, in other words, the existence ofa non-constant solution v to theover-determined problemv = v , v/n| = 0,v | = 1) implies that is a ball.

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 4 / 22

Questions and Motivation

Main question: study () and its relations to the eigenvalues.Known links between N () and spectral theory: two unsolved problems

Pompeius Problem

Let M(d) be a group of rigidmotions of Rd , and be abounded simply connected domainwith piecewise smooth connectedboundary. Prove that the existenceof a non-zero continuous functionf : Rd Rd such thatm() f (x) dx = 0 for all

m M(d) implies that is a ball.

Schiffers conjecture

The existence of an eigenfunction v(corresponding to a non-zeroeigenvalue ) of a NeumannLaplacian on a domain such thatv const along the boundary (or, in other words, the existence ofa non-constant solution v to theover-determined problemv = v , v/n| = 0,v | = 1) implies that is a ball.

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 4 / 22

Questions and Motivation

Main question: study () and its relations to the eigenvalues.Known links between N () and spectral theory: two unsolved problems

Pompeius Problem

m M(d) implies that is a ball.

Schiffers conjecture

M Levitin (Reading) Fourier transform, ..., Hanover, 19 July 2010 4 / 22

Motivation (contd.)

It is known that

the positive answer to the Pompeiu problem Schiffers conjecture 6 and r > 0 such that NC() { Cd :

dj=1

2j = r

2} .

Thus, the interest in NC(). Also, it is of importance for inverse problems determining the shape of . A lot of publications, e.g. Agranovsky,Aviles, Berens

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