[PPT] - Point distributions on the sphere: energy minimization, discrepancy, PowerPoint Presentation

SLIDE 1

Point distributions on the sphere: energy minimization, discrepancy, and more.

Dmitriy Bilyk University of Minnesota ICERM Semester Program on “Point Configurations in Geometry, Physics and Computer Science” Workshop “Optimal and Random Point Configurations” Providence, RI February 27, 2018

Dmitriy Bilyk Points on the sphere

SLIDE 2

Discrepancy

U: a set with a natural probability measure µ (e.g., [0, 1]d, Td, Sd, etc.)

Dmitriy Bilyk Points on the sphere

SLIDE 3

Discrepancy

U: a set with a natural probability measure µ (e.g., [0, 1]d, Td, Sd, etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps)

Dmitriy Bilyk Points on the sphere

SLIDE 4

Discrepancy

U: a set with a natural probability measure µ (e.g., [0, 1]d, Td, Sd, etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Let Z = {z1, . . . , zN} ⊂ U be an N-point set.

Dmitriy Bilyk Points on the sphere

SLIDE 5

Discrepancy

U: a set with a natural probability measure µ (e.g., [0, 1]d, Td, Sd, etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Let Z = {z1, . . . , zN} ⊂ U be an N-point set. Discrepancy of Z with respect to A: DA(Z) = sup

A∈A

#(Z ∩ A)

N − µ(A)

.

Dmitriy Bilyk Points on the sphere

SLIDE 6

Discrepancy

U: a set with a natural probability measure µ (e.g., [0, 1]d, Td, Sd, etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Let Z = {z1, . . . , zN} ⊂ U be an N-point set. Discrepancy of Z with respect to A: DA(Z) = sup

A∈A

#(Z ∩ A)

N − µ(A)

.

Optimal discrepancy wrt A: DN(A) = inf

#Z=N DA(Z).

Dmitriy Bilyk Points on the sphere

SLIDE 7

Discrepancy

U: a set with a natural probability measure µ (e.g., [0, 1]d, Td, Sd, etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Let Z = {z1, . . . , zN} ⊂ U be an N-point set. Discrepancy of Z with respect to A: DA(Z) = sup

A∈A

#(Z ∩ A)

N − µ(A)

.

Optimal discrepancy wrt A: DN(A) = inf

#Z=N DA(Z).

sup → L2-average: L2 discrepancy.

Dmitriy Bilyk Points on the sphere

SLIDE 8

Disk (ball) disrepancy on Td

Theorem (Montgomery; Beck; 80’s) For any N-point set Z = {z1, . . . , zN} ⊂ T2 ≃ [0, 1)2 there exists a disk D ⊂ T2 of radius 1/4 or 1/2 such that

# {1 ≤ i ≤ N : zi ∈ D}

N − |D|

N− 3

4 . Dmitriy Bilyk Points on the sphere

SLIDE 9

Disk (ball) disrepancy on Td

Theorem (Montgomery; Beck; 80’s) For any N-point set Z = {z1, . . . , zN} ⊂ T2 ≃ [0, 1)2 there exists a disk D ⊂ T2 of radius 1/4 or 1/2 such that

# {1 ≤ i ≤ N : zi ∈ D}

N − |D|

N− 3

4 .

Higher-dimensional version for Z ⊂ Td holds with N− 1

2 − 1 2d . Dmitriy Bilyk Points on the sphere

SLIDE 10

Disk (ball) disrepancy on Td

Theorem (Montgomery; Beck; 80’s) For any N-point set Z = {z1, . . . , zN} ⊂ T2 ≃ [0, 1)2 there exists a disk D ⊂ T2 of radius 1/4 or 1/2 such that

# {1 ≤ i ≤ N : zi ∈ D}

N − |D|

N− 3

4 .

Higher-dimensional version for Z ⊂ Td holds with N− 1

2 − 1 2d .

Is one radius enough? Still an open question!

Dmitriy Bilyk Points on the sphere

SLIDE 11

Disk (ball) disrepancy on Td

Theorem (Montgomery; Beck; 80’s) For any N-point set Z = {z1, . . . , zN} ⊂ T2 ≃ [0, 1)2 there exists a disk D ⊂ T2 of radius 1/4 or 1/2 such that

# {1 ≤ i ≤ N : zi ∈ D}

N − |D|

N− 3

4 .

Higher-dimensional version for Z ⊂ Td holds with N− 1

2 − 1 2d .

Is one radius enough? Still an open question! Sharp up to logarithms: jittered sampling.

Dmitriy Bilyk Points on the sphere

SLIDE 12

Disk (ball) disrepancy on Td

Theorem (Montgomery; Beck; 80’s) For any N-point set Z = {z1, . . . , zN} ⊂ T2 ≃ [0, 1)2 there exists a disk D ⊂ T2 of radius 1/4 or 1/2 such that

# {1 ≤ i ≤ N : zi ∈ D}

N − |D|

N− 3

4 .

Higher-dimensional version for Z ⊂ Td holds with N− 1

2 − 1 2d .

Is one radius enough? Still an open question! Sharp up to logarithms: jittered sampling. Sharp in L2 sense: lattice.

Dmitriy Bilyk Points on the sphere

SLIDE 13

L2 discrepancy: lattice vs jittered sampling

Denote D2

L2(Z) =

Td
# {Z ∩ Br(x)}

N − |Br|

2

dx

Dmitriy Bilyk Points on the sphere

SLIDE 14

L2 discrepancy: lattice vs jittered sampling

Denote D2

L2(Z) =

Td
# {Z ∩ Br(x)}

N − |Br|

2

dx Let LM = r1 M , . . . , rd M

: ri = 0, 1, . . . , M − 1
⊂ Td

Dmitriy Bilyk Points on the sphere

SLIDE 15

L2 discrepancy: lattice vs jittered sampling

Denote D2

L2(Z) =

Td
# {Z ∩ Br(x)}

N − |Br|

2

dx Let LM = r1 M , . . . , rd M

: ri = 0, 1, . . . , M − 1
⊂ Td and set

Dlattice(M) = DL2

LM
and

Djittered(M) = E DL2

Ljittered

M

.

Dmitriy Bilyk Points on the sphere

SLIDE 16

L2 discrepancy: lattice vs jittered sampling

Denote D2

L2(Z) =

Td
# {Z ∩ Br(x)}

N − |Br|

2

dx Let LM = r1 M , . . . , rd M

: ri = 0, 1, . . . , M − 1
⊂ Td and set

Dlattice(M) = DL2

LM
and

Djittered(M) = E DL2

Ljittered

M

.

Theorem (Chen, Travaglini, ’08) For d = 1 or 2, Dlattice(M) < Djittered(M).

Dmitriy Bilyk Points on the sphere

SLIDE 17

L2 discrepancy: lattice vs jittered sampling

Denote D2

L2(Z) =

Td
# {Z ∩ Br(x)}

N − |Br|

2

dx Let LM = r1 M , . . . , rd M

: ri = 0, 1, . . . , M − 1
⊂ Td and set

Dlattice(M) = DL2

LM
and

Djittered(M) = E DL2

Ljittered

M

.

Theorem (Chen, Travaglini, ’08) For d = 1 or 2, Dlattice(M) < Djittered(M). For large d, Dlattice(M) > Djittered(M).

Dmitriy Bilyk Points on the sphere

SLIDE 18

L2 discrepancy: lattice vs jittered sampling

Denote D2

L2(Z) =

Td
# {Z ∩ Br(x)}

N − |Br|

2

dx Let LM = r1 M , . . . , rd M

: ri = 0, 1, . . . , M − 1
⊂ Td and set

Dlattice(M) = DL2

LM
and

Djittered(M) = E DL2

Ljittered

M

.

Theorem (Chen, Travaglini, ’08) For d = 1 or 2, Dlattice(M) < Djittered(M). For large d, Dlattice(M) > Djittered(M). unless d ≡ 1 mod 4 and d > 1.

Dmitriy Bilyk Points on the sphere

SLIDE 19

Montgomery’s lower bound

Let Z = {z1, . . . , zN} ⊂ T2. Then # {Z ∩ Br(x)} N − |Br| = (1Br ∗ DZ) (x), where DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). D2

L2(Z) =

n∈Z2

| 1Br(n)|2 · | DZ(n)|2

1Br(n) =

r |n|J1(2π|n|r)

Bessel function of the first kind

J1(t) =

2

πt cos(t − 3π/4) + O(t−3/2)

Dmitriy Bilyk Points on the sphere

SLIDE 20

Montgomery’s lower bound

Let Z = {z1, . . . , zN} ⊂ T2. Then # {Z ∩ Br(x)} N − |Br| = (1Br ∗ DZ) (x), where DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). D2

L2(Z) =

n∈Z2

| 1Br(n)|2 · | DZ(n)|2

1Br(n) =

r |n|J1(2π|n|r)

Bessel function of the first kind

J1(t) =

2

πt cos(t − 3π/4) + O(t−3/2)

Dmitriy Bilyk Points on the sphere

SLIDE 21

Montgomery’s lower bound

Let Z = {z1, . . . , zN} ⊂ T2. Then # {Z ∩ Br(x)} N − |Br| = (1Br ∗ DZ) (x), where DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). D2

L2(Z) =

n∈Z2

| 1Br(n)|2 · | DZ(n)|2

1Br(n) =

r |n|J1(2π|n|r)

Bessel function of the first kind

J1(t) =

2

πt cos(t − 3π/4) + O(t−3/2)

Dmitriy Bilyk Points on the sphere

SLIDE 22

Montgomery’s lower bound

Let Z = {z1, . . . , zN} ⊂ T2. Then # {Z ∩ Br(x)} N − |Br| = (1Br ∗ DZ) (x), where DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). D2

L2(Z) =

n∈Z2

| 1Br(n)|2 · | DZ(n)|2

1Br(n) =

r |n|J1(2π|n|r)

Bessel function of the first kind

J1(t) =

2

πt cos(t − 3π/4) + O(t−3/2)

Dmitriy Bilyk Points on the sphere

SLIDE 23

Montgomery’s lower bound

Let Z = {z1, . . . , zN} ⊂ T2. Then # {Z ∩ Br(x)} N − |Br| = (1Br ∗ DZ) (x), where DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). D2

L2(Z) =

n∈Z2

| 1Br(n)|2 · | DZ(n)|2

1Br(n) =

r |n|J1(2π|n|r)

Bessel function of the first kind

J1(t) =

2

πt cos(t − 3π/4) + O(t−3/2)

Dmitriy Bilyk Points on the sphere

SLIDE 24

Montgomery’s lower bound

Let Z = {z1, . . . , zN} ⊂ T2. Then # {Z ∩ Br(x)} N − |Br| = (1Br ∗ DZ) (x), where DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). D2

L2(Z) =

n∈Z2

| 1Br(n)|2 · | DZ(n)|2

1Br(n) =

r |n|J1(2π|n|r)

Bessel function of the first kind

J1(t) =

2

πt cos(t − 3π/4) + O(t−3/2)

Dmitriy Bilyk Points on the sphere

SLIDE 25

Bessel functions

J1(t)

Dmitriy Bilyk Points on the sphere

SLIDE 26

Bessel functions

t · J2

1(t)

Dmitriy Bilyk Points on the sphere

SLIDE 27

Bessel functions

t · J2

1(t) and t · J2 1(2t)

Dmitriy Bilyk Points on the sphere

SLIDE 28

Bessel functions

t · (J2

1(t) + J2 1(2t))

Dmitriy Bilyk Points on the sphere

SLIDE 29

Bessel functions

J2

1(t) + J2 1(2t) 1 t

Thus | 1B1/4(n)|2 + | 1B1/2(n)|2

1 |n|3

Dmitriy Bilyk Points on the sphere

SLIDE 30

Exponential sums

DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). For n = 0,

DZ(n) = 1

N

i=1

e−2πi n·zi. Montgomery’s esimate:

n≤X
N
i=1

e−2πin·zi

2

X2N and taking X ≈ N1/2 leads to the discrepancy bound. Refinement (Steinerberger, ’17):

n≤X
N
i=1

e−2πin·zi

2
N
i,j=1

X2 1 + X4zi − zj4

Dmitriy Bilyk Points on the sphere

SLIDE 31

Exponential sums

DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). For n = 0,

DZ(n) = 1

N

i=1

e−2πi n·zi. Montgomery’s esimate:

n≤X
N
i=1

e−2πin·zi

2

X2N and taking X ≈ N1/2 leads to the discrepancy bound. Refinement (Steinerberger, ’17):

n≤X
N
i=1

e−2πin·zi

2
N
i,j=1

X2 1 + X4zi − zj4

Dmitriy Bilyk Points on the sphere

SLIDE 32

Exponential sums

DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). For n = 0,

DZ(n) = 1

N

i=1

e−2πi n·zi. Montgomery’s esimate:

n≤X
N
i=1

e−2πin·zi

2

X2N and taking X ≈ N1/2 leads to the discrepancy bound. Refinement (Steinerberger, ’17):

n≤X
N
i=1

e−2πin·zi

2
N
i,j=1

X2 1 + X4zi − zj4

Dmitriy Bilyk Points on the sphere

SLIDE 33

Exponential sums

DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). For n = 0,

DZ(n) = 1

N

i=1

e−2πi n·zi. Montgomery’s esimate:

n≤X
N
i=1

e−2πin·zi

2

X2N and taking X ≈ N1/2 leads to the discrepancy bound. Refinement (Steinerberger, ’17):

n≤X
N
i=1

e−2πin·zi

2
N
i,j=1

X2 1 + X4zi − zj4

Dmitriy Bilyk Points on the sphere

SLIDE 34

Exponential sums

DZ = 1 N

N

i=1

δzi − λ2 (discrepancy measure). For n = 0,

DZ(n) = 1

N

i=1

e−2πi n·zi. Montgomery’s esimate:

n≤X
N
i=1

e−2πin·zi

2

X2N and taking X ≈ N1/2 leads to the discrepancy bound. Refinement (Steinerberger, ’17):

n≤X
N
i=1

e−2πin·zi

2
N
i,j=1

X2 1 + X4zi − zj4

Dmitriy Bilyk Points on the sphere

SLIDE 35

Refined discrepancy estimate

For any N-point set Z = {z1, . . . , zN} ⊂ T2 there exists a disk D ⊂ T2 of radius 1/4 or 1/2 such that

Dmitriy Bilyk Points on the sphere

SLIDE 36

Refined discrepancy estimate

For any N-point set Z = {z1, . . . , zN} ⊂ T2 there exists a disk D ⊂ T2 of radius 1/4 or 1/2 such that Montgomery, ’89:

# {zi ∈ D}

N − |D|

N− 3

4 . Dmitriy Bilyk Points on the sphere

SLIDE 37

Refined discrepancy estimate

For any N-point set Z = {z1, . . . , zN} ⊂ T2 there exists a disk D ⊂ T2 of radius 1/4 or 1/2 such that Montgomery, ’89:

# {zi ∈ D}

N − |D|

N− 3

4 .

Steinerberger, ’17:

# {zi ∈ D}

N − |D|

N− 7

4

 

N

i,j=1

N 1 + N2zi − zj4  

1 2

.

Dmitriy Bilyk Points on the sphere

SLIDE 38

Spherical cap discrepancy

For x ∈ Sd, t ∈ [−1, 1] define spherical caps: C(x, t) = {y ∈ Sd : x, y ≥ t}.

Dmitriy Bilyk Points on the sphere

SLIDE 39

Spherical cap discrepancy

For x ∈ Sd, t ∈ [−1, 1] define spherical caps: C(x, t) = {y ∈ Sd : x, y ≥ t}. For a finite set Z = {z1, z2, ..., zN} ⊂ Sd define Dcap(Z) = sup

x∈Sd,t∈[−1,1]

#
Z ∩ C(x, t)
N

− σ

C(x, t)
.

Dmitriy Bilyk Points on the sphere

SLIDE 40

Spherical cap discrepancy

For x ∈ Sd, t ∈ [−1, 1] define spherical caps: C(x, t) = {y ∈ Sd : x, y ≥ t}. For a finite set Z = {z1, z2, ..., zN} ⊂ Sd define Dcap(Z) = sup

x∈Sd,t∈[−1,1]

#
Z ∩ C(x, t)
N

− σ

C(x, t)
.

Theorem (Beck, ’84) There exist constants cd, Cd > 0 such that cdN− 1

2− 1 2d ≤

inf

#Z=N Dcap(Z) ≤ CdN− 1

2 − 1 2d

log N.

Dmitriy Bilyk Points on the sphere

SLIDE 41

Spherical cap discrepancy: refinement of lower bound

Theorem (Beck, ’84) For any Z = {z1, . . . , zN} ⊂ Sd Dcap(Z) N− 1

2− 1 2d . Dmitriy Bilyk Points on the sphere

SLIDE 42

Spherical cap discrepancy: refinement of lower bound

Theorem (Beck, ’84) For any Z = {z1, . . . , zN} ⊂ Sd Dcap(Z) N− 1

2− 1 2d .

Theorem (DB, Dai, Steinerberger, ’17) For any Z = {z1, . . . , zN} ⊂ Sd Dcap(Z) N− 1

2 − 1 2d

  1 N

N

i,j=1

log (2 + N1/dzi − zj) (1 + N1/dzi − zj)d+1  

1/2

.

Dmitriy Bilyk Points on the sphere

SLIDE 43

Discrepancy and energy: Stolarsky Principle

Define the spherical cap L2 discrepancy D2

cap,L2(Z) =

Sd

1

−1

#
Z ∩ C(x, t)
N

− σ

C(x, t)
2

dt dσ(x).

Dmitriy Bilyk Points on the sphere

SLIDE 44

Discrepancy and energy: Stolarsky Principle

Define the spherical cap L2 discrepancy D2

cap,L2(Z) =

Sd

1

−1

#
Z ∩ C(x, t)
N

− σ

C(x, t)
2

dt dσ(x). Theorem (Stolarsky invariance principle) For any finite set Z = {z1, ..., zN} ⊂ Sd 1 N2

N

i,j=1

zi−zj + cd

DL2,cap

2 = const =

Sd
Sd x − y dσ(x)dσ(y).

Dmitriy Bilyk Points on the sphere

SLIDE 45

Discrepancy and energy: Stolarsky Principle

Define the spherical cap L2 discrepancy D2

cap,L2(Z) =

Sd

1

−1

#
Z ∩ C(x, t)
N

− σ

C(x, t)
2

dt dσ(x). Theorem (Stolarsky invariance principle) For any finite set Z = {z1, ..., zN} ⊂ Sd cd

Dcap,L2(Z)

2 = =

Sd
Sd x − y dσ(x)dσ(y) − 1

N2

N

i,j=1

zi − zj.

Dmitriy Bilyk Points on the sphere

SLIDE 46

Discrepancy and energy: Stolarsky Principle

Define the spherical cap L2 discrepancy D2

cap,L2(Z) =

Sd

1

−1

#
Z ∩ C(x, t)
N

− σ

C(x, t)
2

dt dσ(x). Theorem (Stolarsky invariance principle) For any finite set Z = {z1, ..., zN} ⊂ Sd cd

Dcap,L2(Z)

2 = =

Sd
Sd x − y dσ(x)dσ(y) − 1

N2

N

i,j=1

zi − zj. Stolarsky ’73, Brauchart, Dick ’12, DB, Dai, Matzke ’17

Dmitriy Bilyk Points on the sphere

SLIDE 47

Spherical caps: L2 Stolarsky Principle

Define the spherical cap discrepancy of fixed height t:

D(t)

L2,cap(Z)

2 :=

Sd
1

N

j=1

1C(x,t)(zj) − σ

C(x, t)
2

dσ(x)

Dmitriy Bilyk Points on the sphere

SLIDE 48

Spherical caps: L2 Stolarsky Principle

Define the spherical cap discrepancy of fixed height t:

D(t)

L2,cap(Z)

2 :=

Sd
1

N

j=1

1C(x,t)(zj) − σ

C(x, t)
2

dσ(x)

D(t)

L2,cap(Z)

2 = 1 N2

N

i,j=1

σ

C(zi, t) ∩ C(zj, t)
−
σ
C(p, t)

2 .

Dmitriy Bilyk Points on the sphere

SLIDE 49

Spherical caps: L2 Stolarsky Principle

Define the spherical cap discrepancy of fixed height t:

D(t)

L2,cap(Z)

2 :=

Sd
1

N

j=1

1C(x,t)(zj) − σ

C(x, t)
2

dσ(x)

D(t)

L2,cap(Z)

2 = 1 N2

N

i,j=1

σ

C(zi, t) ∩ C(zj, t)
−
σ
C(p, t)

2 . Averaging over t ∈ [−1, 1] 1

−1

σ

C(x, t) ∩ C(y, t)
dt

= 1 − Cdx − y

Dmitriy Bilyk Points on the sphere

SLIDE 50

Spherical caps: L2 Stolarsky Principle

Define the spherical cap discrepancy of fixed height t:

D(t)

L2,cap(Z)

2 :=

Sd
1

N

j=1

1C(x,t)(zj) − σ

C(x, t)
2

dσ(x)

D(t)

L2,cap(Z)

2 = 1 N2

N

i,j=1

σ

C(zi, t) ∩ C(zj, t)
−
σ
C(p, t)

2 . Averaging over t ∈ [−1, 1] 1

−1

σ

C(x, t) ∩ C(y, t)
dt

= 1 − Cdx − y Taking t = 0 (i.e. hemispheres) σ

C(x, 0) ∩ C(y, 0)
= 1

2

1 − d(x, y)
,

where d(x, y) = arccos(x·y)

π

(normalized geodesic distance).

Dmitriy Bilyk Points on the sphere

SLIDE 51

Stolarsky principle for hemispheres

Theorem (DB, Dai, Matzke ’17, Skriganov ’17) [DL2,hem(Z)]2 = [D(0)

L2,cap(Z)]2

= 1 2  

Sd
Sd

d(x, y) dσ(x) dσ(y) − 1 N2

N

i,j=1

d(zi, zj)   .

Dmitriy Bilyk Points on the sphere

SLIDE 52

Stolarsky principle for hemispheres

Theorem (DB, Dai, Matzke ’17, Skriganov ’17) [DL2,hem(Z)]2 = [D(0)

L2,cap(Z)]2 = 1

2   1 2 − 1 N2

N

i,j=1

d(zi, zj)   .

Dmitriy Bilyk Points on the sphere

SLIDE 53

Stolarsky principle for hemispheres

Theorem (DB, Dai, Matzke ’17, Skriganov ’17) [DL2,hem(Z)]2 = [D(0)

L2,cap(Z)]2 = 1

2   1 2 − 1 N2

N

i,j=1

d(zi, zj)   . Corollary (DB, Dai, Matzke ’17) For any Z = {z1, ..., zN} ⊂ Sd 1 N2

N

i,j=1

d(zi, zj) ≤ 1 2 with equality if and only if Z is symmetric. (This solves a 1959 conjecture of Fejes T´

th.)

Dmitriy Bilyk Points on the sphere

SLIDE 54

Discrete energy

Let Z = {z1, ..., zN} ⊂ Sd and let F : [−1, 1] → R. Discrete energy: EF (Z) = 1 N2

N

i,j=1

F(zi · zj) Questions: What are the minimizing configurations? Almost minimizers? Lower bounds?

Dmitriy Bilyk Points on the sphere

SLIDE 55

Energy integral

Let µ be a Borel probability measure on Sd. Energy integral IF (µ) =

Sd
Sd

F(x · y) dµ(x)dµ(y). i.e. EF (Z) = IF 1 N

δzi
Questions:

What are the minimizers? Is σ a minimizer? Is it unique?

Dmitriy Bilyk Points on the sphere

SLIDE 56

Spherical harmonics and energy minimization

Gegenbauer polynomials form an orthogonal basis on the space L2([−1, 1], wλ) with weight wλ(t) = (1 − t2)λ− 1

2 :

F(t) ∼

∞

n=0
F(n; λ)n + λ

λ Cλ

n(t)

Dmitriy Bilyk Points on the sphere

SLIDE 57

Spherical harmonics and energy minimization

Gegenbauer polynomials form an orthogonal basis on the space L2([−1, 1], wλ) with weight wλ(t) = (1 − t2)λ− 1

2 :

F(t) ∼

∞

n=0
F(n; λ)n + λ

λ Cλ

n(t)

IF (µ) =

∞

n=0
F(n; λ)n + λ

λ

Sd
Sd

Cλ

n(x · y)dµ(x)dµ(y)

Dmitriy Bilyk Points on the sphere

SLIDE 58

Spherical harmonics and energy minimization

Gegenbauer polynomials form an orthogonal basis on the space L2([−1, 1], wλ) with weight wλ(t) = (1 − t2)λ− 1

2 :

F(t) ∼

∞

n=0
F(n; λ)n + λ

λ Cλ

n(t)

IF (µ) =

∞

n=0
F(n; λ)n + λ

λ

Sd
Sd

Cλ

n(x · y)dµ(x)dµ(y)

= F(0; λ) +

∞

n=1
F(n; λ)bn,µ,

where bn,µ ≥ 0,

Dmitriy Bilyk Points on the sphere

SLIDE 59

Spherical harmonics and energy minimization

Gegenbauer polynomials form an orthogonal basis on the space L2([−1, 1], wλ) with weight wλ(t) = (1 − t2)λ− 1

2 :

F(t) ∼

∞

n=0
F(n; λ)n + λ

λ Cλ

n(t)

IF (µ) =

∞

n=0
F(n; λ)n + λ

λ

Sd
Sd

Cλ

n(x · y)dµ(x)dµ(y)

= F(0; λ) +

∞

n=1
F(n; λ)bn,µ,

where bn,µ ≥ 0, ≥ F(0; λ) = IF (σ), if

F(n; λ) ≥ 0.

Dmitriy Bilyk Points on the sphere

SLIDE 60

Positive definite functions on the sphere

Lemma Let F ∈ C[−1, 1]. IF (µ) is minimized by σ iff F(n, λ) ≥ 0 for all n ≥ 1.

Dmitriy Bilyk Points on the sphere

SLIDE 61

Positive definite functions on the sphere

Lemma Let F ∈ C[−1, 1]. IF (µ) is minimized by σ iff F(n, λ) ≥ 0 for all n ≥ 1. σ is the unique minimizer of IF (µ) iff F(n, λ) > 0 for all n ≥ 1.

Dmitriy Bilyk Points on the sphere

SLIDE 62

Positive definite functions on the sphere

Lemma Let F ∈ C[−1, 1]. IF (µ) is minimized by σ iff F(n, λ) ≥ 0 for all n ≥ 1. σ is the unique minimizer of IF (µ) iff F(n, λ) > 0 for all n ≥ 1. A function F ∈ C[−1, 1] is called positive definite on the sphere Sd if for any set of points Z = {z1, ..., zN} ⊂ Sd, the matrix

F(zi · zj)

N

i,j=1 is positive semidefinite, i.e.

i,j

F(zi · zj)cicj ≥ 0 for all ci ∈ R.

Dmitriy Bilyk Points on the sphere

SLIDE 63

Positive definite functions on the sphere

Lemma For a function F ∈ C[−1, 1] the following are equivalent:

i F is positive definite on Sd.

Dmitriy Bilyk Points on the sphere

SLIDE 64

Positive definite functions on the sphere

Lemma For a function F ∈ C[−1, 1] the following are equivalent:

i F is positive definite on Sd. ii Gegenbauer coefficients of F are non-negative, i.e.

F(n, λ) ≥ 0 for all n ≥ 0.

Dmitriy Bilyk Points on the sphere

SLIDE 65

Positive definite functions on the sphere

Lemma For a function F ∈ C[−1, 1] the following are equivalent:

i F is positive definite on Sd. ii Gegenbauer coefficients of F are non-negative, i.e.

F(n, λ) ≥ 0 for all n ≥ 0.

iii For any signed measure µ ∈ B the energy integral is

non-negative: IF (µ) ≥ 0.

Dmitriy Bilyk Points on the sphere

SLIDE 66

Positive definite functions on the sphere

Lemma For a function F ∈ C[−1, 1] the following are equivalent:

i F is positive definite on Sd. ii Gegenbauer coefficients of F are non-negative, i.e.

F(n, λ) ≥ 0 for all n ≥ 0.

iii For any signed measure µ ∈ B the energy integral is

non-negative: IF (µ) ≥ 0.

iv There exists a function f ∈ L2 wλ[−1, 1] such that

F(x · y) =

Sd f(x · z)f(z · y) dσ(z),

x, y ∈ Sd, i.e. F is the spherical convolution of f with itself.

f(n, λ)2 =

F(n, λ)

Dmitriy Bilyk Points on the sphere

SLIDE 67

Generalized Stolarsky principle

Define the L2 discrepancy of a Borel probability measure µ w.r.t. the function f : [−1, 1] → R as D2

L2,f(µ) =

Sd
Sd

f(x · y)dµ(y) −

Sd

f(x · y)dσ(y)

2

dσ(x).

Dmitriy Bilyk Points on the sphere

SLIDE 68

Generalized Stolarsky principle

Define the L2 discrepancy of a Borel probability measure µ w.r.t. the function f : [−1, 1] → R as D2

L2,f(µ) =

Sd
Sd

f(x · y)d

µ − σ
(y)
2

dσ(x).

Dmitriy Bilyk Points on the sphere

SLIDE 69

Generalized Stolarsky principle

Define the L2 discrepancy of a Borel probability measure µ w.r.t. the function f : [−1, 1] → R as D2

L2,f(µ) =

Sd
Sd

f(x · y)d

µ − σ
(y)
2

dσ(x). Theorem (DB, R. Matzke, F. Dai, ’17) Generalized Stolarsky principle: Let F be positive definite and f as in (iv), then IF (µ) − IF (σ) = D2

L2,f(µ).

Dmitriy Bilyk Points on the sphere

SLIDE 70

Generalized Stolarsky principle

Define the L2 discrepancy of a Borel probability measure µ w.r.t. the function f : [−1, 1] → R as D2

L2,f(µ) =

Sd
Sd

f(x · y)d

µ − σ
(y)
2

dσ(x). Theorem (DB, R. Matzke, F. Dai, ’17) Generalized Stolarsky principle: Let F be positive definite and f as in (iv), then IF (µ) − IF (σ) = D2

L2,f(µ).

Important ingredient: IF (µ) − IF (σ) = IF (µ − σ).

Dmitriy Bilyk Points on the sphere

SLIDE 71

Discrepancy/energy bounds

Theorem (DB, F. Dai, ’17) Assume that F is positive definite and f as in (iv). Let Z = {z1, ..., zN} ⊂ Sd and µ = 1 N

N

i=1

δzi. Upper bound: inf

#Z=N D2 L2,f(µ) 1

N max

0≤θN− 1

d

F(1) − F(cos θ)
.

Dmitriy Bilyk Points on the sphere

SLIDE 72

Discrepancy/energy bounds

Theorem (DB, F. Dai, ’17) Assume that F is positive definite and f as in (iv). Let Z = {z1, ..., zN} ⊂ Sd and µ = 1 N

N

i=1

δzi. Upper bound: inf

#Z=N D2 L2,f(µ) 1

N max

0≤θN− 1

d

F(1) − F(cos θ)
.

Lower bound: D2

L2,f,N

min

1≤kN1/d

F(k, λ).

Dmitriy Bilyk Points on the sphere

SLIDE 73

Refined lower bounds (DB, Dai, Steinerberger, ’17)

Montgomery-type lemma:

L

n=0

dn

k=1
N
j=1

Yn,k(xj)

2

Ld

N

i,j=1

log(2 + Lzi − zj) (1 + Lzi − zj)d+1 .

Dmitriy Bilyk Points on the sphere

SLIDE 74

Refined lower bounds (DB, Dai, Steinerberger, ’17)

Montgomery-type lemma:

L

n=0

dn

k=1
N
j=1

Yn,k(xj)

2

Ld

N

i,j=1

log(2 + Lzi − zj) (1 + Lzi − zj)d+1 . Discrepancy bound: D2

L2,f(Z) 1

N · min

1≤kN1/d

F(n, λ)·

N

i,j=1

log(2 + N1/dzi − zj) (1 + N1/dzi − zj)d+1

Dmitriy Bilyk Points on the sphere