A Generalized Fejrs Theorem for Locally Compact Groups Huichi Huang - - PowerPoint PPT Presentation

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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A Generalized Fejér’s Theorem for Locally Compact Groups

Huichi Huang April 22, 2016 Shanghai Jiao Tong University

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Outline

Motivation Preliminaries The Main Theorem Some Special Cases

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Motivation

Theorem (Fejér, 1900) Let T = [0, 1) be the unit circle. For f in L1(T), if both the left limit and the right limit of f(x) exist at x0 in T (denoted by f(x0+) and f(x0−) respectively), then lim

n→∞ Kn ∗ f(x0) = 1

2[f(x0+) + f(x0−)]. Here Kn(x) = sin2(n+1)πx

(n+1) sin2 πx is the Fejér’s kernel.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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f(x0−) x0 f(x0+) Question: Could Fejér’s theorem be generalized to Td?

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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A partial answer: Theorem (A multivariable Fejér’s theorem) Assume that f belongs to L∞(Td) with d ≥ 2. If for x ∈ Td, the limit f(x, k) exists for every k in {0, 1}d, then lim

n→∞ Kd n ∗ f(x) = 1

2d ∑

k∈{0,1}d

f(x, k). Here Kd

n(x1, · · · , xd) = d

∏

j=1

Kn(xj) is the multivariable Fejér’s kernel.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Roughly speaking f(x, k) is the limit of f(y) when y approaches x via one of the 2d directions. x T2 = [0, 1)2

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Observations

{Kd

n}∞ n=1 is an approximate identity of L1(Td);

f(x, k)＝ lim

y→0 y∈Jk

f(x − y) for a Borel subset Jk of Td such that {Jk}k∈{0,1}d is a fjnite partition of Td and every Jk ∩ N is nonempty for any neighborhood N of 0; lim

n→∞

∫

Jk

Kd

n(x) dx = 1

2d . So it is possible to get a generalized Fejér’s theorem for locally compact groups.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Preliminaries

G-a locally compact Hausdorfg group with a fjxed left Haar measure µ. eG-the identity of G. L1(G)-the space of integrable functions (with respect to µ) on G. L∞(G)-the space of essentially bounded functions (with respect to µ) on G. The convolution f ∗ g for f in L1(G) ∪ L∞(G) and g in L1(G) is given by f ∗ g(x) = ∫

G

f(y)g(y−1x) dµ(y) for every x ∈ G.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Defjnition (Approximate identity) An approximate identity is a family of functions {Fθ}θ∈Θ in L1(G) such that

1 ∥Fθ∥L1(G) ≤ C for all θ; 2 ∫

G Fθ(x) dµ(x) = 1 for all θ;

3

lim

θ

∫

N c |Fθ(x)| dµ(x) = 0 for any neighborhood N of eG.

There always exists an approximate identity in L1(G).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Defjnition (Local partition) A fjnite collection {A1, A2, · · · , Ak} of Borel subsets of G is called a local partition (at eG) if the following are true:

1

Ai ∩ Aj = ∅ for 1 ≤ i ̸= j ≤ k;

2 µ(G \

k

∪

i=1

Ai) = 0;

3 each Aj ∩ N ̸= ∅ for any neighborhood N of eG.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Picture of a local partition

eG N G

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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A generalized Fejér’s theorem for locally compact groups

Theorem (H. 2015) Consider a locally compact group G with a fjxed left Haar measure µ. Let {Fθ}θ∈Θ be an approximate identity of L1(G). Assume that there exists a local partition {A1, A2, · · · , Ak} of G such that lim

θ

∫

Aj

Fθ(y) dµ(y) = λj for every 1 ≤ j ≤ k. For f in L∞(G), if there exists x in G such that lim

y→eG y∈Aj

f(y−1x) (denoted by f(x, Aj)) exists for every 1 ≤ j ≤ k, then lim

θ Fθ ∗ f(x) = k

∑

j=1

λjf(x, Aj).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Theorem (Continued) Moreover if lim

θ

sup

y∈N c |Fθ(y)| = 0 for any neighborhood N of

eG, then for every f in L1(G) ∪ L∞(G) such that each f(x, Aj) exists at some x in G, we have lim

θ Fθ ∗ f(x) = k

∑

j=1

λjf(x, Aj).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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d-torus Td

For k = (k1, · · · , kd) in {0, 1}d, defjne Ik =

d

∏

j=1

Ikj with I0 = (0, 1

2) and I1 = ( 1 2, 1).

Then ∫ 1

2

0 Kn(t) dt =

∫ 1

1 2 Kn(t) dt = 1

2 for all n ≥ 0.

So {Ik}k∈{0,1}d is a local partition of Td = [0, 1)d with ∫

Ik

Kd

n(x) dx = d

∏

j=1

∫

Ikj

Kn(xj) dxj = 1 2d for all k ∈ {0, 1}d and n ≥ 0.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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For f in L1(Td), defjne f(x, Ik) = lim

y→0 y∈Ik

f(x − y) for every k ∈ {0, 1}d. Corollary Let d ≥ 2. For f in L∞(Td) and x in Td, if each f(x, Ik) exists, then lim

n→∞ Kd n ∗ f(x) = 1

2d ∑

k∈{0,1}d

f(x, Ik).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Euclidean spaces Rd

The Poisson kernel Pθ(t) is given by Pθ(t) = 1 πθ(1 + t2

θ2 )

for all t ∈ R and θ > 0. Then {Pθ(t)}θ>0 is an approximate identity of L1(R) and lim

θ→0 sup t∈N c |Pθ(t)| = 0 for every neighborhood N of 0 in R.

Defjne Pd

θ (x) = d

∏

j=1

Pθ(xj) For any positive integer d, then {Pd

θ (x)}θ>0 is an approximate identity of L1(Rd).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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For k = (k1, · · · , kd) in {0, 1}d, defjne Jk =

d

∏

l=1

Jkl with J0 = (−∞, 0) and J1 = (0, ∞). Note that ∫ 0

−∞ Pθ(t) dt =

∫ ∞

0 Pθ(t) dt = 1 2 for all θ > 0.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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So {Jk}k∈{0,1}d is a local partition of Rd such that ∫

Jk

Pd

θ (x) dx = d

∏

l=1

∫

Jkl

Pθ(xl) dxl = 1 2d for all k ∈ {0, 1}d and θ > 0. For f in L1(Rd) or L∞(Rd), defjne f(x, Jk) = lim

y→0 y∈Jk

f(x − y) for every k ∈ {0, 1}d.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Corollary For f in L1(R) ∪ L∞(R) and x0 in R, if both the left and the right limits of f(x) at x0 exist (denoted by f(x0−) and f(x0+) respectively), then lim

θ→0 Pθ ∗ f(x0) = f(x0+) + f(x0−)

2 . On the other hand, for f in L∞(Rd) with d ≥ 2, if every f(x, Jk) exists for x ∈ Rd, then lim

θ→0 Pd θ ∗ f(x) = 1

2d ∑

k∈{0,1}d

f(x, Jk).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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The Wigner semicircle kernel Wθ(t) is given by Wθ = {

2 πθ2

√ θ2 − t2 when −θ ≤ t ≤ θ,

• therwise

for all t ∈ R and θ > 0.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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For λ in (0, 1), by shifting the graph of Wθ(t) along the t-axis, we get a new function Wθ,λ(t) satisfying ∫

(−∞,0)

Wθ,λ(t) dt = λ. Also note that every Wθ,λ(t) is compactly supported in a closed interval shrinking to {0} as θ goes to 0. Defjne Wd

θ,λ(x) = d

∏

j=1

Wθ,λ(xj) for any positive integer d, then {Wd

θ,λ(x)}θ>0 is an approximate identity of L1(Rd) and satisfjes

that lim

θ→0 sup x∈N c |Wd θ,λ(x)| = 0 for every neighborhood N of 0 in

Rd.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Corollary Fix λ in (0, 1). For f in L1(Rd) ∪ L∞(Rd), if every f(x, Jk) exists for x ∈ Rd, then lim

θ→0 Wd θ,λ ∗ f(x) =

∑

k∈{0,1}d d

∏

j=1

λ1−kj(1 − λ)kjf(x, Jk).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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The ax + b group

The ax + b group (denoted by F) is the group of affjne transformations on R consisting of maps x → ax + b with a > 0 and b in R. Fix a left Haar measure da db

a2

• f F = (0, ∞) × (−∞, ∞).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Defjne a family of functions {Φθ}0<θ<1 on F as follows: Φθ(a, b) = {

a2 πθ3

√ θ2 − b2 a ∈ (1 − θ, 1 + θ), b ∈ [−θ, θ];

• therwise

for all a > 0, b in R and 0 < θ < 1. Then {Φθ} is an approximate identity of F and lim

θ→0

sup

(a,b)∈N c |Φθ(a, b)| = 0 for every neighborhood N of

eF = (1, 0).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Consider the partition {A1, A2, A3, A4} of F with A1 = (0, 1) × (−∞, 0), A2 = (0, 1) × (0, ∞), A3 = (1, ∞) × (−∞, 0) and A4 = (1, ∞) × (0, ∞). Then {A1, A2, A3, A4} is a local partition of F and ∫

Aj

Φθ(a, b)da db a2 = 1 4 for every 1 ≤ j ≤ 4.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Corollary For f in L1(F) ∪ L∞(F), if every f(x, Aj) exists for x in F, then lim

θ→0 Φθ ∗ f(x) = 1

4

4

∑

j=1

f(x, Aj).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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The Heisenberg group H

The continuous Heisenberg group H = { ( 1 a b

0 1 c 0 0 1

) |a, b, c ∈ R}. As a topological space H = R3 and we fjx a Haar measure dadbdc of H.

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Defjne W3

θ(a, b, c) = Wθ(a)Wθ(b)Wθ(c) for every θ > 0 and

a, b, c in R. Then {W3

θ}θ>0 is an approximate identity of L1(H) and

lim

θ→0

sup

(a,b,c)∈N c |W3 θ(a, b, c)| = 0 for every neighborhood N of

eH = ( 1 0 0

0 1 0 0 0 1

) .

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Let {Jk}k∈{0,1}3 be the local partition defjned for R3. Then ∫

Jk W3 θ(a, b, c) dadbdc = 1 8 for every k ∈ {0, 1}3.

Corollary For f in L1(H) ∪ L∞(H), if every f(x, Jk) exists for x in H, then lim

θ→0 W3 θ ∗ f(x) = 1

8 ∑

k∈{0,1}3

f(x, Jk).

A Generalized Fejér’s Theorem for Locally Compact Groups 黄辉斥 Motivation Preliminaries The Main Theorem Some Special Cases

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Thank you.