Approximate Identities

1.2 Convolution and Approximate Identities

1.2.4 Approximate Identities

Fig. 1.2 The Fej´er kernel F5 plotted on the interval [−¹₂,¹₂].

1 π Z

|x|≥δ

1 ε

1

(x/ε)²+1dx=1−2

πarctan(δ/ε)→0 asε→0, for allδ>0. The functionPεis called thePoisson kernel.

Example 1.2.18.On the circle groupT¹let F_N(t) =

N j=−N

∑

1− |j|

N+1

e^{2πi jt}= 1 N+1

sin(π(N+1)t) sin(πt)

2

. (1.2.27) To check the previous equality we use that

sin²(x) = (2−e^2ix−e^−2ix)/4,

and we carry out the calculation.F_N is called theFej´er kernel. To see that the se- quence{F_N}_N is an approximate identity, we check conditions (i), (ii), and (iii) in Definition 1.2.15. Property (iii) follows from the expression givingF_N in terms of sines, while property (i) follows from the expression givingF_Nin terms of exponen- tials. Property (ii) is identical to property (i), sinceF_Nis nonnegative.

Next comes the basic theorem concerning approximate identities.

Theorem 1.2.19.Let k_ε be an approximate identity on a locally compact group G with left Haar measureλ.

(1) If f ∈L^p(G)for1≤p<∞, then

kε∗f−f

L^p(G)→0asε→0.

(2) When p=∞, the following is valid: If f is continuous in a neighborhood of a a compact subset K of G, then

kε∗f−f

L^∞(K)→0asε→0.

Proof. We start with the case 1≤p<∞. We recall that continuous functions with compact support are dense in L^p of locally compact Hausdorff spaces equipped with measures arising from nonnegative linear functionals (see Hewitt and Ross [125], Theorem 12.10). For a continuous function with compact supportgwe have

|g(h⁻¹x)−g(x)|^p≤(2 g

L^∞)^p forhin a relatively compact neighborhood of the origine, and by the Lebesgue dominated convergence theorem we obtain

-0.4 -0.2 0.2 0.4

1 2 3 4 5 6

Z

G

|g(h⁻¹x)−g(x)|^pdλ(x)→0 (1.2.28) ash→e. Now approximate a givenf inL^p(G)by a continuous function with com- pact supportgto deduce that

Z

G

|f(h⁻¹x)−f(x)|^pdλ(x)→0 as h→e. (1.2.29) Because of (1.2.29), given aδ >0 there exists a neighborhoodV ofesuch that

h∈V =⇒ Z

G

|f(h⁻¹x)−f(x)|^pdλ(x)<

δ 2c

p

, (1.2.30)

wherecis the constant that appears in Definition 1.2.15 (i). Sincek_ε has integral one for allε>0, we have

(k_ε∗f)(x)−f(x) = (k_ε∗f)(x)−f(x) Z

G

k_ε(y)dλ(y)

= Z

G

(f(y⁻¹x)−f(x))k_ε(y)dλ(y)

= Z

V

(f(y⁻¹x)−f(x))k_ε(y)dλ(y) +

Z

V^c

(f(y⁻¹x)−f(x))k_ε(y)dλ(y).

(1.2.31)

Now takeL^pnorms inxin (1.2.31). In view of (1.2.30),

Z

V

(f(y⁻¹x)−f(x))k_ε(y)dλ(y)

L^p(G,dλ(x))

≤ Z

V

f(y⁻¹x)−f(x)

L^p(G,dλ(x))|k_ε(y)|dλ(y)

≤ Z

V

δ

2c|k_ε(y)|dλ(y)<δ 2,

(1.2.32)

while Z

V^c

(f(y⁻¹x)−f(x))k_ε(y)dλ(y)

L^p(G,dλ(x))

≤ Z

V^c

2 f

L^p(G)|k_ε(y)|dλ(y)<δ 2,

(1.2.33)

provided we have that Z

V^c

|k_ε(x)|dλ(x)< δ 4

f _Lp

. (1.2.34)

Chooseε0>0 such that (1.2.34) is valid forε<ε0by property (iii). Now (1.2.32) and (1.2.33) imply the required conclusion.

The case p=∞follows similarly. Since f is uniformly continuous onK, given δ >0 find a neighborhoodV ofe∈Gsuch that

h∈V =⇒ |f(h⁻¹x)−f(x)|< δ

2c for allx∈K, (1.2.35)

wherecis as in Definition 1.2.15(i), and then find anε0>0 such that for 0<ε<ε0

we have

Z

V^c

|k_ε(y)|dλ(y)< δ 4

f _L∞

. (1.2.36)

Using (1.2.35) and (1.2.36), we easily conclude that sup

x∈K

|(k_ε∗f)(x)−f(x)|

≤ Z

V

|kε(y)|sup

x∈K

|f(y⁻¹x)−f(x)|dλ(y) + Z

V^c

|kε(y)|sup

x∈K

|f(y⁻¹x)−f(x)|dλ(y)

≤ δ 2+δ

2 =δ,

which shows thatkε∗f converges uniformly to f onKasε→0.

Remark 1.2.20.Observe that if Haar measure satisfies (1.2.12), then the conclusion of Theorem 1.2.19 also holds for f∗k_ε.

A simple modification in the proof of Theorem 1.2.19 yields the following vari- ant.

Theorem 1.2.21.Let kεbe a family of functions on a locally compact group G that satisfies properties (i) and (iii) of Definition 1.2.15 and also

Z

G

k_ε(x)dλ(x) =a∈C, for allε>0. Let f ∈L^p(G)for some1≤p≤∞.

(a) If1≤p<∞, then

k_ε∗f−a f

L^p(G)→0asε→0. (b) If p=∞and f is continuous on a compact K⊆G, then

k_ε∗f−a f

L^∞(K)→0 asε→0.

Remark 1.2.22.With the notation of Theorem 1.2.21, if f is continuous and tends to zero at infinity, then

k_ε∗f−a f

L^∞(G) →0. To see this, simply observe that outside a compact subset ofG, bothkε∗f,a f have smallL^∞norm, while inside a compact subset ofG, uniform convergence holds.

Exercises

1.2.1.LetGbe a locally compact group and let f,ginL¹(G)be supported in the subsetsA andBof G, respectively. Prove that f∗g is supported in the algebraic product setAB.

1.2.2.For a function f on a locally compact groupGandt∈G, let^tf(x) = f(tx) and f^t(x) =f(xt). Show that

tf∗g=^t(f∗g) and f∗g^t= (f∗g)^t whenever f,g∈L¹(G), equipped with left Haar measure.

1.2.3.LetGbe a locally compact group with left Haar measure. Let f ∈L^p(G) and ˜g ∈L^p0(G), where 1< p<∞; recall that g(x) =e g(x⁻¹). For t,x∈G, let

tg(x) =g(tx). Show that for any ε>0 there exists a relatively compact symmet- ric neighborhood of the originU such thatu∈U implies

^ueg−eg

L^p0(G)<ε and therefore

|(f∗g)(v)−(f∗g)(w)|<

f _Lpε whenevervw⁻¹∈U.

1.2.4.LetGbe a locally compact group and let 1≤p≤∞. Let f ∈L^p(G)andµ be a finite Borel measure onGwith total variation

µ . Define

(µ∗f)(x) = Z

G

f(y⁻¹x)dµ(y).

Show that ifµis an absolutely continuous measure, then the preceding definition extends (1.2.4). Prove that

µ∗f

L^p(G)≤ µ

f

L^p(G).

1.2.5.Show that a Haar measureλ for the multiplicative group of all positive real numbers is

λ(A) = Z ∞

0

χ_A(t)dt t .

1.2.6.LetG=R²\ {(0,y): y∈R}with group operation(x,y)(z,w) = (xz,xw+y).

[Think ofGas the group of all 2×2 matrices with bottom row(0,1)and nonzero top left entry.] Show that a left Haar measure onGis

λ(A) = Z +∞

−∞

Z +∞

−∞ χ_A(x,y)dx dy x² , while a right Haar measure onGis

ρ(A) = Z _+∞

−∞

Z _+∞

−∞ χ_A(x,y)dx dy

|x| .

1.2.7.(Hardy [118], [119]) Use Theorem 1.2.10 to prove that Z _∞

0

1 x

Z _x 0

|f(t)|dt p

dx ¹_p

≤ p p−1

f

L^p(0,∞), Z _∞

0

Z _∞ x

|f(t)|dt p

dx ¹_p

≤p Z _∞

0

|f(t)|^pt^pdt ¹_p

, when 1<p<∞.

Hint:On the multiplicative group(R⁺,^dt_t )consider the convolution of the function

|f(x)|x^1p with the functionx⁻

1 p0

χ_[1,∞)and the convolution of the function|f(x)|x^1+1p withx^1pχ_(0,1].

1.2.8.(G. H. Hardy) Let 0<b<∞and 1≤p<∞. Prove that _{Z ∞}

0

_{Z x}

0

|f(t)|dt p

x^−b−1dx ¹_p

≤ p b

_{Z ∞}

0

|f(t)|^pt^p−b−1dt ¹_p

,

_{Z ∞}

0

_{Z ∞}

x

|f(t)|dt p

x^b−1dx ¹_p

≤ p b

_{Z ∞}

0

|f(t)|^pt^p+b−1dt ¹_p

.

Hint:On the multiplicative group(R⁺,^dt_t )consider the convolution of the function

|f(x)|x^1−bp withx^−bpχ_[1,∞)and of the function|f(x)|x^1+bp withx^bpχ_(0,1].

1.2.9.OnRⁿletT(f) = f∗K, whereKis a positive L¹function and f is inL^p, 1≤p≤∞. Prove that the operator norm ofT:L^p→L^pis equal to

K L¹. Hint:Clearly,kTk_Lp→L^p≤ kKk_L1. Conversely, fix 0<ε<1 and letNbe a positive integer. LetχN=χ_B(0,N)and for anyR>0 letK_R=Kχ_B(0,R), whereB(x,R)is the ball of radiusRcentered atx. Observe that for|x| ≤(1−ε)N, we haveB(0,Nε)⊆ B(x,N); thus^R_RnχN(x−y)K_Nε(y)dy=^R_RnK_Nε(y)dy=

K_Nε

L¹. Then K∗χN

p L^p

kχ_Nk_L^pp

≥

K_Nε∗χ_N

p

L^p(B(0,(1−ε)N)

χ_N

p L^p

≥ K_Nε

p

L¹(1−ε)ⁿ. LetN→∞first and thenε→0.

1.2.10.On the multiplicative group(R⁺,^dt_t)letT(f) = f∗K, whereKis a positive L¹function and f is inL^p, 1≤p≤∞. Prove that the operator norm ofT:L^p→L^p is equal to theL¹norm ofK. Deduce that the constantsp/(p−1)andp/bare sharp in Exercises 1.2.7 and 1.2.8.

Hint:Adapt the idea of Exercise 1.2.9 to this setting.

1.2.11.LetQ_j(t) =c_j(1−t²)^jfort∈[−1,1]and zero elsewhere, wherec_jis cho- sen such that^R₋₁¹ Q_j(t)dt=1 for all j=1,2, . . ..

(a) Show thatc_j<√ j.

(b) Use part (a) to show that{Q_j}_jis an approximate identity onRas j→∞.

(c) Given a continuous function f onRthat vanishes outside the interval[−1,1], show that f∗Q_jconverges tof uniformly on[−1,1].

(d) (Weierstrass) Prove that every continuous function on[−1,1]can be approxi- mated uniformly by polynomials.

Hint:Part (a): Consider the integral^R_|t|≤n−1/2Q_j(t)dt. Part (d): Consider the func- tiong(t) =f(t)−f(−1)−^t+1₂ (f(1)−f(−1)).

1.2.12.(Christ and Grafakos [51]) LetF ≥0,G≥0 be measurable functions on the sphereSⁿ⁻¹and letK≥0 be a measurable function on[−1,1]. Prove that

Z

Sⁿ⁻¹ Z

Sⁿ⁻¹

F(θ)G(ϕ)K(θ·ϕ)dϕdθ≤C F

L^p(Sⁿ⁻¹)

G

L^p0(Sⁿ⁻¹),

where 1≤p≤∞,θ·ϕ=∑ⁿ_j=1θ_jϕ_jandC=^R_Sn−1K(θ·ϕ)dϕ,which is independent ofθ. Moreover, show thatCis the best possible constant in the preceding inequality.

Using duality, compute the norm of the linear operator F(θ)7→

Z

Sⁿ⁻¹

F(θ)K(θ·ϕ)dϕ fromL^p(Sⁿ⁻¹)to itself.

Hint:Observe that^R_Sn−1R

Sⁿ⁻¹F(θ)G(ϕ)K(θ·ϕ)dϕdθ is bounded by the quan- tityR

Sⁿ⁻¹

R

Sⁿ⁻¹F(θ)K(θ·ϕ)dθp

dϕ

1

pkGk

L^p0(Sⁿ⁻¹).Apply H¨older’s inequality to the functions F and 1 with respect to the measure K(θ·ϕ)dθ to deduce that R

Sⁿ⁻¹F(θ)K(θ·ϕ)dθ is controlled by _Z

Sⁿ⁻¹

F(θ)^pK(θ·ϕ)dθ 1/p_Z

Sⁿ⁻¹

K(θ·ϕ)dθ 1/p⁰

.

Use Fubini’s theorem to bound the latter by kFk_Lp(Sⁿ⁻¹)kGk_Lp0

(Sⁿ⁻¹) Z

Sⁿ⁻¹

K(θ·ϕ)dϕ.

Note that equality is attained if and only if bothFandGare constants.