LATEX-macros in [NuHAG-TeX] database:created on 2019-01-03 14:55:37

a \aa A \AA

(A, ·, ‖ · ‖A) \AaN (A, ·, ‖ · ‖A) \AaspN

A \Ab (A,H,A′) \ABGTr∣∣1∣∣ \abs∣∣1∣∣ \absbig

|〈1, 2〉| \absip∣∣ \absl

|Z| \absmcZ∣∣ \absr

|t|2 = t21 + · · · t2d \abstsq aZ× bZ \abZ

aZ× bZ \abZst aZ× bZ \abZZ

1bZd × 1

aZd \abZZadj aZd × bZd \abZZd

1bZd × 1

aZd \abZZdo

1bZ× 1

aZ \abZZo

[~a1 | · · · | ~an ] \Acoln A = [−→a1 ,−→a2 , . . . ,

−→an] \Acols

Ac(R) \AcR Ac(Rd) \AcRd

Ac \Acsp Ad \Ad

As

p,q \Adspq ∀ε > 0 ∃δ > 0 \aeed

a.e. t \aet a.e. t, ω \aeto

∀f ∈ B \afB ∀f ∈ C0(Rd) \afCORd

A(G) \AG a ∈ A \aiA

α ∈ I \aiI (A1, ‖ · ‖(1)) \AiN

a ∈ A \ainA α ∈ I \ainI

A1(R) \AiR A1\Aisp

akZ× bkZ \akbkZZ akZd × bkZd \akbkZZd

−→ak \akvec α \al

α α0 \alalo α, α′ \alalp

A− λI \Alam A(Λ) \ALamd

1

Ac(T) \alext α∈I \alinI

‖a‖A ≤ 1 \alqone A \Alsp

Alt \Alt ∀µ ∈Mb(Rd) \amuMbRd

∀µ ∈M(Rd) \amuMRd (A, ·, ‖ · ‖A) \AN

‖a‖A \anA f 7→ (〈f, gi〉) \analgi

‖]1‖A \Anorm#1 A+\AP

|(Ψ| → 0 \aPsitoz A′ \APsp

A′ \Apsp (A′, ‖ · ‖A′) \APspN

(A′, ‖ · ‖A′) \ApspN A(R) \AR

A(Rd) \ARd(A(Rd), ‖ · ‖A

)\ARdN(

A(R), ‖ · ‖A)

\ARN A(Rn) \ARn

(A(Rn), ‖ · ‖A(Rn)) \ARnN ∀ρ 6= 0 \arno

∀σ ∈ S′0(Rd) \asiSOPRd a \asp

A \Asp Ad\Aspd

(A, ‖ · ‖A) \AspN A0 \Aspo

A′ \AspP (A′, ‖ · ‖A′) \AspPN

(A′, ‖ · ‖A′) \AspPN Asp,q \Aspq

As,αp,q \Aspqa A \Ast

∗1 \asti ∗Λ \astLam

∗1 \astli ∗2 \astlt

∗′ \astp As,τp,q \Astpq

As,τp,q, unif \Astpqu ∗pw \astpw

∗2 \astt∗

\astup

A(T) \AT a · b \atb

2

a · b = 1 \atbc a · b < 1 \atbo

a · b > 1 \atbu A(Td) \ATd

A \Atil (A, ‖ · ‖A) \AtilN

(A, ‖ · ‖A) \AtilNn A \Atilsp(A(T), ‖ · ‖A

)\ATN (A2, ‖ · ‖(2)) \AtN

‖]1‖A \Atnorm#1 AT \ATsp

A2\Atsp Aunif \Aunif

Aut \Aut−→a \avec

Anti−Wick \aw Aw(Λ) \AwLc(Aw(Λ), ‖ · ‖Aw

)\AwLcN Aw(R) \AwR

Aw \Awsp Aw(T) \AwT

Aw(T2) \AwTt A ∗ x = b \Axb∑nk=1 xk

−→ak \Axlc A ∗ x = 0 \Axo

A(Z) \AZ aZ× bZ \aZbZ

aZd× bZd \aZdbZd A0Q(R) \AzQR

A0(R) \AzR A00(R) \AzzR

Bc(T) \balext B1(2) \Ball

Bα\Balph (Y, ‖ ‖Y ) \bany

(Yd, ‖ ‖Yd) \banydis B \bas

A \basa A \basA

A \basAt B \basB

B •C \basbC B \basBt

C \basC C \basCt

B∗ \basd D \basD

3

E \basE B1 \basi

B = b1, . . .bn \basin B = b1, . . .bn ⊂ V \basinV

B1B2 \basit B2 ← B1 \basitt

BA \BAsp 123 \basrep

12 \basrepi 1B2

B3\basrepn

B2 \bast B3 \bastr

BA\BAusp B \BB

b \bb C \bbC

(B,H,B′) \bBGTr b ∈ B \bbinB

B(B1,B2) \BBit N \bbN

R \bbR T \bbT

Z \bbZ C \bC

B = [−→b1 ,−→b2 , . . . ,

−→bn ] \Bcols B′ \Bd

Bδ(0) \Bdn(Bd ‖ · ‖Bd

)\BdN

B \Bdot Bd \Bdsc

Bd \Bdsp (Bd, ‖ · ‖Bd) \BdspN

Bδ(x) \Bdx β \be

β β0 \bebeo β∈J \beinJ

Bε(0) \Ben Bε(0) \Beps

Bε(0) \Bepso Bε(x) \Bex

α \bfal β \bfbe

η \bfeta f \Bff

g \Bfg 1J \bfj

l \bfl λ \bflam

4

m \bfm n \bfn

ν \bfnu 0 \bfnull

0 \bfnulls 11 \bfo

ω \bfom 1 \bfone

P \bfP p \bfp

pq \bfpq Q \bfQ

q \bfq s \bfs

t \bft u \bfu

U \bfU v \bfv

w \bfw x \bfx

ξ \bfxi y \bfy

z \bfz ζ \bfze

g \bg G \bG

(B,H,B′) \BGTr H \bH

B(H ) \Bh (B,H ,B′) \BHBP

(B,HS,B′) \BHSBp b ∈ B \biB∣∣∣1∣∣∣ \Bigabs∣∣1∣∣ \bigabs(

|1)| \biggabs

(\biggl

(‖1)‖ \biggnorm

((1

)) \biggparen

)\biggr \Biggskip

∥∥∥1∥∥∥ \Bignorm

∥∥1∥∥ \bignorm

∣∣∣∣∣∣ \bigON

(1)

\Bigparen(1)

\bigparen

1

\bigset1

\Bigset (B1,H1,B′1) \BiGTr

5

β ∈ J \biJ (B1, ‖ · ‖(1)) \BiN

b ∈ B \binB inf \binf

β ∈ J \binJ (]1 + ]2)]3 =∑]3

k=0

(]3

)k]1k]2]3−k \binomf#1#2#3

(]1 + ]2)]3 =∑]3

k=0

(]3

)k]1]3−k]2k \binomfa#1#2#3 (1 + 2)3 =

∑3k=0

(3k

)1k23−k

\binomth

B1(0) \BiO B′1 \BiPsp

B1\Bisp (B1, ‖ · ‖(1)) \BispN

B1 → B2\Bitot

−→bj \bjvec

(B1′, ‖ · ‖(1)′) \BkPN (B1, ‖ · ‖(1)) \BkspN

(B1′, ‖ · ‖(1)′) \BkspPN−→bk \bkvec

BA \BlA BAG \BlAG

BA,G \BlAlG⟨

\blan

BAG

\BlAuG BGA \BlGA

BGA

\BlGuA ‖b‖B ≤ 1 \blqone

BM \BM BMa \BMasp

[1]

\bmat BM(G1 ×G2) \BMGit

(BM(G1 ×G2), ‖ · ‖BM ) \BMGitN(BMO, ‖ · ‖BMO

)\BMON

BMO(Rd) \BMORd(BMO(Rd), ‖ · ‖BMO

)\BMORdN

BMO \BMOsp BM \BMsp(B, ‖ · ‖B

)\BN N \bN

‖b‖B \bnB (B0, ‖ · ‖B) \BNO

‖]1‖B \Bnorm#1 B0(H) \BOHilb

BΩ\BOm B0,0 \BOOsp

B0 \BOsp 1[−1/2,1/2] \boxcar

1[−.5,.5] \boxcarf (B′, ‖ · ‖B′) \BPN

6

B′ \BPsp (B′, ‖ · ‖B′) \BPspN

PW \bPW Q \bQ

R \bR 〈]1| \bra#1

〈]1|]2〉 \braket#1#2⟩

\bran

B(Rd) \BRd \ \bs

BA \BsA BA\BsAsp

B(S0) \BSO B(S0, S0) \BSOSO

B(S0, S′0) \BSOSOS B \Bsp

b \bsp (B,H,B′) \BspGTr

(B, ‖ · ‖B) \BspN B′ \BspP

(B′, ‖ · ‖B′) \BspPN Bsp,q \Bspq

bsp,q \bspq Bs

p,q \BspqHom

Bsp,q(Rd) \BspqRd (Bs

p,q(Rd), ‖ · ‖Bsp,q

) \BspqRdN

Bsp,q \Bspqsp Bs

q \Bsq

B \Bst b∗ \bst

(B∗, ‖ · ‖B∗) \BstN B∗ \Bstsp

Bs(x)p,q \Bsxpq T \bT

(B2,H2,B′2) \BtGTr B \Btil

(B, ‖ · ‖B) \BtilN B \Btilsp

B2M \BtM B2

M1M2\BtMM

(B2, ‖ · ‖(2)) \BtN B′2 \BtPsp

B3′\BtrPsp B3

\Btrsp

(B3, ‖ · ‖(3)) \BtrspN B2\Btsp

(B2, ‖ · ‖(2)) \BtspN u \bu

7

BA\BuA BAG

\BuAG

BAG

\BuAlG BA,G\BuAuG

BGA\BuGA BG

A \BuGlA

BG,A\BuGuA • \bull

•1 \bulli •1 \bullli

•2 \bulllt •′ \bullp

•q \bullq •2 \bullt

(B, ‖ · ‖B) \BuN B0,c(R) \BuOcR

B0,c \BuOcsp B′0(R) \BuOPR

B′0(Rd) \BuOPRd B′0 \BuOPsp

B0(R) \BuOR B0(Rd) \BuORd

B0 \BuOsp B(R) \BuR

B(Rd) \BuRd(B(Rd), ‖ · ‖B

)\BuRdN(

B(R), ‖ · ‖B)

\BuRN B \Busp

v \bv−→b \bvec

(BV (I), ‖ · ‖BV ) \BVIN (BV , ‖ · ‖BV ) \BVN

(BV (R), ‖ · ‖BV ) \BVRN BV \BVsp

BV(T) \BVT B(V,W) \BVW

=∆

\bytri Z \bZ

A \cA C([a, b]) \Cab(C([a, b]) ‖ · ‖∞

)\CabN A←A \cAcA

A←B \cAcB A \calA

B \calB D \calD

G \calG H \calH

8

O \calO P \calP

Q \calQ R \calR

S \calS U \calU

1(2) \car B \cB

C \Cb B←A \cBcA

B←B \cBcB Cb(G) \CbG

Cb(G) \CbGd(Cb(G), ‖ · ‖∞

)\CbGN

(B,H ,B′) \cBGTr Cb(R) \CbR

Cb(Rd) \CbRd(Cb(Rd), ‖ · ‖∞

)\CbRdN(

Cb(R), ‖ · ‖∞)

\CbRN Cb(R2d) \CbRtd(Cb(R2d), ‖ · ‖∞

)\CbRtdN Cb \Cbsp

B s,τp,q \cBstpq C \cC

C \CC c \cc

C \Cc Cc(G ) \CccG

Cc(G) \CcG Cc(Hd) \CcHd

C∞c \Ccinf C∞c \Ccinsp

CM\CCM Cm

\CCm

Cn\CCn CN

\CCN

(C, ‖ · ‖L1) \CcNi (C, ‖ · ‖L2) \CcNt

C = [−→c1 ,−→c2 , . . . ,

−→cn ] \Ccols Cc(R) \CcR

C r\cCr Cc(Rd) \CcRd

(Cc(Rd), ‖ · ‖1) \CcRdiN (Cc(Rd), ‖ · ‖∞) \CcRdinN

(Cc(Rd), ‖ · ‖L1) \CcRdNi (Cc(Rd), ‖ · ‖L2) \CcRdNt

(Cc(Rd), ‖ · ‖2) \CcRdtN Cc \Ccsp

9

cc \ccsp D \cD

Cd\Cdst · \cdto

E \cE d1e \ceil

C−∑

\Cessum F \cF

for \cfc G \cG

G \cGd G \cGh

G \cGhat Cg,Λ \CgLam

H \cHX

\chck

X\checkm χ \CHI

χJ,M \CHJM 1X \chop

CR\ChR I \cI

C∞ \ci (C(I), ‖ · ‖∞) \CIN

C∞c \Cinfc C(I) \CIsp

J \cJ K \cK

C(k)(Rd) \CkRd(ck)k∈Zd \ckZd

CΛ \CL L \cL

(cλ)λ∈Λ \cLam cλ \clam

CΛ(f) \CLf clos \clos

Lp \cLp L r\cLr

L rp \cLrp sp \clsp

span \clspan M \cM

Cµ1 \Cmi Cµ2 \Cmt

N \cN cn,m \cnm

‖]1‖C \Cnorm#1 O \cO

10

C0 \co C0(G ) \COcG(C0(G ), ‖ · ‖∞

)\CocGN C \coef

C \coeff CB\coefv

2[1] \cofb (〈f, gi)i∈I \coffgi

C0(G) \COG C0(G) \COGd(C0(G), ‖ · ‖∞

)\COGdN

(C0(G), ‖ · ‖∞

)\COGN

C ′0(G) \COGP(C0′(G), ‖ · ‖∞

)\COGPN

C∞0 \coi Col \Col

c0(Λ) \cOLa co(Λ) \coLam

[−→a1 ,−→a2 , . . . ,

−→an] \colsA Col \Colsp

B0(H) \compHilb C \Compl

B0(B1,B2) \compopit(c0, ‖ · ‖∞

)\cON(

C0, ‖ · ‖∞)

\CON cond \cond

1 \conjug const \const

1 ∗ 2 \conv 1 ∗L 2 \convL

M(1, ∗) \convMult C00 \COO

C00G) \COOG [1]2 \coord

[1]2 \coordd Co \Coosp

Co(Y ) \CooY C ′0(G) \COPG(C ′0(G), ‖ · ‖C′0

)\COPGN C ′0(R) \COPR

C ′0(Rd) \COPRd(C ′0(Rd), ‖ · ‖C′0

)\COPRdN

C ′0 \COPsp(c0, · , ‖ · ‖∞

)\cOptN(

C0(Rd), ·, ‖ · ‖∞)

\COptRdN C0(R) \COR

C0(Rd) \CORd(C0(Rd), ‖ · ‖∞

)\CORdN

11

C0(Rd)′

\CORdp(C0′(Rd), ‖ · ‖M

)\CORdPN(

C0(R), ‖ · ‖∞)

\CORN CoSi \CoSi

c0 \cOsp C0 \COsp

cosp \cosp(c0, ‖ · ‖∞

)\cOspN(

C0, ‖ · ‖∞)

\COspN C0⊗C0 \COtCO(C0⊗C0), ‖ · ‖C0⊗C0

)\COtCON c0(Z) \cOZ

P \cP C ′0 \CPsp

\CR R \cR

S \cS C∗ \CS

SH \cSH c \csp

C \Csp C \Csp

(C, ‖ · ‖C) \CspN C \Cst

T \cT C(T) \CT(C(T), ‖ · ‖∞

)\CTN U \cU

Cub(G) \CubG 213 + 312 + 41 + 5 \cubpol

Cub(Rd) \CubRd(Cub(Rd), ‖ · ‖∞

)\CubRdN

Cub(G) \CuG(Cub(G), ‖ · ‖∞

)\CuGN

Cub(R) \CuR Cub(Rd) \CuRd

Cub(Rd) \CuRd(Cub(Rd), ‖ · ‖∞

)\CuRdN(

Cub(R), ‖ · ‖∞)

\CuRN Cub \Cusp

V \cV W \cW

X \cXcX \CX

(C(X), ‖ · ‖∞) \CXN C(X) \CXsp

(C(X), ‖ · ‖∞) \CXspN Z \cZ

12

C([0, 1]) \Czo(C([0, 1]) ‖ · ‖∞

)\CzoN

d \d D \D

(dα)α∈I \dalph G \dcG

D′ \DcP D′(R) \DcPR

D′(Rd) \DcPRd D′ \DcPsp

D \Dcsp D \DD

d \dd δ \de

d \debar D(1, L2, 3) \decompsp

D(Q, Lp, Y ) \decQLY D(Q, Lp, Y ) \DecQLY

\degr ∆Kron \DelKron

δ > 0 \delo δt \delt

δΛ \deltLc δ → 0 \deltz

δx \delx δxi \delxi

Der \Der G \dG

~ \dia diag \diag

d \diff D12 \Dil

D]1]2 \Dil#1#2 D1

λ \Dila

D \Dilat d \dilat

D(2)1 \Dilt dim \Dim

dim \dim D∞λ \Dinla∫∫\dint D′ \DiPsp∑

λ∈Λ δλ \DiracLam D \Disp

\disps dist \dist

δ \dl Dλ \Dnla

13

Dρ \Dnrh [ 1, 2, 3 ] \dorowv

1 · 2 \dotp

123

\dovec

(1, 2, 3

)\dovect DΦ \DPhi

DΦ \DPhi DΦµ \DPhimu

Dpλ \Dpla D′(R) \DPR

D′(Rd) \DPRd D′(R) \DPRsp

DΨ \DPsi DΨµ \DPsimu

D+Ψ \DPsipl diam(Ψ)→ 0 \dPsitoz

D(Rd) \DRd Dρ \Drho

D1/ρ \Drhoi D1/ρ \Drhoinv

Dρ2·ρ1 \Drhoit Dρ1 \Drhoo

Dρ2·ρ1 \Drhoot Dρ2 \Drhot

Dρ \Dro \dro

D1 \Drop D(R) \DRsp

D \Dsp D \Dst

D2λ \Dtla eα \eal

eα∗ \ealpa (eα)α∈I \ealph

· \ebbes E \Ecsp

e \ee E \EE

e.g. \eg e.g., \egc

e2πi12\eip

−→ek \ekvec

Eλ \Elam Eλ′ \Elamp

(`1, `2, `∞) \ellBGTr (`1, `2, `∞) \ellGTr

`(p)(r)\ellpr (`1, `2, `∞) \elltGTr

14

→ \embb → \embed

e−2πi12\emip \endofproof

E ′(R) \EPR ε \eps

ε− δ \epsdel ε > 0 \epso

= \EQ E \Esp

ess \ess ess sup \esssup

ess sup \esssupOLD η \etao

Euc \Euc eit = cos(t) + i sin(t) \euler

−→e \evec Expi \Expi

e2πist\expist e2πisx

\expisx

e−2πist\expmist e−2πisx

\expmisx

∀f ∈ S0(Rd) \fafiSORd ∀f ∈ S0(Rd) \fafSORd

(]1α)α∈I \famaI#1 (]1i)i∈I \fambiI#1

(]1β)β∈J \fambJ#1 (ei)i∈I \FAMeiI

(fi)i∈I \FAMfiI (fl)l∈L \FAMflL

(gi)i∈I \FAMgiI (gλ)λ∈Λ \FAMglaLa

(]1i)i∈I \famiI#1 (]1j)j∈J \famjJ#1

(LhiG)i∈I \FAMLhiGI (π(λ)g)λ∈Λ \FAMpilacgLac

(π(λ)g)λ∈Λ \FAMpilagLa (ψi)i∈I \FAMpsiiI(Th φ

)h∈H \FAMThPhiH ∀σ ∈ S′0(Rd) \fasiSOPRd

∀f ∈ S0 \fasoSO ‖f‖B \fBN

F \Fc (〈f, χs〉) \fchis

f ∈ C0(Rd) \fCORd F \FF

f \ff ∀f ∈ C0(Rd) \ffCO

15

〈f, fk〉 \ffk∑

i∈I〈f, gi〉gi \fframd

fft \fft 〈f, gλ〉 \fgdlam

〈f, gi〉 \fgi (〈f, gi〉)i∈I \fgiI

〈f, gλ〉 \fglam (〈f, gλ〉)λ∈Λ \fgLam

〈f, π(λ)g〉π(λ)g \fglamg f \fh

f \fhat fn \fhatn

F 1α \Fial 1 \field

(fi)i∈I \fiI f ∈ L2\fiLt

f ∈ B \finB]

\fis

f ∈ S0(Rd) \fiSORd fk \fkhat

〈f, gλ〉 \flamg 〈f, gλ〉 \flamgd

〈f, gλ〉 \flamog 〈f, gλ〉 \flamogd

FL1\FLi FL 1

α \FLial

FL1(G) \FLiG FL1(G) \FLiGmc

FL1(G) \FLiGmcd FL∞ \FLin(FL1, ‖ · ‖FL1

)\FLiN FL∞ \FLinf

FL∞(Rd) \FLinfRd FL∞(Rd) \FLinRd

FL∞ \FLinsp 1− \flip(FL1(Rd), · , ‖ · ‖FL1

)\FLiptRdN FL1(R) \FLiR

FL1(Rd) \FLiRd(FL1(Rd), ‖ · ‖FL1

)\FLiRdN(

FL1(R), ‖ · ‖FL1

)\FLiRN FL1

\FLisp

b1c \floor FLp \FLp

FLp(Rd) \FLpRd(FLp(Rd), ‖ · ‖p

)\FLpRdN

FLp \FLpsp FLq \FLq

16

FLq(Rd) \FLqRd FL2\FLtsp

FMb \FMb FMb(Rd) \FMbRd(FMb(Rd), ‖ · ‖FMb

)\FMbRdN FM (Rd) \FMRd(

FM (Rd), ‖ · ‖FM)

\FMRdN FM \FMsp

F \four F−1\fouri

〈f, π(λ)g〉 \fpilg 〈f, π(λ)g〉π(λ)g \fpilgg

〈f, π(λ)g〉 \fpilglam1

2π\fppi

for diam(Ψ)→ 0 \fPsitoz1N

\fracn

〈f, fi〉 \framcof∑

i∈I |〈f, gi〉|2 \framengI∑i∈I〈f, gi〉gi \framf Fro \Fro

Fs \Fs Fs \FS

f ∈ S \fSc f ∈ S(R) \fScR

f ∈ S(Rd) \fScRd ‖1‖2 \fsn

‖]1‖]2 \fsn#1#2 ‖1 | 2‖ \fsnt

f ∈ S0 \fSO f ∈ S0(Rd) \fSORd

F \Fsp f \fsp

‖]1‖]2 \fspn#1#2 F sp,q \Fspq

f sp,q \fspq Fs

p,q \FspqHom

F sp,q(Rd) \FspqRd (F s

p,q(Rd), ‖ · ‖F sp,q) \FspqRdN

F sp,q \Fspqsp F \Fst

F \FT Fα \FTalph

Fβ \FTbet F~ \FTh

F−1\FTI (f(ti))i∈I \fti

f \ftil f \ftild

17

(fi)i∈I \ftiliI (f(t′i))i∈I \ftip

FΛ \FTLam FMb \FTMbsp

FM \FTMsp Fs \FTs

Fsymp \FTsymp f (u) \fu

Sp \funcclass F \Fur

〈f, fk〉 \fwfk FW (Rd) \FWRd

FW \FWsp (f(xi))i∈I \fxi

Γ \Ga∑

λ∈Λ cλπ(λ)g \gabexp

(π(λ)g)λ∈Λ \GabfrLam∑

λ∈Λ cλπ(λ)g \gabsumLam

(γ, g,Λ) \gagLam γ∈K \gainK

γ(u,v) \gamuv G \Gc

g \gd G \GD

G \Gd gi \gdi

(gi)i∈I \gdiI gλ \gdlam

(gλ)λ∈Λ \gdLam gλ \gdlamo

≥ \GE (B,HS,B′) \GelfBHSB

(B,HS,B′) \GelfBHSBP(`1(Λ), `2(Λ), `∞(Λ)

)\GelfLam(

`1(Λ), `2(Λ), `∞(Λ))

\GelfliLa(S0(Rd),L2(Rd),S′0(Rd)

)\GelfSORd(

S0(R2d),L2(R2d),S′0(R2d))

\GelfSORtd

(S0(Rd × Rd),L2(Rd × Rd), S ′0(Rd × Rd)

)\GelfSOTFd

(S0,L2,S′0) \GETRI (B,H,B′) \GETRIb

(`1, `2, `∞) \GETRId generalizedfunction \gfun

G \GG G \GG

G(g, a, b) \Ggab G× G \GGd

(g, g,Λ) \ggdL G × G \GGG

18

G×G \GGgrp G× G \GGhat

(G × G))/Λ \GGL Gg,m \Ggm

G \Ggrp G \Ggrph

G×G \GGst G \Ghat

g \ghat g−x,ρ \ghmxro

gρ \ghro gs,ρ \ghsro

(gi)i∈I \giI g ∈ L2\giLt

g ∈ S0 \giSO GL \GL

(g,Λ) \gL (gλ)λ∈Λ \gLam

gλ \glam gλ \glamc

(gλ)λ∈Λ \gLamc gλ \glamd

gλ,µ \glammud gλ \glamo

GLd(R) \GLdR (gλ)λ∈Λ \glLam

GL(n,C) \GLnC GL(n,K) \GLnK

GL(n,R) \GLnR GL(R, d) \GLRd

GL(R, n) \GLRn gρ \glro

GL(R2d) \GLRtd GL \GLsp

gs,ρ \glsro GL(2d,R) \GLtdR

GL(2n,R) \GLtnR gx,ρ \glxro

Gm \Gm G \Gmc

G \Gmcd Gg,Λ,m \GmgLam

Ggr,Ir \Gmgr Gm,Λ \GmL

G \GN ‖g‖2 = 1 \gni

g0(t) = e−π|t|2

\godef g0(t) = eπ|t|2

\got

19

grad \grad G \group

G \Gsp g∗ \gst

gt \gt G×G \GtG

G \Gtil g \gtil

g \gtilde ‖g‖2 = 1 \gto

‖g‖2 = 1 \gtone HA(A) \HAA

HA(A,A) \HAAA HA(A,B) \HAAB

HA(B) \HAB HA(B,B) \HABB

HA(B1,B2) \HABiBt Hq,α \Halq

H2,α \Halt HA(L2,L2) \HALtLt

H1\Hardy H1

\Hasp

f \hatf G \hatg

G \hatG K \hatK

µ \hatmu ∗ \hatoconv

ϕ \hatphi σ \hatsi

σ \hatsig ⊗ \hattensor

G × G \hattf H \Hb

hβ \hbe hβ· \hbeta

(hβ)β∈J \hbetJ H(B1,B2) \HBiBt

G \hcG H \Hcirc

Hd\Hd H \heis

f \hf g \hg

HG(B1,B2) \HGBiBt HG(C0(G)) \HGCOG

HRd(C0(Rd)) \HGCORd (HRd(C0(Rd)), |‖ · |‖ ) \HGCORdN

20

HG \HGsp H \HH

h \hh h \hhat

(hi)i∈I \hiI H1w \Hiiw

(H1w)∠ \HiiwA (H1

w,H,H1w′) \HiiwBGTr

(H1w)′ \HiiwP (H1

w)′ \HiiwS

H1w \Hiiwsp H \Hilb

H1 \Hilbi (H, ‖ · ‖H) \HilbN

(H, 〈 · 〉) \Hilbscal H \Hilbsp

H2 \Hilbt h∈H \hinH

H1(Rd) \HiRd(H1(Rd), ‖ · ‖H1

)\HiRdN

H1\Hisp H1

\Hisp

(H1w,H ,H1

w

′) \HiwBGTr (H1

w, ‖ · ‖H1w) \HiwN

(H1w)′ \HiwP H11

w

′\HiwPsp

(H1w)′ \HiwS H1

w \Hiwsp

HL1(B1,B2) \HLiBiBt HL1(C0,C0) \HLiCoCo

HL1(L1) \HLiLi HL1(L1,L1) \HLiLiLi

HL1(Lp) \HLiLp HL1(Lp,Lp) \HLiLpLp

HL1(Lp,Lq) \HLiLpLq HL1(Lp′,Lp

′) \HLiLppLpp

HL1(Lq,Lq)) \HLiLqLq HL1(L2) \HLiLt

HL1(L2,L2) \HLiLtLt ‖h‖∞ ≤ 1 \hlinf

HL1(S0,S′0) \HLiSOSOP ‖1‖H \Hnorm

‖2‖H1 \Hnorm2 Hom \Hom

H⊥ \Hperp Hsp \Hps

→ \hra HRd \HRdsp

21

HS \HS Hs \Hs

(B,HS,B′) \HSGTr(Hs, ‖ · ‖Hs

)\HsN

L(S0,S0) \HSO L(S′0,S′0) \HSOP

L(S′0(Rd),S′0(Rd)) \HSOPRd L(S′0(Rd),S0(Rd)) \HSOPSORd

L(S0(Rd),S0(Rd)) \HSORd L(S0,S0) \HSOSO

L(S0,S′0) \HSOSOP L(S0(Rd),S′0(Rd)) \HSOSOPRd

H \Hsp Hps(Rd) \HspRd

H′s(Rd) \HsPRd Hs(Rd) \HsRd

(Hs(Rd), ‖ · ‖Hs)

\HsRdN Hs(R2d) \HsRtd

HS \HSsp Hs \Hssp

H \Hst H \Htil

H2\Htsp H(A) \HuAsp

H(B) \HuBsp H \Hunderl

H1w \Hwi (H1

w,H ,H1w

′) \HwiBGTr

(H1w)′ \HwiP H1

w \Hwisp

Id \Id id \id

Id \Idbf IdH \IdH

1Qk \idQk IdV \IdV

i.e. \ie i.e., \iec

⇐⇒ \IFF F−1\IFour

F−1\IFT F−1

Λ \IFTLam

I \II i∈I \iii

i∈I \iinI∫Rd

∫Rd

\iintRdRd

∫∫R2

\iintRt 1 ≤ j ≤ n \ijn

22

1 ≤ 1 ≤ 2 \illeq 1 ≤ p <∞ \ilplin

1 ≤ p ≤ ∞ \ilplinf 1 ≤ p, q <∞ \ilpqlin

1 ≤ p, q ≤ ∞ \ilpqlinf 1 ≤ p <∞ \ilpsinf

1 ≤ q <∞ \ilqlin 1 ≤ q ≤ ∞ \ilqlinf

1 ≤ q <∞ \ilqvlin im \im

imag \imag ∈ \IN

1 \indicatorf 11 \indicof∫∞−∞ \infint ‖1‖∞ \infn

‖1‖∞ \infnorm 〈1, 2〉 \inner⟨1, 2⟩

\innerBig ∞, 1 \inon

→ \inonto ‖ · ‖1 \inorm∫ ba

\intab∫

D\intcd∫

G\intG

∫∞−∞ \Intinf∫ ∞

−∞ \intinf → \into∫ 1

0\Intoi

∫Q1/α

\intqa∫R \intR

∫∞−∞ \IntR∫

Rd \intRd∫Rn \intRn∫

R2d \intRtd∫R×R \intTF∫

G× G \intTFcG∫Rd×Rd \intTFd∫

G× G \intTFG∫ 1

0\intzo

−1\inv

1α

\inval

1β

\invbe F−1\invF

−1/2\invsq 〈1, 2〉 \ip

1 ≤ p ≤ ∞ \ipinf 1 ≤ p <∞ \iplinf

23

1/p+ 1/q = 1 \ipqi1p

+ 1q

= 1 \ipqif

1 ≤ p, q ≤ ∞ \ipqinf 1 ≤ p, q <∞ \ipqlinf

1 ≤ q ≤ ∞ \iqinf 1 ≤ q <∞ \iqlinf

Isom \Isom I \Isp

1 < p, q <∞ \ispqsinf 1 < p <∞ \ispsinf

1∨ \iv 1∗ \iva

1∨ \ivch 1∗ \ivs

1∨ \ivsm 1? \ivst

(x) \ixx∑

λ∈Λ w(λ) \jansliw

j ∈ I \jiI j∈J \jinJ

j ∈ N,m ∈ Zn \JMnd κ \kahat

κ \katil K(B1,B2) \KBit

Ker \Ker |]1〉 \ket#1

K(G) \KGKk=1 \kiK

nk=1 \kin

∞k=1 \kinf

k ∈ N \kinN k ∈ Z \kiZ

k ∈ Zd \kiZd k ∈ Zm \kiZm

k ∈ Zn \kiZn K \KK

Kk[x] \Kkx Kσ \kn

σ1 \kohnink=0 \kon

K(1) \KOP K(Rd) \KRd

K(G) \KsG K(Hd) \KsHd

K(R) \KsR K(Rd) \KsRd

K \Kssp K \Kst

24

k ×m \ktm k ×m \ktn

k ∈ Zd \kZdnk=0 \kzn

Λ \La λ∈Λ \lacinLac

Λ \Lacirc λ \lacirc

λ \lahat λ∈Λ \lainLa

λ ∈ Λ \lainLam λ ∈ Λ \lainLamc

λ⊥ ∈ Λ⊥ \lainLamp λ ∈ Λ \laL

λ ∈ Λ \laLa λ ∈ Λ \laLam

λ ∈ Λ \laLamo Λ \Lamc

λ \lamc Λ \Lamd

(λi)i∈I \lamiiI λ− λ′ \lamilap

Λ⊥1 \Lamip ΛC R2d\LamiRtd

Λ \Lamo λ \lamo

Λ⊥ \Lamp λ⊥ \lamp

λ = (t, ω) \lamtom Λ⊥2 \Lamtp

(λ,v) \lamv 〈 \lan

L \Lanbas λ0, . . . , λn−1 \laon

Λ⊥ \Lap λ′ \lap

Λ⊥ \Laperp λ′∈Λ \laPinLa

λ = (t, ω) \lato L(B) \LB

L(B1,B2) \LBiBt L(B1,B2) \LBit

L(L2(G),L2(G)) \LbLtGLtG L(L2,L2) \LbLtLt

L(L2(Rd),L2(Rd)) \LbLtRdLtRd (L(B), |‖ · |‖ ) \LBN

L(S0(G),S′0(G)) \LbSOGSOPG L(S0(Rd),S′0(Rd)) \LbSORdSOPRd

25

L(S0,S′0) \LbSOSOP L \Lbsp

Λ \Lc λ \lc

L(B1,B2) \LcBit L(Cb(Rd)) \LCbRd

λ ∈ Λ \lcinLc Λ \Lcirc

λ \lcirc λ ∈ Λ \lcLc

0 6= λ ∈ Λ \lcLcno L(C0(G)) \LCOG

L(C0(Rd)) \LCORd (LRd(C0(Rd)), |‖ · |‖ ) \LCORdN

L(Cub(Rd)) \LCuRd ≤ \LE

L1(Rd) \Le ≤ ε \leps

5 \leqq < ε \lesp

L2(G) \LG L(H) \LH

L(H1,H2) \LHiHt L(H) \LHilb

(L(H), |‖ · |‖ ) \LHN L(H2,H1) \LHtHi

L1at \Liat L1

at(Rd) \LiatRd

(L1at(Rd), ‖ · ‖at−1) \LiatRdN L1

bc \Libc

L1bc(R) \LibcR L1

bc(Rd) \LibcRd

L1(G ) \LicG(L1(G ), ‖ · ‖1

)\LicGN

L1 ∩ C0 \LiCO L1 ∩ C0(G) \LiCOG

L1 ∩C0(Rd) \LiCORd (L1 ∩C0(Rd), ‖ · ‖1 + ‖ · ‖∞) \LiCORdN

L1 ∩C0 \LiCOsp `1(G) \liG

L1(G) \LiG L1(G) \LiGmcd(L1(G), ‖ · ‖1

)\LiGN (`1, `2, `∞) \liGTr

(L1,L2,L∞) \LiGTr `1(H) \liH

L1(Hd) \LiHd L1(Rd) \LiiRd

26

λ ∈ Λ \liL `1(Λ) \liLa

`1(Λ) \liLac `1(Λ) \liLam(`1(Λ), ‖ · ‖1

)\liLamN `1(Λ) \liLamo

λ′ ∈ Λ⊥ \liLampl,iλ′−λ \lilapla

`1(Λ) \liLc L1loc \Liloc

L1loc(G) \LilocG L1

loc(R) \LilocR

L1loc(Rd) \LilocRd L1

loc \Lilocrm

limα \lima limβ \limb

limγ \limg limh→0 \limhO

limk→∞ \limkinf limα µα \limma

limβ µβ \limmb limγ µγ \limmg

limn→∞ \limninf limρ→0 \limrho

limρ→1 \limri limρ→∞ \limroi

limρ→0 \limroz limα σα \limsa

limβ σβ \limsb limγ σγ \limsg

lim τ→0 \limtau limt→∞ \limtin

limt→0 \limtO limx→∞ \limxin

lim|x|→∞ \limxinf limx→0 \limxO

limy→∞ \limyin limy→0 \limyO

`1(N) \liN(L1, ‖ · ‖1

)\LiN

L∞(G) \LinfGmcd `∞(I) \linfI

`∞(Λ) \linfL `∞(Λ)− norms \linfL-norms

`∞(Λ) \linfLam `∞(Λ) \linfLamo(`∞, ‖ · ‖∞

)\linfN

(L∞, ‖ · ‖∞

)\LinfN

27

‖1‖∞ \linfnorm ‖1‖∞ \linfnorm

‖1‖∞ \Linfnorm ‖ · ‖∞ \linfnorme

L∞(R) \LinfR L∞(Rd) \LinfRd(L∞(Rd), ‖ · ‖∞

)\LinfRdN L∞(R2d) \LinfRtd

L∞ \Linfsp `∞ \linfsp(`∞, ‖ · ‖∞

)\linfspN L∞(Rd × Rd) \LinfTFd

`∞ \linfty L∞ \Linfty

`∞(G) \linG L∞(G) \LinG(L∞(G), ‖ · ‖∞

)\LinGN `∞(H) \linH

LH(1) \linh `∞(Λ) \linLa

`∞(Λ) \linLac `∞(Λ) \linLam(L∞, ‖ · ‖∞

)\LinN ‖ebbes‖∞ \linnm

`∞(N) \linoN ‖ · ‖1 \Linorm

‖1‖1 \linorm L∞(R) \LinR

L∞(Rd) \LinRd(L∞(Rd), ‖ · ‖∞

)\LinRdN(

L∞(R), ‖ · ‖∞)

\LinRN L∞(R2) \LinRt

L∞(R2d) \LinRtd `∞ \linsp

L∞ \Linsp(L∞, ‖ · ‖∞

)\LinspN(

`∞, ‖ · ‖∞)

\linspN L∞(T) \LinT

L∞(G× G) \LinTFG L∞(Rd× Rd) \LinTFRd

`∞ → `∞ \lintolin L∞w (G) \LinwG(L∞w (G), ‖ · ‖∞,w

)\LinwGN `∞w (H) \linwH

`∞w (Λ) \linwLc(`∞w (Λ), ‖ · ‖1,w

)\linwLcN

L∞w (R) \LinwR L∞w (Rd) \LinwRd

28

(L∞w (Rd), ‖ · ‖∞,w

)\LinwRdN

(L∞w (R), ‖ · ‖∞,w

)\LinwRN

L∞w (R2) \LinwRt `∞w \linwsp

L∞w \Linwsp `∞w (Z) \linwZ

`∞w (Zd) \linwZd `∞(Z) \linZ

`∞(Zd) \linZd `∞(ZN) \linZN

L∞,1 \Lio Lip \Lip

Lip (α) \Lipa (Lip (α), ‖ · ‖Lip (α)) \LipaN

Lip \Lipsp L1(R) \LiR

L1(Rd) \LiRd(L1(Rd), ‖ · ‖1

)\LiRdN

L(Rm) \LiRm(L1(R), ‖ · ‖1

)\LiRN

L1(R2) \LiRt L1(R2d) \LiRtd

`1\lisp L1

\Lisp(`1, ‖ · ‖1

)\lispN

(L1, ‖ · ‖1

)\LispN(

L1(Rd), ∗, ‖ · ‖1

)\ListRdN L1(T) \LiT

L1(Td) \LiTd L1(Rd × Rd) \LiTFd

L1(G× G) \LiTFG L1(Rd× Rd) \LiTFRd(L1(T), ‖ · ‖1

)\LiTN `1 → `1

\litoli

L1vs \Livs L1

vs(R) \LivsR

L1w(G) \LiwG

(L1w(G), ‖ · ‖1,w

)\LiwGN

`1w(H) \liwH `1

w(Λ) \liwLc(`1w(Λ), ‖ · ‖1,w

)\liwLcN

(`1w, ‖ · ‖1,w

)\liwN

L1w(R) \LiwR L1

w(Rd) \LiwRd(L1w(Rd), ‖ · ‖1,w

)\LiwRdN

(L1w(R), ‖ · ‖1,w

)\LiwRN

L1w(R2) \LiwRt `1

w \liwsp

29

L1w \Liwsp `1

w \liwsp

`1w(Z) \liwZ `1

w(Zd) \liwZd

`1(Z) \liZ `1(Zd) \liZd

`1(ZN) \liZN `1(ZdN) \liZNd

L1([0, 1]) \Lizo (L1([0, 1]), ‖ · ‖1) \LizoN

`1(Z2d) \liZtdl,jλ \ljlam

L \LL L(L∞(Rd)) \LLinRd

L(L1(Rd)) \LLiRd L1 \Llisp

L(Lp) \LLp L(L2,L2) \LLt

L(L2(G),L2(G)) \LLTG L(L2(Rd)) \LLtRd

L(L2(Rd),L2(Rd)) \LLTRd Lpm \Lmp

Lp,qm \Lmpq `p,qm \lmpq

L(M (Rd)) \LMRd 7−→ \lms

LMZ

(1)

\LMZ LMZ

(]1)

\LMZ#1

lndx \lndx `1\lo

loc \loc L1loc \Loneloc

L1loc(G) \LonelocG L(1) \LOP

L(]1, ]2) \LOPs#1#2 L(B1,B2) \LOPsit

L1(Rn) \LoRn lowM \lowndM

lowM (B) \lowndMB lowW \lowndW

lowW (B) \lowndWB Λ⊥ \Lperp

`p(G) \lpG Lp(G) \LpG

Lp(G) \LpGmcd(Lp(G), ‖ · ‖p

)\LpGN

`p(H) \lpH (`p(I), ‖ · ‖p) \lpIN

30

`p(Λ) \lpLam Lp(`q) \Lplq

`pm \lpm `pm \lpm

Lpm \Lpmsp(`p, ‖ · ‖p

)\lpN(

Lp, ‖ · ‖p)

\LpN ‖1‖p \lpnorm

`p(N) \lponN Lp′(Rd) \LppRd

Lp′

\Lppsp `p′

\lppsp

`p′(Z) \lppZ Lp(R) \LpR

Lp(Rd) \LpRd(Lp(Rd), ‖ · ‖p

)\LpRdN(

Lp(R), ‖ · ‖p)

\LpRN Lp(R2) \LpRt

Lp(R2d) \LpRtd `p \lpsp

Lp \Lpsp(Lp, ‖ · ‖p

)\LpspN(

`p, ‖ · ‖p)

\lpspN Lp \Lpss

Lp\Lpst Lp(T) \LpT(

Lp(T), ‖ · ‖p)

\LpTN `p → `p \lptolp

Lpw(G) \LpwG(Lpw(G), ‖ · ‖p,w

)\LpwGN

`pw(H) \lpwH Lpw(R) \LpwR

Lpw(Rd) \LpwRd(Lpw(Rd), ‖ · ‖p,w

)\LpwRdN(

Lpw(R), ‖ · ‖p,w)

\LpwRN Lpw(R2) \LpwRt

`pw \lpwsp Lpw \Lpwsp

`pw(Z) \lpwZ `pw(Zd) \lpwZd

`p(Z) \lpZ `p(Zd) \lpZd(Lq(G), ‖ · ‖q

)\LqGN `q(Lp) \lqLp

(Lq, `p)α \Lqpa (Lq, `p)α \Lqpal

(Lq, `p)α(Rd) \LqpalRd (Lq, `p)α(Rd) \LqpaRd

31

Lq(R) \LqR Lq(Rd) \LqRd(Lq(Rd), ‖ · ‖q

)\LqRdN

(Lq(R), ‖ · ‖q

)\LqRN

Lq(R2) \LqRt Lq(R2d) \LqRtd

Lqs \Lqs `qs \lqs

Lq \Lqsp `q \lqsp(Lq, ‖ · ‖q

)\LqspN

(`q, ‖ · ‖q

)\lqspN

`s(x)q (Lp) \lqsxLp Lqw \Lqw

Lqws \Lqws `qws \lqws

Lqws(Rd) \LqwsRd L2(R) \lr

L r\Lr =⇒ \Lra

−→ \lra ⇔ \Lrarr

L rAsp,q \LrAspq L2(Rd) \lrd

L rp \Lrp Lr(Rd) \LrRd(

Lr(Rd), ‖ · ‖q)

\LrRdN Lr \Lrsp

`r \lrsp L(S0) \LSO

L(S ′0, S0) \LSOdSO (L(S′0,S0),HS,L(S0,S′0)) \LSOGTr

L(S′0,S0) \LSOpSO L(S′0,S0) \LSOPSO

L(S′0,S′0) \LSOpSOp L(S′0,S

′0) \LSOPSOP

Lw∗(S′0,S0) \LSOPwSO L(S0,S0) \LSOSO

L(S0, S′0) \LSOSOd L(S0,S

′0) \LSOSOp

L(S0,S′0) \LSOSOP L (S0(G),S′0(G)) \LSOSOpG

L(S0(Rd),S′0(Rd)

)\LSOSOpRd L(S0,S

′0) \LSOSOS

` \lsp L \Lsp(L1, ‖ · ‖1

)\LspN L

(S (Rn),S ′(Rn)

)\LSSp

32

L \Lst < \LT

L2(G ) \LtcG(L2(G ), ‖ · ‖2

)\LtcGN

LTFATM− files : \ltfatml `2(G) \ltG

L2(G) \LtG L2(G) \LtGd(L2(G, ‖ · ‖2

)) \LtGdN L2(G) \LtGg

L2(G × G) \LtGGD L2(G) \LtGmcd(L2(G), ‖ · ‖2

)\LtGN `2(H) \ltH

`2(I) \ltI (`2(I), ‖ · ‖2) \ltIN

`2(Λ) \ltLa `2(Λ) \ltLac

`2(Λ) \ltLam `2(Λ) \ltLamo

`2(Λ) \ltLc `2(Λ) \ltLL

`2(N) \ltN ‖ · ‖2 \Ltnorm

‖1‖2 \ltnorm L2(R) \LtR

L2(R) \LTR L2(Rd) \LtRd

L2(Rd) \LTRd(L2(Rd), ‖ · ‖2

)\LtRdN

L2(Rm) \LtRm(L2(R), ‖ · ‖2

)\LtRN

L2(Rn) \LtRn L2(R2) \LtRt

L2(R2d) \LtRtd(L2(R2d), ‖ · ‖2

)\LtRtdN

`2\ltsp L2

\Ltsp(`2, ‖ · ‖2

)\ltspN L2(T) \LtT

L2(R× R) \LtTF L2(Rd × Rd) \LtTFd

L2(G× G) \LtTFG L2(Rd× Rd) \LtTFRd(L2(T), ‖ · ‖2

)\LtTN `2 → `2

\lttolt

L2w(G) \LtwG L2

w(G) \LtwGd

33

(L2w(G), ‖ · ‖2,w

)\LtwGN `2

w(H) \ltwH

L2w(R) \LtwR L2

w(Rd) \LtwRd(L2w(Rd), ‖ · ‖2,w

)\LtwRdN

(L2w(R), ‖ · ‖2,w

)\LtwRN

L2w(R2) \LtwRt L2

ws \Ltws

`2w \ltwsp L2

w \Ltwsp

L2ws(R

d) \LtwsRd `2w(Z) \ltwZ

`2w(Zd) \ltwZd `2(Z) \ltZ

LTZ

(]1)

\LTZ#1 `2(Zd) \ltZd

`2(ZN) \ltZN (L2([0, 1]) \Ltzo

(L2([0, 1]), ‖ · ‖2) \LtzoN `2(Z2) \ltZt

`2(Z2d) \ltZtd L(V ) \LV

L(V,W) \LVW L(W ′,V ′) \LWVP

L(W ∗,V ∗) \LWVst 〈x, y〉 \lxyr

L1(R) \LYR µα \ma

Mac \Macsp materialtakendirectly − sofar − fromcoursescript \mat15(a −bb a

)\matabba

(a cb d

)\matabcd

(1 −22 1

)\matcnum

(1 2−2 1

)\matcnumneg

[~11 | · · · |~12

]\matcols

(1 32 4

)\matfour

[1]23 \matfrt

(p rq s

)\matpqrs

[1]B←B \matrbb [1]3←2 \matrbbb

2[1]2 \matrbbT 2[1]3 \matrbTb

C [1]B \matrBTC [1]23 \matrfrt

[1]3←2 \matrhbb [1]−13←2 \matribbb

2[1]−13 \matribTb B[1]−1

C \matriBTC

34

[1]B2←B1 \matrit

(cos(1) − sin(1)sin(1) cos(1)

)\matrotx

[~11 ; · · · ; ~12

]\matrows [T ]C←B \matrTBC

[T ]B←B \matT [T ]B←B \matTB

[T ]B2←B1 \mattit

(u −vv u

)\matuvvu

µβ \mb Mb(G) \MbG

(Mb(G), ‖ · ‖Mb) \MbGN M(B1,B2) \MBit

Mb(R) \MbR Mb(Rd) \MbRd

(Mb(Rd), ‖ · ‖Mb) \MbRdN Mb \Mbsp

MβlTαk \mbta µ \mc

M (G ) \McG H \mcH

Mχ \Mchi L \mcL

L \McL M c(Rd) \McRd

Mcs(Rd) \McsRd Mcs \Mcssp

Z \mcZ Md(G) \MdG

M d(G) \MdGc Md(R) \MdR

Md(Rd) \MdRd Md \Mdsp

C : /xampp/htdocs/nuhag/media/files/ \MEDIA |1| \mes

1 \mfd1

2\mfrac

µγ \mg Mb(G ) \MG

Mb(G) \MGd (Mb(G), ‖ · ‖Mb) \MGN

−1/2\mhalf M 1(G ) \MicG(

M 1(G ), ‖ · ‖M1

)\MicGN M 1(G) \MiG(

M 1(G), ‖ · ‖M)

\MiGN M 1,10 (Rd) \MiiRd

M 1,1w (R) \MiiwR M 1,1

w (Rd) \MiiwRd

35

M 1,1w \Miiwsp M∞

\Minf

M∞,∞\Minfinf M∞(Rd) \MinfRd

(M∞(Rd), ‖ · ‖M∞) \MinfRdN M∞−s(Rd) \MinfsRd

M∞,1\Mini M∞,1(Rd) \MiniRd(

M∞,1(Rd), ‖ · ‖M∞,1)

\MiniRdN m∈N \minN

M∞s \Mins M∞

s (Rd) \MinsRd

− \minus m∈Z \minZ

m∈Zd \minZd M∞,1\Mio

M 1(R) \MiR M 1(Rd) \MiRd

(M 1(Rd), ‖ · ‖M1) \MiRdN M 1\Misp

M 1vs(R

d) \MisRd M 1vs \Mivs

M 1vs(R

d) \MivsRd M 1w(Rd) \MiwRd

Mkα \Mka Mkβ \Mkb

Mk/α \Mkba Mk/β \Mkbb

mλ \mlam (mλ)λ∈Λ \mLam

Ml/α \Mlba (m(λ)) \mlL

M \MM m \mm

MM×M(C) \MMC M|G×G|(C) \MMG

M|G×G|(C) \MMGG M1,2 \Mmn

MM×N(C) \MMNC M1,2(C) \MmnC

M1,2(K) \MmnK M1,2(R) \MmnR

M p,qm \Mmpq Mm

p,q(Rd) \MmpqRd

Mmp,q \Mmpqsp Mnα \Mna

Mnβ \Mnb Mn/α \Mnba

36

Mn/β \Mnbb MN×N(C) \MNC

Mn,n(C) \MnC M1,1(C) \MnnC

M1,1(R) \MnnR Mn,n(R) \MnR

M 1\Mo M1

\modii

M11⊗v \modiiiv M1

v (G) \modiivg

M \Modul m \modul

M1,∞\Moi Mω \Mom

M \monbas M1(R) \MondR

[x2, x, 1] \Mont [x3, x2, x, 1] \Montr

M∞,∞\Moo M 1(R) \MoR(

M 1(R), ‖ · ‖M1

)\MoRN M 1

s \Mos

M 1s(Rd) \MosRd MωTt \MoTt

MωnTtn \MoTtn µ0 = w∗- limµα \mowma

µ0 = limw∗µα \mowsma µ0 = w∗− \mowsman

µ0 = w∗- limµα \mowsmas M p1(Rd) \MpiRd

M pm \Mpm M p

m(Rd) \MpmRd

M p,p\Mpp M p,p(Rd) \MppRd(

M p,p(Rd), ‖ · ‖Mp,p

)\MppRdN M p,p

\Mppsp

M p,q\Mpq M p1,q1 \Mpqi

Mp,qm (Rd) \MpqmRd M p,q(Rd) \MpqRd(

M p,q(Rd), ‖ · ‖Mp,q

)\MpqRdN M p,q

\Mpqsp

M p2,q2 \Mpqt Mp,qvs (Rd) \Mpqvs

M p(Rd) \MpRd(M p(Rd), ‖ · ‖Mp

)\MpRdN

M p\Mpsp M p2(Rd) \MptRd

37

M pvs(R

d) \MpvsRd M q(Rd) \MqRd(M q(Rd), ‖ · ‖Mq

)\MqRdN M q

\Mqsp

M(Rd) \MRd (M(Rd), ‖ · ‖M ) \MRdN

M (R2d) \MRtd Mr(R) \Mrx

M s,αp,q \Msapq M s1,α1

p1,q1\Msapqi

M s,αp,q (Rd) \MsapqRd M s,α

p,q (Rn) \MsapqRn

M s2,α2p2,q2

\Msapqt Msc \Mscsp

M(S0) \MSO m \msp

M \Msp M sp,p \Mspp

M sp,q(Rd) \MsppRd M s

p,q \Mspq

M s1p1,q1

(Rd) \Mspqi M sp,q(R) \MspqR

M sp,q(Rd) \MspqRd

(M s

p,q(Rd), ‖ · ‖Msp,q

)\MspqRdN(

M sp,q(R), ‖ · ‖Ms

p,q

)\MspqRN M s

p,q \Mspqsp

M s2p2,q2

(Rd) \Mspqt M \Mst

(Mb(G), ∗, ‖ · ‖Mb) \MstGN M s

2,2(Rd) \MsttRd

M2,2 \Mtbt M 2 = t2, t1, t0 \Mth

M2 \Mtl M2 = 1, t, t2 \Mtld

m×m \mtm m× n \mtn

MωTt \MTomt MωnTtn \MTomtn

M3,3 \Mtrbtr M 2(Rd) \MtRd

M 3\Mtrh M 3 = t3, t2, t1, t0 \Mtrhd

M3 = 1, t, t2, t3 \Mtrl M3 = 1, t, t2, t3 \Mtrld

M 3\Mtru M 3 = t3, t2, t1, t0 \Mtrud

M 2\Mtu M 2 = t2, t1, t0 \Mtud

38

MuTv \MTuv |µ| \muabs

µα \mual∑

k∈F ckδtk \mudisc

µ \muhat µ1 ∗ µ2 \muit

(µλ)λ∈Λ \muLam µ∈Mb(Rd) \muMbRd

µ− \mumi ‖µ‖M \muMN

µ \mumo µ∈M (Rd) \muMRd

µ+\mupl µψi \mupsi

M(S0) \MuSO M \Musp

M 1v \mv1 Mvs \Mvs

M vs1 (Rd) \MvsiRd Mvs(Rd) \MvsRd

Mw1,1(R) \MwiiR Mw

1,1(Rd) \MwiiRd

Mw1,1 \Mwiisp Mξ Tx \MxiTx

M1,1(R) \MxxR \myhat

] \mysharp ‖ · |A(Rn)‖ \nARn

A \nAsp \ \nat

N \Nb (1α)α∈I \ndalph

- \ndash Nd\Ndst

‖ · ‖ \nebbes \negfive

\negthree ‖1‖2 \nfs

‖]1‖]2 \nfs#1#2 ‖]1‖]2 \nfsp#1#2

‖1 | 2‖ \nfst ‖1 | 2‖ \NFST

0 6= g ∈ S0(Rd) \ngSORd G \nGsp

NuHAGM− files : \nhgml D : /NuHAGall/NuHAGTEX/ \NHGTEX

∞n=1 \niinf n ∈ N \niN

39

∞n=1 \ninf ‖∞ \ninfty

n∈I \ninI n∈N \ninN

n∈Z \ninZ n∈Zd \ninZd

\nix∑

k∈F |ck| \nmudisc

N \NN n \nn

\nnnth \nnth

Nd0 \NOdst ‖[‖1]‖1‖ \norm∥∥∥1∥∥∥ \normBig

∥∥1∥∥ \normbig

‖ · ‖(4)\normfe ‖ · ‖(4)

\normfs

‖1‖(1)\normi ‖ · ‖(1)

\normie

‖1‖∞ \norminf ‖1‖`2,1 \normmix

‖]1‖ \normo#1 |‖1|‖2 \normop

‖1‖(2)\normt ‖]1‖]2 \normta#1#2

‖ · ‖(2)\normte ‖ · ‖(3)

\normtre

‖]1‖ \normv#1 ‖[1]W‖ \normW

N0 \NOst Φ,Λ \nPhiL

φ,Λ \nphiL 0 \nset

N \Nsp (]1, ‖ · ‖]1) \NSP#1

(]1, ‖ · ‖]1) \NSPB#1 (]1, ‖ · |]1‖) \NSPT#1

N \Nst \nth

n→∞ \ntinf n× k \ntk

n×m \ntm n× n \ntn

N ×N \NtN null \nullsp

Null \Nullsp ν \nutil

40

o.B.d.A. \obda OC \OC

O = [−→o1 ,−→o2 , . . . ,

−→on] \Ocols ∗ \oconv

∗ \oconvhat O ′C(Rn) \OCp

O′C \OCP |1| \ofabs

|]1| \ofabs#1 [1] \ofb

[]1] \ofb#1 1 \ofcb

]1 \ofcb#1 ‖1‖ \ofnorm

(1) \ofp (]1) \ofp#1

(x, ξ) \ofpxxi (Rd) \ofRd

(Rd) \ofRd (tt ) \ofshahf

[0, 1]d \oid lim \olim

Ω \Om OM \OM

ωδ \omdel Ω0 \OmO

O′M \OMP OM(Rn) \OMRn

||| \ON 1Rd \oneRd

\onto Op \Op

(B,H,B′) \OPGTr |‖]1|‖ \opn#1

|‖]1|‖B1→B2 \opnit#1 |‖]1|‖]2 \opnorm#1#2

|‖]1|‖ \opnormi#1 |‖]1 | ]2|‖ \opnots#1#2

Opw \Opw O`1 \orbli

O`1 \orblib O`1(g,Λ) \orbligL

O \ord O \Ord

osc \osc oscδ \oscd

oscδ(f) \oscdf−→o \ovec

41

pβ· \pbeta (pβ)β∈J \pbetJ

p1(t) \pbt p1(z) \pbz

P(C) \PC P3(C) \PcC

Pcol(A) \Pcol P3(R) \PcR

P1(C) \PdC P1(R) \PdR

⊥⊥⊥\perpcomb Φ \Phid

ϕ \phid ϕj \phidj

ϕja \phidja Φ = (φj)j∈J \Phifam

Φ \Phihat ϕj \phijd

(ϕkf)∨

\phikf ϕΛ \phiLam

ϕλ \philam φ \phitil

φ \phitilhat (ϕj f)

∨\phjf

ϕx,y \phxy •π \pibul

•π \pibull π(λ) \pil

π(λ) \pila π(λ) \pilac

π(Λ) \piLac π(λ)g \pilacg

π(λ)γ \pilacga π(λ)g \pilacgd

π(λ)gt \pilacgt π(λ)g \pilag

π(λ)γ \pilaga π(λ)g \pilagd

π(λ), λ ∈ Λ \pilaLa π(λ) \pilam

(π(λ))λ∈Λ \piLam π(λ)g \pilamg

(π(λ)g)λ∈Λ \piLamg π(λ)g \pilamgd

π(λ)g0 \pilamgo (π(λ)g0)λ∈Λ \piLamgo

π(λ)g \pilg (π(λ)g)λ∈Λ \pilgdLam

42

(π(λ)g)`1(Λ) \pilgdLamo (π(λ)g)λ∈Λ \pilgLam

(π(λ)g)`1(Λ) \pilgLamo π(λ) \pilo

(π(λ)g)λ∈Λ \pilogdLamo (π(λ)g)λ∈Λ \pilogLamo

p ∈ [1,∞) \pinf pinv \pinv

π ⊗ π∗ \pipist π ⊗ π∗(λ) \pipistlam

(π ⊗ π∗) \pistar π(x) \pix

Pk(C) \PkC Pk(R) \PkR

Pλ \Plam (Pλ)λ∈Λ \PLam

(Pλ)λ∈Λ \PlL + \plus

PM \PM(1)

\pma

PM(G) \PMG PM1M2 \PMM(PM , ‖ · ‖PM

)\PMN PM \PMsp

PM (T) \PMT Pn(C) \PnC

C : /NuHAGALL/NuHAGPNGs/ \PNGs ‖1‖p \pnorm

Pn(R) \PnR P0 \PO

(ai)ni=0 \polcofs P(R) \PolR

p(1) =∑2

k=0 ak1k

\polst p(x) =∑n

k=0 akxk

\polxnst

P \PP p \pp∑rk=0 ckA

k\ppAA 2π \ppi

Ψ = (ψi)i∈I \PpsiI∑r

k=0 ckxk

\ppxx

P(R) \PR L2(Ω,Σ, P ) \Probsp

∗ \projconv ⊗ \projtens

Prow(A) \Prow Pr(R) \PrR

R[t] \PRT P2(C) \PsC

43

44

Ψ \Psibig Ψ = Ψ |Ψ is a BUPU \Psibigdef

Ψδ \Psibigdel Ψδ = Ψ |Ψ is a BUPU with |Ψ| ≤ δ \Psibigdeldef

Ψ = (ψi)i∈I \Psifam ψ \psih

ψ \psihat ψ(j)\psihj

(ψi)i∈I \psiI ψ(j)\psij

ψl \psilhat ψ∗ \psis

ψ∗ \psish ψ∗(j) \psishj

ψ∗(j) \psisj |Ψ| → 0 \PsitoO

diam(Ψ)→ 0 \Psitoz P \Psp

P2(R) \PsR P \Pssp

P \Pst pτ (x) \ptaux

P2(C) \PtC P2(I) \PtI

P2(R) \PtR P3(C) \PtrC

P3(I) \PtrI P3(R) \PtrR

M(1, · ) \ptwMult⊥

\pup

Pu \Puu Pu(x) \Pux

PWω \PWo PWω(√L) \PWoL

PW rp (Rd) \PWRd PW \PWsp

P1(C) \PxC P1(R) \PxR

p~1(t) \pxt p~1(z) \pxz

and \qand and \qandq

1 \qbox 1 \qboxq

q1(t) \qbt q1(z) \qbz

, \qcom , \qcq

45

⇔ \qeq ∀f ∈ C0(Rd) \qffCO

Q[√]1] \Qfld#1 for \qfor

∀ \qforall ∀ \qforallq

for \qforq Q \Qhat

QJ,M \QJM Qλ \Qlam

(Qλ)λ∈Λ \QLam (Qλ)λ∈Λ \QlL

⇔ \qLRq ‖]1‖q,p,α \qpalnorm#1

Q \QQ ⇒ \qRarrq

Q ⊂ R ⊂ C \QRC Qρ \Qro

QΨ,ρ \QroPsi Q∆,ρ \QroTri

; \qscq Q \Qsp

Q[√

1] \QSQ Q[√]1] \QSQ#1

Qs(Rd) \QsRd(Qs(Rd), ‖ · ‖Qs

)\QsRdN

Qs \Qssp Q \Qst

“1 ′′ \qu ⇔ \quadarr

212 + 31 + 4 \quadpol Q(G) \QuG

Q(R) \QuR Q(Rd) \QuRd

Q \Qusp with \qwith

with \qwithq q~1(t) \qxt

q~1(z) \qxz RA \RA

R(G) \RaG 〉 \ran

Ran \Ran Rng \range

rank \rank R(R) \RaR

R(Rd) \RaRd ⇒ \Rarr

46

R \Rasp R \Rb

Rd\Rd Rd

\Rdh

Rd \Rdhat Rd×Rd×T \RdRdhTst

Rd×Rd×T \RdRdTst [Rd− specific!] \Rdspec

Rd\Rdst Rd

\Rdsth

real \real R \recon

rect \rect red \red

red(Λ) \redLam (??) \refeq

R(G) \RG Rg,Λ \RgLam

•ρ \rhobul ρ \rhohat

ρ \rhoo ρ′ \rhop

ρ→∞ \rhotinf ρ→ 0 \rhotz

RR\RhR Rk

\Rkst

Rm\Rm a(j) \rmaj

a(k, r) \rmakr a(λ) \rmala

a(l, s) \rmals b(j) \rmbj

b(k, r) \rmbkr b(λ) \rmbla

b(l, s) \rmbls C|G| \rmCCG

C|G×G|\rmCCGG CM

\rmCCM

CN\rmCCN Cg,Λ \rmCgLa

ZN \rmdZZN f(j) \rmfj

G \rmG G× G \rmGG

Gg,Λ \rmGgLa IN \rmIN

k \rmk M \rmM

47

N \rmN π(k, r) \rmpikr

π(l, s) \rmpils r \rmr

Sg,Λ \rmSgLa Rm\Rmst

Vgf \rmVgf Vgf(k, r) \rmVgfkr

Vgf(λ) \rmVgfla Vgg \rmVgg

Vgg(k, r) \rmVggkr Vgg(λ) \rmVggla

ZN \rmZZN ZN × ZN \rmZZNN

Rn\Rn Rn×n

\Rnn

Rn\Rnst Rν

\Rnu

Row \Row Row \Rowsp

R∗ \Rpl R∗+ \Rplst

R+\Rplus r‖]1‖q, p \rpqnorm#1

r|‖]1|‖q, p \rpqnormc#1 r‖]1‖q, p \rqpnorm#1

r|‖]1|‖q, p \rqpnormc#1 R \RR

r \rr Rd\RRd

rref \rref Rm\RRm

Rn\RRn Rn × Rm

\RRnm

Rρ \Rro Rρf \Rrof

Rρ(σ) \Rrosi R \RRR

R2d\RRtd R2n

\RRtn

(s, r) \rs Rn \Rsn

R \Rsp R \Rst

R \Rsth R∗+ \Rstpl

R(Z, ~ξ ) \Rsumxi R(Zj, ~ξ ) \Rsumxij

48

R2d\Rtd R2d

\Rtdst

R2m\Rtmst R3

\Rtr

R2\Rtst σα \sa

| \sabs σβ \sb

B1→B2 \sBiBt B1→B3 \sBiBtr

B2→B1 \sBtBi S(G) \ScG

S(Γ) \ScGAM S(G) \ScGd

(S,L2,S ′) \ScGTr (S,L2,S ′)(Rd) \ScGTrRd

S \Sch S \sch

S ′(R) \SchpR S ′(Rd) \SchpRd

S ′(Rm) \SchpRm S ′(Rn) \SchpRn

S ′(Rn × Rm) \SchpRnm S ′(R2d) \SchpRtd

S ′(R2n) \SchpRtn S (R) \SchR

S (Rd) \SchRd S (Rm) \SchRm

S (Rn) \SchRn S (Rn × Rm) \SchRnm

S \Schw S ′\Schwp

S ′(R) \SchwpR S ′(Rm) \SchwpRm

S ′(Rn) \SchwpRn S ′(R2d) \SchwpRtd

S ′(R2n) \SchwpRtn S (R) \SchwR

(√L) \ScLB S ′ \ScP

〈]1, ]2〉 \scp#1#2

⟨1⟩

\scpBig⟨1⟩

\scpbig S ′(G) \ScPG

S ′(R) \ScPR S ′(R) \ScpR

S ′(Rd) \ScPRd S ′(Rd) \ScpRd

49

S ′(Rn) \ScPRn S ′ \ScPsp

S(R) \ScR S(Rd) \ScRd

S ′(Rd) \ScRdp , \scs

S \Scsp ∗′ \sdcp

σ(D,X) \sdx (S, ‖ · ‖S) \SegN

sep

1∣∣

\sep sep (1) \sepp

(]1n)∞n=1 \sequinf#1 S \SEsp1∣∣ 2 \set C \setC

N \setN Q \setQ

R \setR 1 \sett

Z \setZ [S]B←A \SfBA

σγ \sg Sγ,g \Sgag

Sγ,g,Λ \SgagLa (Sgf(λ))L \SgdfLam

S0(G × G) \SGG Sg,γ \Sgga

Sg,γ,Λ \SggaL Sg,γ,Λ \SggaLa

S−1g,γ,Λ \SggaLi S0(G × G) \SGGD

Sg,γ,Λ \SggLa Sg,Λ \SgL

Sg,Λ \SgLac Sg,Λ \SgLam

Sg,Λ \SgLc sgn \sgn

tt \sha tttρ \shadrho

tt \Shah tt 1 \shah1

tt a \Shaha tt α \Shahal

tt b \Shahb ttt \shahd

ttt \shahdoub tttρ \shahdrho

50

tt \shahf tt α \shahfal

tt β \shahfbe ttp \shahfp

ttq \shahfq ttH \shahH

ttH \ShahH ttH⊥ \shahHp

ttH⊥ \ShahHp ttΛ \ShahLam

ttΛ \shahLam ttΛ \ShahLamc

ttΛ⊥ \ShahLamp ttΛ1 \ShahLi

ttΛ2 \ShahLt tt aN \ShahNa

tt a \Shahua tt b \Shahub

ttNa \ShahuNa ttNb \ShahuNb

ttX \shahX tt Zd \ShahZd

tt Λ \shaLam tt Λ \shaLamc

tt Λ⊥ \shaLamp ttρ \shar

tttρ \shard tttρ \shardrho

(J 1,J 2,J∞) \ShatGTR Qs \Shubs

σ \sig σα \sigmal

(σα) \sigmaln σ0 = w∗ − limασα \sigmalo

sign \sign (σn)n∈N \signN

σ0 = w∗ − limnσn \signo σ \sigo

sinc \sinc sinc \SINC

sincM \sincM S−1\Sinv

σ ∈ S ′ \siScP σ ∈ S ′(R) \siScPR

σ ∈ S ′(Rd) \siScPRd σ ∈ S′0 \siSOP

σ ∈ S′0(Rd) \siSOPRd M∞,1\Sjo

51

M∞,1(Rd) \SjoRd(M∞,1(Rd), ‖ · ‖M∞,1

)\SjoRdN

〈1, 2〉 \skalp 〈~]1, ~]2〉 · ~]2 \skalproj#1#2

〈~]1, ~]2〉 \skalvv#1#2 〈1, 2〉 \skpro

〈1, 2〉 \Skpro 〈v, v〉 \skvv

〈x,x〉 \skxx 〈x,y〉 \skxy

SΛ,g \Slamg [ \slb

(√L) \SLB \slcb

( \slp SL \SLsp

SL(2,R) \SLtR ‖ \snorm

S0 \SO S0 \So

S0 \so Hs \Sobol

S0(G ) \SOcG(S0(G ), ‖ · ‖S0

)\SOcGN

S0,c(R) \SOcR S0,c(Rd) \SOcRd

S0,c \SOcsp S′′0 \SOdoubleprime

S0(G) \SOG S′0(G) \SOGd

S0(G×G) \SOGG S0(G) \SOGg

S0(G × G) \SOGGD S0(G) \SOGhat(S0(G), ‖ · ‖S0

)\SOGN S′0(G) \SOGp

S′0(G) \SOGP (S′0(G), ‖ · ‖S′0(G)) \SOGpN

(S0,L2,S′0) \SOGTr (S0,L

2,S′0)(G) \SOGTrG

(S0,L2,S′0)(Rd) \SOGTrRd (S0,L

2,S′0)(R2d) \SOGTrRtd

(S0,L2,S′0)(Rd × Rd) \SOGTrTFd (S0,L

2,S′0) \SoLSop

(S0,L2,S′0) \SOLSOP ‖1‖S0 \sonorm

‖1‖S0 \SOnorm S′0 \SOp

52

S′0 \SOP S′0(G ) \SOPcG

S′0(G ) \SOPcGd(S′0(G), ‖ · ‖S′0

)\SOPcGN

S′0(G) \SOPG S′0(G) \SOpG

S′0(G×G) \SOpGG S′0(G×G) \SOPGG

S′0(G) \SOPGg(S′0(G), ‖ · ‖S′0

)\SOPGN

(S′0, ‖ · ‖S′0) \SOPN ‖1‖S′0 \sopnorm

‖1‖S′0 \SOPnorm S′0(R) \SOPR

S′0(Rd) \SOPRd (S′0(Rd), ‖ · ‖S′0) \SOPRdN

S′0 \SOprime (S′0(R), ‖ · ‖S′0) \SOPRN

S′0(R2d) \SOPRtd S′0 \SOPsp

(S′0, ‖ · ‖S′0) \SOPspN S′0(R× R) \SOPTF

S′0(G× G) \SOPTFG S′0(G× G) \SOpTFG(S′0(G× G), ‖ · ‖S′0

)\SOPTFGN S0(R) \SOR

S0(R) \SoR S0(Rd) \SORd

S0(Rd) → L2(Rd) → S′0(Rd) \SORdembed(S0(Rd), ‖ · ‖S0

)\SORdN

S′0(Rd) \SORdp(S0(R), ‖ · ‖S0

)\SORN

S0(R∗) \SORpl S0(R2d) \SORtd(S0(R2d), ‖ · ‖S0

)\SORtdN S′0(R2d) \SORtdp

S′0(Rd × Rd) \SORTFd S0 \SOsp

(S0, ‖ · ‖S0) \SOspN S0(R× R) \SOTF

S0(Rd × Rd) \SOTFd S ′0(Rd × Rd) \SOTFdp

S0(G× G) \SOTFG(S0(G× G), ‖ · ‖S0

)\SOTFGN

S′0(G× G) \SOTFGp (S′0(G× G), ‖ · ‖S′0) \SOTFGpn

S0(Rd× Rd) \SOTFRd S0[V , g] \SOVg

53

S0[V , g0] \SOVgO S0[V , g0] \SOVgo

S0[W , ϕ] \SOWphi Sp \Sp(1, ‖ · ‖1

)\SpacN span \Span

spec \spec spec\0 \speco

Spin \Spin span(1) \spn

SpΦ \SpPhi SpΦ(f) \SpPhif

SpΨ \SpPsi Sp′Ψ \SpPsiP

S ′(R) \SPR 〈1, 2〉 \spr

S ′(Rd) \SPRd |Ψ| \sPsi

(]1k)nk=1 \sqkin#1 (]1k)

∞k=1 \sqkinf#1

(]1n)∞n=1 \sqninf#1 ] \srb

\srcb S(Rd) \SRd

) \srp S rp,q \Srpq

; \sscs ∼ \ssim

σ ∈ S′0(Rd) \sSOPRd S \Ssp

; \ssp St \St

V \STFT Vg0 \STFTgO

Stρ \Str St \Stret

St(p)ρ \Strhal Stρ \Strho

St(α)ρ \Strhoal St(β)

ρ \Strhobet

St(β)ρn \Strhobetn St1/ρ \Strhoinv

St(p)ρ \Strhop St(p)

ρ1\Strhopi

St(p)ρ1·ρ2

\Strhopit St(p)ρ2

\Strhopt

Stρ \Stro ⊂ \SUBSET

54

( \SUBSETNEQ∑

λ∈Λ cλgλ \sumcg∑i∈I cigi \sumcgi

∑λ∈Λ cλgλ \sumcgLam∑

λ∈Λ〈f, gλ〉gλ \sumfglamg∑

λ∈Λ 〈f, π(λ)g〉π(λ)g \sumfpilg∑h∈H \sumH

∑h∈H \sumHc∑

h∈H \sumhcHc∑

h∈H \sumhH∑h′∈H⊥ \sumhHp

∑i∈F \sumiF∑

i∈I \sumiI∑∞

j=1 \sumjin∑k∈F \sumkF

∑dk=1 \sumkid∑d

k=1 \sumkid∑n

k=1 \sumkin∑∞k=1 \sumkinf

∑rk=1 \sumkir∑

k,l∈Zd \sumklZ∑n

k=1 |xk|2 \sumknxx∑nk=1 xkyk \sumknxy

∑k 6=0 \sumknz∑

(k,n)∈Z2 \sumknZt∑∞

k=0 \sumkoinf∑k∈Z \sumkZ

∑k∈Zd \sumkZd∑

k∈Zd\0 \sumkZdO∑∞

k=0 \sumkzinf∑k∈Zm \sumkZm

∑k∈Zn \sumkZn∑

k∈Zd\0 \sumkZO∑

k 6=0 \sumkZOO∑k∈Zd\0 \sumkZtdnz

∑λ∈Λ \sumL∑

λ∈Λ \sumla∑

λ∈Λ \sumlacLac∑λ∈Λ \sumlaLa

∑λ∈Λ \sumLam∑

λ∈Λ \sumLamo∑

λ∈Λ⊥ \sumlap∑λ∈Λ \sumLc

∑λ∈Λ cλπ(λ) \sumLccpi∑

λ∈Λ \sumlcLc∑

h⊥∈H⊥ \sumlHp∑λ∈Λ \sumlL

∑λ∈Λ \sumlLo

55

∑λ⊥∈Λ⊥ \sumlLp

∑l∈Zd \sumlZ∑

l∈Zd \sumlZd∑

l∈Zm \sumlZm∑∞m=−∞ \summ

∑∞n=−∞ \sumn∑∞

n=1 \sumniin∑∞

n=1 \sumniinf∑∞n=−∞ \sumninf

∑n=−∞∞ \sumninin∑

(n,k)∈Z2 \sumnkZt∑

(n,k)∈Z×Z \sumnkZZ∑n∈N \sumnN

∑n∈Z \sumnZ∑

n∈Zd \sumnZd∑

(u,v) \sumuv

supess \supess ‖1‖∞ \supnorm

supp \supp | \suth

| \suthat Symm \Symm

T \Tau τ \tauo

T \Tb T \Tbar

[T ]C←B \TBC B1→B2 \TBiBt

B1→B3 \TBiBtr 2× 2 \tbt

B2→B3 \TBtBtr Td \Tdst

⊗ \tensor R× R \TF

[T ]A←A \TfAA [T ]A←B \TfAB

[T ]B←A \TfBA [T ]B←B \TfBB

[T ]B1←B2 \TfBiBt [T ]B1←B3 \TfBiBtr

[T ]B2←B3 \TfBtBtr G × G \TFcG

Rd × Rd\TFd Rd × Rd

\TFdst

ν \TFelement G× G \TFG

G × G \TFGc MξTx \tfop

56

R2\tfp G × G \tfpl

R× R \TFR Rd× Rd\TFRd

MωTx \TFsh ‖1 | 2‖ \tfst

ZN × ZN \TFZN θ \thetahat

θ \thetao f \tif

g \tig h \tih

g \tilg Tkα \Tka

Tkβ \Tkb Tk/α \Tkba

Tk/β \Tkbb(Tkϕ

)k∈Zd \TkphiZd

Tlα \Tla Tλ \Tlam

Tλϕ \Tlamphi Tl/α \Tlba

T \Tmsp (T , ‖ · ‖T ) \TmspN

Tnα \Tna Tnβ \Tnb

Tn/α \Tnba Tn/β \Tnbb

(Tnϕ)n∈Zd \TnphiZ →∞ \toinf

→ 0 \tozero (Tλϕ)λ∈Λ \TphidLam

(TλϕΛ)λ∈Λ \TphiLam Tr \Tr

tr(1) \tr trace \trace

T \Trans t \trans

3× 3 \trbtr 3× 3 \trt

3× 3 \trtitr T \Tsp

T \Tst T \TT

Td \TTd T \Ttil

2× 2 \ttit∑∞

i=0 \tttest

57

T : V 1 → V 2 \TVit T : V →W \TVW

1 \ 2 \twc ∗tw \twconv(12

)\twovec Tx \Tx

U \Ub uβ \ubet

U = [−→u1,−→u2, . . . ,

−→un] \Ucols u \uhat

−→uk \ukvec lim \ulim

a(j) \ulrmaj a(k, r) \ulrmakr

a(l, s) \ulrmals b(j) \ulrmbj

b(k, r) \ulrmbkr b(l, s) \ulrmbls

k \ulrmk l \ulrml

π(k, r) \ulrmpikr π(l, s) \ulrmpils

r \ulrmr s \ulrms

Un \Un undx \undx

unif \unif U \Unsp

uppM \uppndM uppM (B) \uppndMB

uppW \uppndW uppW (B) \uppndWB

U \Usp U \Ust

u \uu U \UU

U \UUU (u,v) \uv

\v3 (vα)α∈I \valnet

(vφ(β))β∈J \valphinet (vβ)β∈J \valsnet

ε \vareps ϕ \varphih

ϕ∗ \varphis ϕ∗ \varphish

ϕ∗ \varphist ϕ \varphit

58

ϑ \varthetah ϑ \varthetat

VB \VB v = B • x \vbasx

VBb \VBb V BΦ,Λ \VBPhiLam

[v]A \vcA [~1, · · · , ~12] \vcolbn

[~11, · · · ,~12] \vcoln ~1 \vecb

~1 \vecbn ~v \vecv

~w \vecw ∨ \vees

~e1 \vei ε \veps

|||]1||| \vertiii#1 ~e2 \vet

Vgg \Vgdgd Vgg|Λ \VgdgdLc

Vgg(λ) \Vgdgdlc (Vgf(λ))λ∈Λ \VgdLam

Vgf????? \Vgf Vgf \vgf

(Vg(f)(λ))λ∈Λ \VgfLam Vgg|Λ \VggLc

Vg,Λ \VgL (Vgf(λ))λ∈Λ \VgLam

Vg,Λ \VgLo Vg0 \vgo

Vg0 \Vgo Vg0f \vgof

Vg0g0 \vgogo Vg0g0 \Vgogo

Vg0(g0) \Vgogob Vg0σ \vgosi

Vg0σ \Vgosig Vg0σn \vgosin

Vg0σ0 \vgosio Vg0σ \Vgsi

V ∗g \Vgst (V 1, ‖ · ‖(1)) \ViN

V 1 \Visp v ∈ V \viV

V ⊥K \VKp (V, ‖ · ‖) \Vn

(V, ‖ · ‖) \VN ‖]1‖ \vnorm#1

59

V0 \VO vol \vol

V 0 \VOn V ∗0 \VOP

V0 \VOsp V ∗ \VP

Vϕ(f) \Vpf Vϕ \Vphi

V ϕ,a \Vphia V Φ,Λ \VPhiLam

V φ,Λ \VphiLam V 2ϕ \Vphit

V pm(ϕ) \vpm V p(Φ) \VpofPhi

Vϕ(ϕ) \Vpp V pΦ \VpPhi

V pΦ,Λ \VpPhiLam V p

ϕ(Λ) \VpphiLam

V ∗ \VPsp(V ∗, ‖ · ‖V ∗

)\VPspN

v \vsp V \Vsp

V ∗ \Vspd Vj \Vspj

Vj+1 \Vspji VM \VspM

(V , ‖ · ‖V ) \VspN V 0 \Vspo

V ∗ \VspP Vs(Rd) \VsRd

v∗ \vst vn \vtiln

V 21 (Rd) \VtiRd (V 2, ‖ · ‖(2)) \VtN

V 2Φ \VtPhi V 2

Φ,Λ \VtPhiLam

V 2 \Vtsp ~u1 \vui

~u2 \vut v \vv

V \VV V \VVV

\wafa W (A,L1) \WALi

W (A, `1) \WAli W (A,L∞) \WALin

W (A, `∞) \WAlin(W (A,L1), ‖ · ‖W (A,L1)

)\WALiN

60

(W (A,L∞), ‖ · ‖W (A,L∞)

)\WALinN W (A, `p) \WAlp

WienerAmalgamspaces \WAMs W (A′, `∞) \WAPlin

W (A′,L∞) \WAPLin(W (A′,L∞), ‖ · ‖W (A′,L∞)

)\WAPLinN(

W (A′, `∞), ‖ · ‖W (A′,`∞)

)\WAPlinN C : /NuHAGALL/NuHAGWAV s/ \WAVs

W (B,C) \WBC(W (B,C), ‖ · ‖W (B,C)

)\WBCN

Wsp(Bsp, Lpsp) \WBLp W (B, `p) \WBlp

W (B, `q) \WBlq Wsp(Bsp, Lspq) \WBLq

W (B, `r) \WBlr W (B′,C ′) \WBPCP

W \Wc [w]B \wcB

W (C(k), `1) \WCkli W (C(k),Lp) \WCkLp

W (C(k), `p) \WCklp W (C(k), `2) \WCklt

W (C(k),L2) \WCkLt [1]W \Wcl

W (C0,L1) \WCLi W (C0, `

1) \WCli

W (C0,L1)(G) \WCLiG

(W (C0,L

1) ‖ · ‖W)

\WCLiN

W (C0,L1)(Rd) \WCLiRd W (C0, `

1)(Rd) \WCliRd(W (C0,L

1)(Rd), ‖ · ‖W)

\WCLiRdN WC(l), `2) \WCllt

W (C(l),L2) \WClLt W (C0,Lp) \WCLp

W (C0, `p) \WClp W (C0, `

2) \WClt

‖]1‖W ,Φ \Wcnorm#1 W (C0,L1) \WCOLi

W (C0, `1) \WCOli W (C0, `

1)(G) \WCOliG

W (C0,L1)(G) \WCOLiG

(W (C0,L

1)(G), ‖ · ‖W)

\WCOLiGN(W (C0, `

1)(G), ‖ · ‖W)

\WCOliGN W (C0, `1)(R) \WCOliR

W (C0, `1)(Rd) \WCOliRd

(W (C0, `

1)(Rd), ‖ · ‖W)

\WCOliRdN

W (C0, `1) \WCOlisp W (C0, `

p) \WCOlp

61

W (C0, `2) \WCOlt W (C0, `

2)(Rd) \WCOltRd

W (C0, `2) \WCt w∗- \wdash

(1 + |x|) \wei (1 + |ω|) \weiom

(1 + |t|) \weit (1 + |x|) \weix

WFG \WFG fk \wfk

W (FL1, `1) \WFLili W (FL1,L1) \WFLiLi

W (FL1, `∞) \WFLilin W (FL1, `∞) \WFLilinf

W (FL1, `∞)(Rd) \WFLilinRd W (FL1, `1)(Rd) \WFLiliRd

W (FL1, `p) \WFLilp W (FL1, `2) \WFLilt

W (FL1, `2)(G) \WFLiltG W (FL∞, `1) \WFLinfli

W (FL∞, `1) \WFLinli W (FL∞, `∞) \WFLinlin

W (FL∞, `∞) \WFlinlin W (FLp,Lp) \WFLpLp

W (FLp, `p)(Rd) \WFLplpRd W (FLp,Lq) \WFLpLq

W (FLp, `q) \WFLplq W (FLp, `q)(Rd) \WFLplqRd

W (FL2, `2) \WFLtlt W (FLp, `q) \WFpq

WF p,rG \WFprG W (FW , `1)(Rd) \WFWliRd

W (G ) \WG W(Rd) \WieRd(W(Rd), ‖ · ‖W

)\WieRdN W∞,1

\Wio

C : /ml5/gabml/wks01/ \wks W (L1, `∞) \WLilin

W (L1, `∞)(Rd) \WLilinRd W (L1, `p) \WLilp

W (L1, `2) \WLilt w∗−lim \wlim

w∗−limα \wlima w∗−limβ \wlimb

w∗−limγ \wlimg w∗−limα µα \wlimma

w∗−limβ µβ \wlimmb w∗−limγ µγ \wlimmg

62

w∗−limα σα \wlimsa w∗−limβ σβ \wlimsb

w∗−limγ σγ \wlimsg W (L∞, `1) \WLinfli

W (L∞, `p) \WLinflp W (L∞, `1)(Rd) \WLinliRd

Wsp(L1)(r)\WLir W (Lp, `1) \WLpli

W (Lp,L1) \WLpLi W (Lp, `∞) \WLplinf

W (Lp, `p) \WLplp W (Lp, `q) \WLplq

W (Lp,Lq) \WLpLq W (Lp, `q)(Rd) \WLplqRd

W (Lq, `p) \WLqlp W (Lq′,`p′) \WLqPlpP

W (L2, `1) \WLtli W (L2, `∞) \WLtlin

W (L2, `2) \WLtlt W (M , `∞) \WMlin

W (M , `∞) \WMlinf W (M , `∞)(G) \WMlinG(W (M , `∞)(G), ‖ · ‖W

)\WMlinGN W (M , `∞)(Rd) \WMlinRd(

W (M , `∞)(Rd), ‖ · ‖W)

\WMlinRdN W (M , `p) \WMlp

W (M , `2) \WMlt W pm \Wmp

W (M p,q,Lr) \WMpqLr W (M p,q, `r) \WMpqlr

Wmp,q(Rd) \WmpqRd Wm

p,q \Wmpqsp

‖]1‖W \Wnorm#1 W 1,∞\Woi

W 1,q\Woq ϕ \wphi

W p,1\Wpo W (Lp, `q) \Wpq

ψ \wpsi W (R) \WR

W (Rd) \WRd(W (Rd), ‖ · ‖W

)\WRdN(

W (R), ‖ · ‖W)

\WRN ws \ws

W \Wsp W ∗\Wspd

(W , ‖ · ‖W ) \WspN W \Wspn

63

W sp,p \Wspp W s

p,p(Rd) \WsppRd

W sp,q \Wspq W s

p,q(Rd) \WspqRd

W (Rd) \WspRd(W (Rd), ‖ · ‖W

)\WspRdN

w∗ \wst w∗− \wstd

w∗- lim \wstlim w∗- lim α→∞ \wstlimal

w∗- lim ρ→0 \wstlimroz W st(L∞, L1w) \wstrong

W st(L∞,L1w) \Wstrong 1 \wtilb

1 \wtiln v \wtilvv

W (1, 2) \WTsp W (]1, ]2) \WTsp#1#2

w \ww W \WW

w∗-w∗-continuous \wwcont WweakR (L∞, L1

w) \wweak

Wweak(L∞, L1w) \wweakl W weak(L∞,L1

w) \WweakL

W weakR (L∞,L1

w) \WweakR (W , ‖ · ‖W ) \WWN

w∗-w∗- \wwst w∗-w∗-continuous \wwstc

w∗-w∗- \wwstn W (X, `q) \WXlq

(Xd, ‖ · ‖Xd) \XdN Xd \Xdsp

(xi)i∈I \xiI (xi)i∈I \xiiI

ξ∈G \xiinGh x∈G \xinG

X0 \XOsp X \Xsp

X \Xspbar (X, ‖ · ‖X) \XspN

−→x \xvec x = [x1, x2, . . . , xn−1] \xvml

x \xx (x, ξ) \xxi

X \XXsc xxxxxxxxxxxxxxxxxx \xxx

〈~x, ~y〉 \xyscal (Y d, ‖ · ‖Yd) \YdN

64

Y d \Ydsp Y 0 \Yosp

Y \Ysp Y \Yspbar

Yd \Yspd (Y , ‖ · ‖Y ) \YspN

−→y \yvec y = [y1, y2, . . . , yn−1] \yvml

y \yy Y \YY

Y \YYsc Za \Zak

Z \Zb Zd \Zdst

ZM \ZM Z(modnZ) \Zmodn

ZM \ZMst ZM×ZN \ZMZNst

ZN \ZN Zn \Zn

ZdN \ZNd ZM × ZN \ZNM

ZN \ZNst ZN×ZM \ZNZMst

Z \Zsp Z \Zst

Z2d\Ztd Z2d

\Ztdst

Z2\Ztst Z× Z \ZtZ

Z \ZZ z \zz

Zd \ZZd Zm \ZZm

Zn \ZZn Z \ZZsc

Z× Z \ZZst Z2d\ZZtd

65