Preliminaries and Notation This preliminary chapter fixes notation conventions and normalizations. It also states some basic theorems that will be used later in the book. For those less familiar with Hilbert and Banach spaces, it contains a very brief primer. (This primer should be used mainly as a reference to come back to in those instances when the reader comes across some Hilbert or Banach space language that she or he is unfamiliar with. For most chapters, these concepts are not used. Let us start by some notation conventions. For E E R, we write z for the argest integer not exceeding =max{m∈z;n≤x} For example, 13/2=1,[-3/2=-2, 1-2]=-2 Similarly, [=l is the smallest integer which is larger than or equal to z If a-0(or oo), then we denote by o(a)any quantity that is bounded by a constant times a, by o(a)any quantity that tends to o(or oo) when a does. The end of a proof is always marked with a s; for clarity, many remarks or examples are ended with a o In many proofs, C denotes a "generic"constant, which need not have the same value throughout the proof. In chains of inequalities, I often use C,C,C",……orC1,C2,C3,…… to avoid confusion We use the following convention for the Fourier transform(in one dimension) (万∫)(E)=f()= dxe-1ff(a) (0.0.1) With this normalization, one has )12‖fE 1/p IfM p= dz If(z)IP (0.0.2)
PRELIMINARIES AND NOTATION Inversion of the Fourier transform is then given by ∫(r) V2n/de(F)()=(f)y(x), (0.03 (x)=9(-x) Strictly speaking, (0.0.1),(0.0. 3) are well defined only if f, respectively Ff, are bsolutely integrable; for general L-functions f, e.g. we should define Ff via a limiting process(see also below). We will implicitly assume that the adequate limiting process is used in all cases, and write, with a convenient abuse of no- tation, formulas similar to(0.0. 1)and (0.0.3)even when a limiting process is understood A standard property of the Fourier transform is =(i)4()() drfo()2<∞…/ dE 1512 If()2<∞ with the notation f(e dzf If unction f is compactly supported, i.e. f(=)=0 if r a or I>6, where -oo< a<6<oo. then its Fourier transform f(E)is well defined also for complex 5, and 1(E) s(2m)-1/ dr e(lme)r 1f(z) eb(m)ifIm≥0 (Im E) ifIm5≤0 If f is moreover infinitely differentiable, then the same argument can be applied to f(e), leading to bounds on IEI If(E). For a Coo function f with support a, bl there exist therefore constants CN so that the analytic extension of the Fourier transform of f satisfies (0.0.4) Conversely, any entire function which satisfies bounds of the type(0.0.4)for all NN is the analytic extension of the fourier transform of a Coo function with support in (a, b]. This is the Paley-Wiener theorem We will occasionally encounter (tempered)distributions. These are linear naps T from the set S(R)(consisting of all Coo functions that decay faster than any negative power (1+a)-)to C, such that for all m, n E N, there exists n m for which T()≤ +1fm()
PRELIMINARIES AND NOTATION i hoids, for all f E S(R)The set of all such distributions is called S(R).Any polynomially bounded function F can be interpreted as a distribution, with F()dr F(E)f(a). Another example is the so-called" 6-function"of Dirac, &()=f(O). A distribution T is said to be supported in a, b l if T()=0 for all functions f the support of which has empty intersection with [a, b]. One can define the Fourier transform FT or T of a distribution T by T()=T()(if T is a function, then this coincides with our earlier definition). There exists a version of the Paley-Wiener theorem for distributions: an entire function T(S) is the analytic extension of the Fourier transform of a distribution T in S(R) supported in a, bl if and only if, for some NEN, CN>0, T()|≤Cx(1+|) lmIm≥0 ≤0 The only measure we will use is Lebesgue measure on R and R". We will often denote the(Lebesgue)measure of S by SI; in particular, l(a, bl=b-a (where b>a) Well-known theorems from measure and integration theory which we will use include Fatou,s lemma. In20,fn(r)-f(z)almost everywhere(i.e, the get o f points where pomtuLse convergence fauls has zero measure wnth respect to Lebesgue measure), then dr f(=)s limsup dr/n(=) In particular, if thus lim sup is finite, then f is integrable (The lim sup of a sequence is defined by limsup an= lim [sup ak;k2n; every sequence, even if it does not have a limit(such as an =(-1)"),has a lim sup(which may be oo); for sequences that converge to a limit, the lim sup coincides with the limit. Dominated convergence theorem. Suppose fn(r)-f(=)almost every where.Jf|∫n(x)≤g(x) for all n,and∫drg(x)<∞, then f is integrable,, and dxf(=)=lim dr fn(z) Fubini' g theorem.Jf∫dr∫d|f(x,y)<o,ten ∫(x,y) dy dr f(a,y)
PRELIMINARIES AND NOTATION i. e, the order of the integrations can be permuted In these three theorems the domain of integration can be any measura bset of R(or R- for Fubini) When Hilbert spaces are used, they are usually denoted by H, unless t already have a name. We will follow the mathematician's convention and scalar products which are linear in the first argument (A11+入2u2,u)={u1,t)+A2{u2,t) As usual. we have v,u)={u,u, where a denotes the complex conjugate of a, and (u, u)>0 for all u E H. define the norm Mull of u by In a Hilbert space, ull=0 implies u 0, and all Cauchy sequences respect to‖‖ have limits within the space.( More explicitly,iftn∈Ra Jun -uml becomes arbitrarily small if n, m are large enough-i. e, for all depending on 6, so that llun-umll se if n, m2 n0-, the exists 1∈ H so that the un tend to u for n→oo,ie,limn-∞‖u-tn‖= A standard example of such a Hilbert space is L2 (R),with (,9)=/d∫(x)9(x) Here the integration runs from -oo to oo; we will often drop the integr bounds when the integral runs over the whole real line Another example is e (Z), the set of all square summable sequences of plex numbers indexed by integers, with (c, d) dn Again, we will often drop the limits on the summation index when w over all integers. Both L(R)and e (z) are infinite-dimensional Hilbert Even simpler are finite-dimensional Hilbert spaces, of which C is the sta example, with the scalar product vanik hilbert spaces always have orthonormal bases, i. e, there exist fam ctors e in H #2=∑en)2
PRELIMIN ARIES AND NOTATION for all uE H.( We only consider separable Hilbert spaces, i.e., spaces in which orthonormal bases are countable ) Examples of orthonormal bases'are the Her mite functions in L(R), the sequences en defined by (en),=8n,,, with n,jE Z in e (Z)(i. e, all entries but the nth vanish), or the k vectors el,.,ek in C defined by( ec)m= &L, m, with 1 <l, m<k(We use Kronecker's symbol 6 with the usual meaning:6,=1ⅱfi=j,0ft≠ A standard inequality in a hilbert space is the Cauchy-Schwarz inequality (v,t)≤ lull wll, (0.0.6) easily proved by writing(0.0.5)for appropriate linear combinations of v and w forf,g∈L2(R 1/2 dxlg(x川 c=(cn)n∈z,d=(dn)n∈z∈2(z), 1/2 ∑a≤(∑a of(00.6) (0.0.7) H are linear from H to another Hilbert often咒 Itself. Explicitly, if A is an operator on H, then A(λ1a1+入2u2)=A1Au1+A2A An operator is continuous if Au- Av can be made arbitrarily small by making u-v small. Explicitly, for all e>0 there should exist &( depending on e)so that‖l-ⅶ≤ 8 implies‖Aa-A啡≤∈. If we take v=0,e=1, then we find that, for some b>0,‖Al≤1if|≤b. For any w∈ we can define clearly/l≤ b and therefore Awll=ll‖Anln≤b-l.fr I Aw/lull(w#0)is bounded, then the operator A is called bounded. We have just seen that any continuous operator is bounded; the reverse is also true. The norm‖A‖ of A is defined by ‖A= sup Aull/ll=sup‖Au‖l (0.0.8) t∈,‖u‖10 It immediately follows that, for all uE 7 ‖Atll≤‖Alll Operators from H td c are called "linear functionals. " For bounded linear functionals one has Riesz@representation theorem: for any &: H-C, linear and