3.2 Fourier series
3.2.6 Fourier series for periodic distributions
The Fourier series for periodic distributions is defined analogously to the Fourier transform for distributions. However, one has to bear in mind that the domain and the image of the Fourier transform on the torus are not the same.
3.53 Definition. Tempered distributions on the lattice are elements of the dual space of the Schwartz space on the latticeS1pZnq:“LpSpZnq,Cq.
3.54 Remark. ([RT10, 3.1.7, p. 300]) Any tempered distributionsuPS1pZnq is of the form upψq “ ÿ
ξPZn
fpξqψpξq pψPSpZnqq,
wheref :ZnÑC grows at most polynomially, i.e. there areC ą0,M ă 8with
|fpξq| ďCp1` |ξ|2qM{2 pξ PZnq.
Hence, u is pointwise well-defined. This can be seen if we use the Schwartz function δk:ZnÑC, ξÞÑ
#
1 ξ “k 0 ξ ‰k, which yields
upδkq “ ÿ
ξPZn
fpξqδkpξq “fpkq.
Therefore, we write upkq and meanupδkq for kPZn.
3.55 Definition. For a periodic distribution uPD1pλTnq the Fourier transform is given by FT:D1pλTnq ÑS1pZnq, F
Truspϕq “upF´1
T rϕsqq pϕPSpZnqq.
3.56 Lemma. Let uPD1pλTnq and kPZn. The kth Fourier coefficient can be expressed by
uppkq:“F
Truspkq “u ˆ
exp ˆ
´2π λ k¨
˙˙
. This yields the representation
FTrus:C8pλTnq ÑC, ϕÞÑ ÿ
kPZn
uppkqϕpkq.
Proof. ForuPD1pλTnq andkPZn we have, considering Remark 3.54, FTruspkq “u
´ F´1
T
” δqk
ı¯
“u
¨
˝xÞÑ ÿ
ξPZn
ei2πλxξδ´kpξq
˛
‚“u
´
xÞÑe´i2πλxk
¯ ,
where we have used δqk“δ´k.
3.57 Theorem. The Fourier transform F
T for periodic distributions is a bijection and contin-uous with the contincontin-uous inverse given by
F´1
T is a bijection from SpZnq toC8pλTnq, we can identify eachϕqPSpZnq with exactly oneψPC8pλTnq. Hence, we
Trψs PSpZnq. Therefore, u is well-defined. Furthermore, sincev,F
T and inversion are continuous, hence it isu. Therefore,uPD1pλTnq. Then, we have forϕPSpZnqwithrpϕq:“ϕq
and a similar calculation shows the other identity.
To show that F
T is continuous, it is sufficient to show that F
T is continuous at 0, since F
T is
T is continuous by Definition 2.29.
Interchanging F
T and F´1
T above yields the continuity of F´1
T .
3.2 Fourier series
3.58 Corollary. For uPD1pλTnq, we have the representation u“
ÿ
kPZn
uppkq
” ei2πλk¨
ı ,
where
” ei2πλk¨
ı
denotes the regular distribution induced by the functionxÞÑei2πλkx in D1pλTnq.
Proof. Since F´1
T ˝F
T “idD1pλTnq, we have forϕPC8pλTnq upϕq “F´1
T ˝F
Tuϕ“F
Trus pF
Trϕsq “q ÿ
kPZn
uppkqF
T
” ψq
ı pkq, where we have used Lemma 3.56 for the representation of F
Trus. Furthermore, we have for kPZn
FTrϕspkq “q ż
λTn
e´i2πλkxϕp´xqdx“ ż
λTn
ei2πλkxϕpxqdx“
” ei2πλ¨
ı pϕq.
This yields
upϕq “ ÿ
kPZn
uppkq
” ei2πλk¨
ı pϕq for every ϕPC8pλTnq, which concludes the proof.
4 The Sampling Theorem
In his original publication [Sha49, p. 448], Shannon makes use of the fact, that a compactly supported function can be expanded as a Fourier series. Although he doesn’t take into account the prerequisites for both Fourier transformation and Fourier series expansion, the idea can be turned into a formally valid proof as is done in [Mar01, p. 52ff], [Nat89, p. 56] and [Den89, p. 9, Beweis 1]. In the meantime, many other methods have been developed, of which we will elaborate some. But first, we have to establish the general prerequisites for any kind of sampling theorem - the concept of band limitation.
4.1 Band limited functions and distributions
4.1 Definition. LetW ą0.
(i) For1ďpď 8a function f PLppRnq XCpRnq is called band limited with band-width W or W-band limited for short if there isgPL1pr´W, Wsnqsuch thatf “F´1rgs.
(ii) u P S1pRnq is called band limited with band-width W or W-band limited for short if suppFrus Ď r´W, Wsn.
(iii) An entire functionf :CnÑCis of at mostexponential typeσě0if there isC ą0such that|fpzq| ďCexppσ|Imz|qfor every zPCn.
(iv) BpσpCnq (1 ď p ď 8) is the class of all entire functions of at most exponential type σ whose restriction to Rn belongs toLppRnq.
(v) The Paley-Wiener space PWWpCnqis defined as
PWWpCnq:“ tf :CnÑCentire,DAě0, N PN0:@zPCn:|fpzq| ďAp1`|z|qNeW|Imz|u.
Furthermore, for 1ďpď 8we define
PWpWpCnq:“ tf PPWWpCnq:DgPLppr´W, Wsnq:f|Rn “F´1rgsu.
4.2 Lemma. For 1 ď p ď 8 and σ ě 0, BσppCnq forms a vector space over C by pointwise addition and scalar multiplication.
Proof. Since the space of all entire functions forms a vector space as well as LppRnq does, it remains only to show that the sum of two elements of BpσpCnqagain is of at most exponential type σ. This can be verified using the triangular inequality.
4.3 Lemma. ([BSS88, Eq. 2.2, p. 6]) Let f PBσppCq. Then the following inequalities hold for every hą0:
}f}p ďsup
uPR
˜
?h 2π
8
ÿ
k“´8
|fpu´hkq|p
¸1{p
ď p1`hσq }f}p (4.1)
(Nikol’ski˘i’s inequality) and
›
›
›fprq
›
›
›p ďσr}f}p (4.2)
(Bernstein’s inequality), where fprq denote therth (complex) derivative of f (r PN0).
4.4 Theorem (Paley-Wiener for functions). ([Rud73, 7.22, p.181]) (a) IfϕPDpRnq has its support in rB :“ txPRn:|x| ďru, and if
fpzq “ 1 p2πqn{2
ż
Rn
ϕptqexpp´iztqdt pzPCnq, (4.3) then f is entire and there are constants γN ă 8such that
|fpzq| ďγNp1` |z|q´Ner|Imz| pzPCn, N PN0q. (4.4) (b) Conversely, if an entire function f satisfies the conditions (4.4), then there exists ϕ P
DpRnq, with support in rB such that (4.3) holds.
4.5 Theorem (Paley-Wiener for distributions). ( [Rud73, 7.23, p. 183])
(a) IfuPD1pRnq has orderN PN0, its support is contained in rB :“ txPRn:|x| ďru and if
fpzq:“ p2πq´n{2upexpp´iz¨qq pzPCnq, (4.5) then f is entire, the restriction of f to Rn is the Fourier transform of u and there is a constant γ ă 8such that
|fpzq| ďγp1` |z|qNer|Imz| pzPCnq (4.6) (b) Conversely, if f is an entire function in Cn which satisfies (4.6) for some N PN0 and
some γ, then there exists uPD1pRnq with support in rB, such that (4.5) holds.
4.6 Lemma. Let f PLppRnq XCpRnq be W-band limited. Then f PC8pRnq.
Proof. Let f PLppRnq XCpRnq beW-band limited. By definition, there is gPL1pr´W, Wsnq such that f “F´1rgs. rqgsis a regular distribution with support contained in txPRn :|x| ď
?nWu. rgsshas order 0 as proved in Example 2.55, because it is a regular distribution. Define f˜:Cn ÑC by f˜pzq :“ p2πq´qg{2pexpp´z¨qq. Then, the Paley-Wiener Theorem 4.5 yields that f˜is entire and thatf˜restricted toRnis the Fourier transform ofrqgs. SinceqgPL1pr´W, Wsnq, its Fourier transform exists. By Remark 3.19, we have Frrqgss “ rFrqgss “ rF´1rgss. Hence, we infer f˜pxq “F´1rgs pxq “fpxq for x PRn. Since f˜is entire, it is infinitely differentiable for all xPRn. Therefore,f PC8pRnq.
4.7 Lemma. ([Hig96, 6.3, p. 52]) Let 1ăpď2, 1{p`1{q“1 then PWpσpCnq ĂBσqpCnq. Proof. Let f PPWpσpCnq. Then there isgPLppr´σ, σsnq with
fpzq “ p2πq´n{2 ż
r´σ,σsn
gpxqeizxdx.
4.1 Band limited functions and distributions
forzPRn. Sincef is entire, this holds also forzPCndue to the Identity Theorem 2.41. Using Lppr´σ, σsnq ĂL1pr´σ, σsnq, we get
|fpzq| ď p2πq´n{2 ż
r´σ,σs
|gpxq|e´x|Imz|dxď p2πq´n{2}g}1eσ|Imz|,
hencef is of exponential type at mostσ. By the Hausdorff-Young Theorem 3.14, which is also true for the inverse Fourier transform, we get f PLqpRnq. Therefore,f PBσqpCnq.
Forp“2, we have equality as stated in the following lemma:
4.8 Lemma. Let σ ą0. Then PW2σpCnq “B2σpCnq.
Proof. We only have to showBσ2pCnq ĂPW2σpCnq. Letf PB2σpCnq. Thenf|Rn PL2pRnq. The Hausdorff-Young Theorem 3.14 providesFrfs PL2pRnq. The Paley-Wiener Theorem 4.5 shows that Frfsis a distribution with support in σB Ă r´σ, σsn. Both Fourier transforms coincide by Remark 3.19. Hence, we conclude f|Rn “F´1rgs for some gPL2pr´σ, σsnq. Furthermore, there holds|fpzq| ďCp1` |z|q0eσ|Imz|by definition of Bσ2pCnq. Therefore,f PPW2σpCnq.
4.9 Definition. Foraą0, thesinca-function is defined as sinca:RÑR, xÞÑ
#sinpaxq
ax , x‰0
1, x“0. (4.7)
This definition can be extended to Rnby
sinca:RnÑR, x“ px1, . . . , xnq ÞÑ
n
ź
k“1
sincapxkq.
4.10 Lemma. (i) sincaPL2pRnq and sinca is a-band limited,
(ii) sincapkπ{aq “ δk0 for k P Zn with δk0 being Kronecker’s delta (δk0 “ 1 if k “ 0 and δk0 “0 if k‰0).
Proof. (i) First we considern“1. By definition }sinca}22 “
ż
R
ˇ ˇ ˇ ˇ
sinpaxq ax
ˇ ˇ ˇ ˇ
2
dx“2 żπ{a
0
sin2paxq paxq2 dx`2
ż8
π{a
sin2paxq paxq2 dx.
For the expansion, we exploit the fact that psinpxq{xq2 is symmetric. Since sin2pxq ď1 for all xPR, we get the estimate
ż8
π{a
sin2paxq paxq2 dxď
ż8
π{a
1
paxq2 “ 1
aπ. (4.8)
Assin2pxq{x2 is continuous in r0, π{as, it is bounded by some constant cand we get the estimate
żπ{a 0
sin2paxq
paxq2 ďcπ{a. (4.9)
Using Equations (4.9) and (4.8), we get }sinca}22 ă 8.
Now we have to consider the Fourier transform ofsinca. We already know by Example 3.8 thatF“
χr´a,as‰
“ p?
2πa{πqsinca. Sincesinca, χr´a,as PL2pRqand the Fourier transform is a linear isometry on L2pRq, we can use Corollary 3.12 and get
χr´a,aspxq “χr´a,asp´xq “F2“ for almost every xPR. Hence
Frsincas “ π
?2πaχr´a,as and therefore sinca isa-band limited.
For ną1, we get by an iterated application of Tonelli’s theorem }sinca}22 “ where }sinca}2,1 is the norm of the one dimensional sinca as evaluated above. Hence sinca P L2pRnq. Similarly, the Fourier transform can be evaluated component-wise be-cause the exponential term splits into factors, hence
Frsincas pxq “ for almost every xPRn. Thereforesinca isa-band limited.
(ii) Forn“1, the statement sincapkπ{aq “δk0 is clear, since sincap0q “1by definition and