Ge. Math. Notes, Vol. 3, No., March 20, pp.4-25 ISSN 229-784; Copyright c ICSS Publicatio, 200 www.i-csrs.org Available free olie at http://www.gema.i Fourier Trasform i L p () Spaces, p Devedra Kumar ad Dimple Sigh Departmet of Mathematics esearch ad Post Graduate Studies M.M.H.College, Model Tow, Ghaziabad-2000, U.P.Idia E-mail:d kumar00@rediffmail.com (eceived: 7--0/ Accepted: 3-2-0) Abstract A method for restrictig the Fourier trasform of f L p (), p, spaces have bee discussed by usig the approximate idetities. Keywords: Approximate idetities, covolutio operator, Schwartz space ad atomic measure. Itroductio Let f L ().The Fourier trasform of f(x) is deoted by f(ξ) ad defied by f(ξ) = f(x)e iξx dx, ξ. () 2π If f L () ad f L (), the the iverse Fourier trasform of f is defied by f(x) = f(ξ)e iξx dξ (2) 2π for a.e. x. If f is cotiuous, the(.2) holds for every x. It is kow that several elemetary fuctios, such as costat fuctio, si wt, cos wt, do ot belogs to L () ad hece they do ot have Fourier trasforms. But whe these fuctios are multiplied by characteristic fuctio, the resultig fuctios belogs to L () ad have Fourier trasforms. May
Fourier Trasform i L p () Spaces, p 5 applicatios, icludig the aalysis of statioary sigals ad real time sigal processig, make a effective use of Fourier trasform i time ad frequecy domais. The remarkable success of the Fourier trasform aalysis is due to the fact that, uder certai coditios, the sigal ca be recostructed by the Fourier iversio formula. Thus the Fourier trasform theory has bee very useful for aalyzig harmoic sigals or sigals for which there is o eed for local iformatio. O the other had, Fourier trasform aalysis has also bee very useful i may other areas, icludig quatum mechaics, wave motio ad turbulece. By Lebesgue lemma we have if f L () the lim ξ f(ξ) = 0, it follows that Fourier trasform is a cotiuous liear operator from L () ito C o (), the space of all cotiuous fuctios o which decay at ifiity, that is, f(x) 0 as x. oughly we say that if f L (), it does ot ecessarily imply that f also belogs to L (). Bellow [] ad eihold - Larsso [2] costructed examples of sequece of atural umbers alog which the idividual ergodic theorem holds i some L p spaces (good behavior) ad ot i others (bad behaviour). I particular, well behaved sequeces were perturbed i such a way that good behavior persists oly i certai spaces. I the preset work we provide a method for restrictig the Fourier trasform of f L p () spaces usig the poitwise covergece of covolutio operators for approximate idetities. Defiitio.. Let ϕ L () such that ϕ(0) =. The ϕ ε (x) = ε ϕ(x/ε) is called a approximate idetity if (i) ϕ ε(x)dx = (ii) sup ε>0 ϕ ε(x) dx < +, (iii) lim ε 0 x >δ ϕ ε(x) dx = 0,for all δ > 0. Proof. Properties (i) ad (ii) ca be proved by observig ϕ ε (x)dx = ε ϕ(x/ε)dx = ϕ(x/ε)d(x/ε) =. For (iii), we have x >δ ϕ ε (x)dx = x >δ ε ϕ(x/ε)dx = δ δ ε ϕ(x/ε)dx + ε ϕ(x/ε)dx.
6 Devedra Kumar et al. Substitutig y = x/ε, we get lim ε 0 δ/ε ϕ(y)dy + δ/ε ϕ(y)dy = 0. Defiitio.2. A sequece of fuctios {φ } N such that φ (x) = φ(x) where =,, ε 0 is called a approximate idetity ε if (i) φ (x)dx = for all, (ii)sup φ (x) dx < +, (iii)lim x >δ φ (x) dx = 0 for every δ > 0. I the cosequece of above Defiitio.2, we ca easily prove the followig propositio. Propositio.. A sequece of fuctios{φ } N with φ 0, φ (0) = is a approximate idetity if for every ε > 0 there exists o N so that for all o we have ε ε φ > ε. Let us cosider the class S() of rapidly decreasig C fuctios o i.e., Schwartz class such that S() = {f :, sup x (x dm f)(x) < }, m N (0). dxm It is well kow that if f S() the f S() ad S() L p ().To prove the deseess of S() L p (), we have For p <, ρ S() ρ(x) ρ(x) p dx c + x. c p ( + x ) p < which gives ρ L p (). Defie a sequece {ρ N } such that ρ N (x) = { f(x), if N x N; 0, otherwise ;
Fourier Trasform i L p () Spaces, p 7 ρ N S(), f L p () such that ρ N f p dx 0 as N. Hece S() is dese i L p (). emark.. If 0 φ(x) S() ad φ(0) =. The φ (x) = φ(x) is a approximate idetity. Propositio.2. If f L () ad φ S() the φ f S(). Proof. We have φ f = φ(y)f(x y)dy or d (φ f) = dx φ(y) d f(x y)dy dx x d (φ f) = x f(x y) d dx dy φ(y)dy, substitutig x y = z, we obtai, = f(y) x d φ(x y)dy dx usig x y x + y 3 x 2 = y > x 2, we get f(y) x d φ(x y)dy + dx y x 2 f(y) x d φ(x y)dy 0. dx Propositio.3. If φ (x) is a approximate idetity ad f L p () the φ f f L p (). Proof. Cosider [ (φ f)(x) f(x) p dx] /p = [ = [ dx dx φ (x y)f(y)dy f(x) p ] /p φ (y)f(x y)dy f(x) p ] /p
8 Devedra Kumar et al. usig f(x) = f(x)φ (y)dy i above we obtai [ dx φ (y)(f(x y) f(x))dy p ] /p [ + [ + dx φ (y) p f(x y) f(x) p dy] /p y >δ dx φ (y) p. f(x y) f(x) p ] /p (3) y δ dy φ (y) [ dx f(x y) f(x) p ] /p dy φ (y) [ f(x y) f(x) p dx] /p (4) dy φ (y) (2 f p ) + dy φ (y) sup[ f(x y) f(x) p dx] /p. y δ y δ y >δ y δ y >δ Proceedig limits as, the right had side teds to zero sice sup[ f(x y) f(x) p dx] /p 0. y <δ Hece the proof is completed. Propositio.4. Let φ = α ϕ + ( α )σ, where {ϕ } N, {σ } N are approximate idetities ad 0 α. (a) For p < + ad every f L p (), lim (φ ϕ ) f 0 ad lim (φ σ ) f 0. (b)for every f L (), lim (φ ϕ ) f 0 a.e.. (c) For p <, if ( α ) p lim (φ ϕ ) f 0 a.e.. < +, the for every f L p (), Proof.(a) Set p, ad f L p (). I view of Mikowski s iequality (φ ϕ ) f p ( α )( σ f f p + ϕ f f p ) ad usig Propositio.3 we obtai (φ σ ) f p 0. (b)for f L (), (φ ϕ ) f (φ ϕ ) f 0 by part(a).
Fourier Trasform i L p () Spaces, p 9 (c)for f L p () ( α ) p σ f(x) p dx = ( α )σ f p p ( σ ) p f p p< +. The ( α )σ f 0 a.e.. Similarly (α )ϕ f 0 a.e.. Defiitio.3. A approximate idetity {φ } is called L p good if φ f f a.e. for all f L p (), ad it is called good if it is L p good for every p +. A approximate idetity {φ } is called L p bad if there exists f L p () such that φ f f o a set of positive measure. Defiitio.4. Let {ϕ } N ad {σ } N be approximate idetities, α be a sequece of real umbers with 0 α ad α. We call perturbed approximate idetities ay approximate idetity {φ } N of the form φ ϕ + ( α )σ. 2 Mai esults Theorem 2.. (i)give ay good approximate idetity {ϕ } N there exists a perturbed approximate idetity {φ } N such that f L q () for q p, p [, ) ad (φ f)(ξ) = φ (ξ) f(ξ) ( f(ξ)) φ (ξ) f(x) ( φ (ξ) f(ξ)) f(x) for q < p. (ii)( φ (ξ) f(ξ)) f(x) for q > p ad ( φ (ξ) f(ξ)) f(x) for q p. (iii)( φ (ξ) f(ξ)) f(x) for q =
20 Devedra Kumar et al. ( φ (ξ) f(ξ)) f(x) for q <. Proof. (i) Let g (x) = 2π e ixξ φ (ξ) f(ξ)dξ = e φ ixξ (ξ) e iξy f(y)dydξ 2π 2π = e i(x y)ξ φ (ξ) f(y)dy 2π = φ (x y)f(y)dy 2π or = (φ f)(x) ( φ (ξ) f(ξ)) = e ixξ φ (ξ) f(ξ)dξ = (φ f)(x). 2π Fix q p ad takig α = ( log 2 ) /p Sice Σ ( α ) q < + ad ϕ is a L q good approximate idetity, usig Propositio.4 we obtai that {φ } is also a L q good approximate idetity. Hece for q p, (φ f)(x) f(x). Now we have to prove that for each q < p. there exists f q L q () so that lim sup κ x κ dκ dx κ (φ κ f q ) o a set of positive measure. Set Choose r = f q (x) = (x log 2 (x/2)) /q χ [0,](x) L q (). +/p (log ), a 2/p = r p+ = /p (log ) 2 p(p+), J = [a r, a + r ] ad U = [ a + r, a + + r + ], for sufficietly large ad for all κ, x U κ, φ κ f q (x) ( α κ )σ κ f q (x) (κ log 2 σ κ) /p κ (y)f q (x y)dy. J κ
Fourier Trasform i L p () Spaces, p 2 Now, we get or φ κ f q (x) f q(c rκ (log κ) 2/p+ ) (κ log 2 κ) /p J κ σ κ (y)dy f q (C rκ (log κ) 2/p+ ) = κ /q+/pq (log κ) pq(p+) C /q (log(c/2κ (p+)/p (log κ) 2/p(p+) )). 2/q 2 The φ κ f q (x) Cκ q p + pq Hq (κ) > κ δ δ, where H q (κ) = (log κ) pq(p+) 2 p C /q (log C/2κ (p+)/p (log κ) 2/p(p+) ) 2/q 2 ad 0 < δ < /q /p + /pq. So or d κ dx κ (φ κ f q (x)) C dκ dx κ (κ/q /p+/pq H q (κ)) x d dx (φ f q (x)) x Jκ f q (x y) d dy σ κ(y)dy for κ x κ dκ dx κ (φ κ f q (x)) x d dx δ x d dx ( (x y) pδ ) = x ( ) (pδ + )! (pδ)!(x y) pδ+ x ( ) (pδ + )! (pδ)!cr (log ) 2/p+ (log ) 2δ/p+ as. I view of Sawyer s Priciple [3] there exists a fuctios f L q ([0, )) L q () such that lim sup x d (φ dx f) a.e. o a set of positive measure i, It follows that φ f ot belogs to S() or φ f f or φ (ξ) f(ξ) f(x)
22 Devedra Kumar et al. for q < p. (ii) Let p be a decreasig sequece of real umbers such that p > p 2 >...p >... p. for each p i we ca costruct a perturbatio {φ i } of {ϕ } that is L q good for q p i, ad L q bad for q < p i. Cosider a sequece of blocks {B κ } κ N, where B κ = {φ κ κ +,..., φ κ κ, } ad { κ } is a sequece of positive itegers icreasig to ifiity. Let D κ = { κ +,..., κ }, ad let {φ } = U κ B κ. Now fix q > p. There exists o N so that for all > o we have p < q, κ= o D κ ( α κ ) q κ= o ( ( log 2 ) q/po D κ log 2 )q/po < +. Usig Propositio.4(c) we get φ f f for f L q (), q > p, or φ (ξ) f(ξ) f(x) for q > p. Now cosider a sequece Ci N as i. Sice {φ i } is L q bad for all q < p i, it is also L p bad. These exists f i L p ([0, )) ad λ N i > 0 such that { sup φ i f i (x)} > i > φ i (x)f i (y) p dy J κ Set > C N f i (x λ N i ) p p = 2C N i, [ f i (x λ N i ) p = 2 i, C N = 2 (i )p+ C N i ]. It follows that there exists i > i, so that { sup i < i (φ i f i )} > C N i. f = i f i, the f p i f i p 2. Suppose that {φ }satisfies a weak (p, p) iequality i L p ([0, )). We kow that if µ be a fiite positive Borel measure, the these exists a sequece µ of atomic measure that coverges to µ weakly or if f has compact support the dµ f(x) f(x)dµ or
Fourier Trasform i L p () Spaces, p 23 µ µ weakly. If f L (), dµ = f(x) dx is a fiite Borel measure, so we ca fid µ = N Ci N δ λ N i µ weakly. ı= Cosider {sup(φ i f)} = φ i (y)f(x y) p dy J κ φ i (y)d µ (x y) p dy J κ N f(x λ N i )Ci N p p N i= Ci N i= f(x λ N i ) p p C N o f p p = 2 p C N o. () O the other had, {sup(φ f)} { sup (φ i f(i))} > Ci N (2) i < i Combiig Equatios (2.) ad (2.2) we get C N o > C N i But Ci N as i +. Hece φ f f i L p ([0, )). Sice the spaces L q ([0, )) are ested, {φ } is L q ([0, ))-bad for all q p. Therefore, such a choice of { κ } makes {φ }L q () bad for all q p. This implies that φ (ξ) f(ξ) f(x) for q p. (iii) Let {ϕ } N be a good approximate idetity, ad let {ζ } N be ay approximate idetity. Let {p } be a sequece of real umbers satisfyig p < p 2 <... < p
24 Devedra Kumar et al. Cosider the blocks {B κ }, where each block B κ is related to p κ. for i D κ, let φ i = α κ i ϕ κ i + ( α κ i )σ κ i. Choose κ such that α κ i. The sice {ϕ } is L good, ad Sice ϕ f f a.e. for all f L (), α κ i ϕ κ i f f a.e. for all f L (). σ κ i f(x) f. ( α κ i )σ κ i f 0 a.e. for all f L (). It follows that φ f f a.e, for all f L (). This implies that ( φ (ξ) f(ξ)) f(x) for q =. The approximate idetity {φ κ } is L pm bad for every m {,..., κ}, sice it is L q bad for every q p κ. There exists fm κ L pm ([0, )) with fm(x κ λ κ(n) m ) = 2 κ, λ κ(n) m > 0 ad κ m > m κ so that { sup (φ κ fm)} κ > C N fm(x κ λ κ(n) m ) pm p m κ < κ m = CN κ 2 κpm Let f = κ κ o f κ κ o, the f pκo < 2. So Hece {sup(φ f)} C0 f pκo p κo 2 pκo Co N. (3) {sup(φ f)} { sup (φ κ fκ κ o )} κ < κ usig (2.3) ad (2.4) we get > CN κ 2 κpκo (4).Thus we coclude that C N o > Hece the proof is completed. C N κ + 2κpκo(κ+) φ (ξ) f(ξ) f(x) For q <.
Fourier Trasform i L p () Spaces, p 25 efereces [] A. Bellow, Perturbatio of a sequece, Advaces i Mathematics, 78(989), 3-39. [2] K. eihold-larsso, Discrepacy of behaviour of perturbed sequeces i L p spaces, Proc. Amer. Math. Soc., 20 (994), 865-874. [3] S. Sawyer, Maximal iequalities of weak type, A. of Math., 84 (2)(966), 57-74.