算法随机性与逼近元

1 Introduction

In this paper we propose an effective version of Differentiation Theorem for Approximate Identities.Recent results have shown that algorithmic randomness is a very useful tool to study the effective aspects of classical analysis.Many theorems have been investigated,including the ergodic theorem([1,4]),Lebesgue differentiation theorem([7])and many other topics([3,2]).Generally,these results have this form:

A real xhas a certain property that holds almost everywhere

⇔xsatisfies a certain randomness notion.

Approximate identity is a very important topic in the study of functional and harmonic analysis.The Differentiation Theorem for Approximate Identities is a basic theorem concerning approximate identity.It says that for a k∈L1with a radially decreasing majorant such that kε=ε-1k(ε-1x),we have kε∗f→f,for all xexcept a null set as ε→0.

This theorem also indicate the Lebesgue differentiation theorem.It has been shown in[7]that the Lebesgue differentiation theorem can characterize Schnorr randomness.In this paper,our main result is that the Differentiation Theorem for Approximate Identities also characterize Schnorr randomness.And we will obtain an effective version of Lebesgue differentiation theorem which is slightly different from[7].

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In section 2,we present the necessary preliminaries.Section 3 and 4 contain the proof of the two directions of our main theorem respectively.

两种算法的运行时间以及ERLE参数如表1所示。输入信号为语音情况下的两种算法的失调量如图3所示,噪声输入情况下的失调量如图4所示。两组测试的结果一致,NLMS和FDNLMS算法的收敛时间、失调量、ERLE相差不大,但是,在处理速度方面,FDNLMS算法由于运算量大幅度降低,所用的处理时间只占NLMS算法的8.5%,说明算法的效率高,实用性强。

2 Preliminaries

The classical version of the Differentiation Theorem for Approximate Identities can be found in[5].To introduce this theorem,we begin with some definitions:

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Definition 1(convolution) Let f,gbe in L1(R).The convolution f∗gis

 

Itisknownthatconvolutionsatisfiescommutativity,associativityanddistributivity.

Definition2 Anapproximateidentity(asε→0)isafamilyofL1(R)functionskε(0＜ε≤1)with the following three properties:

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·There exists a constant c＞0 such that for all ε.

· for all ε＞0.

mials gksuch that‖f-gk‖L1≤2-k,

Let B(x,θ)be the neighborhood(x-θ,x+θ)and Vcbe R-V.B(E,θ)denotes{y|∃x∈E(|y-x|＜θ)}where E⊆R.We use χAas the characteristic function of A,where A⊆R.Next comes the basic theorem concerning approximate identities:

Theorem 1 ([5])Let kεbe an approximate identity on R.

(1)If f∈L1,then‖kε∗f-f‖L1→0 as ε→0.

(2)If fis continuous in a neighborhood of a compact set Eand bound on R,then kε∗fconverges uniformly to fon Eas ε→0.

In this paper we focus on a special kind of approximate identity.Let k(x)be a integrable function with integral one.Then we define kε:=ε-1k(ε-1x).It is straightforward to see kεis an approximate identity.

A function fis called radial,if f(x)=f(y)whenever|x|=|y|.If such fis decreasing on[0,∞)and f≥|g|in R,we call fa radial decreasing majorant of g.

Definition 3(Dominated Approximate Identity) Given a function k∈L1(R)such that,we call kεa dominated approximate identityif:

 

·khas a radial decreasing majorant Kwhich is continuous except at a finite number of points.

Definition 4 The function

 

is called the centered Hardy-Littlewood maximal functionof f.Actually,if we define,then we have

 

Here are two classical results which are useful in our proof.We refer to[5]for more details.

Lemma 1(2.1.12,[5]) If kεis a dominated approximate identity with the radially decreasing majorant K,then the estimate

 

is valid for all function f∈L1(R).

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Lemma 2(2.1.6,[5]) If f∈L1(R),then

 

Next comes the differentiation theorem for approximate identities,which can be viewed as a generalization of Lebesgue differentiation theorem:

Theorem2(DifferentiationTheoremforApproximateIdentities,2.1.17,[5]) Letkbea dominated approximate identity with a radially decreasing majorantK.Then kε∗f→fa.e.as ε→0 for all f∈L1(R).

Our job is to find the effective version of this theorem.The first thing we need to do is extending the computable notion from N to R.The next definitions can be found in[8].Let akbe a computable enumeration of all rationals on R.

Definition 5 A real number is computable if there are two computable functions h:N→N and d:N→N s.t.

 

This computable real xis coded by the indices of hand d.Let rebe the computable real on R indexed by e.

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Definition 6 Eis a compact set on R.A function f:E→R is computable if:·fis sequentially computable:there is a computable function h:N→N s.t.f(re)=rh(e);·fis effectively uniformly continuous on E:there is a computable function d:N→Q s.t.for all x,y∈E:

 

A function f:R→R is computable if:

Lemma 5 ([7,6])If Aand Bare Σ1sets andµ(A)andµ(B)are computable,then A∩Band A∪Bare Σ1sets andµ(A∩B)andµ(A∪B)are computable.

·fis sequentially computable:there is a computable function h:N→N s.t.f(re)=rh(e);

·fis effectively uniformly continuous on every compact subset:there is a computable function d:N2→Q s.t.for all x,y∈[-N,N]:

 

The computable function fis coded by the indices of hand d.Notice that for every[-N,N],the functions sup[-N,N](f)andfdxare uniformly computable on N.([8])A function fis compactly supported,if f(x)=0 everywhere outside a compact set.

Definition 7 Eis a compact set of R.A function f:E→R is L1(E)-computable,or L1comp(E)for short if:there is a sequence of uniformly computable functions fkon E→R and a computable function d:N→N s.t.

 

A function f:R→R is L1(R)-computable,or L1comp(R)for short,if there is a sequence of uniformly computable compactly supported functions fkon R→R and a computable function d:N→N s.t.fkis supported on[-k,k]and:

 

f∈L1comp(R)is coded by the indices of fkand d.

It is easy to see that we can extend everyfunction to afunction uniformly.Notice that we actually define an L1-computable function fas an equivalence class on L1.We will select a representativefrom such equivalence class such thathas a certain value when xis Schnorr random.

Definition 8 A sequence of set Un⊆R,n∈N is uniformly effectively open,or uniformly Σ1,if

 

where an,i,bn,iis a double sequence of uniformly computable reals.

Based on the conception of uniformly effectively open sets,we can define different versions of algorithmic randomness.Intuitively,a real is random if it avoids all‘effectively null’sets.The most commonly accepted randomness notions are Martin-Löf randomness and Schnorr randomness.

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Definition 9 A Martin-Löf testis a uniformly Σ1sequence of Un⊆R such thatµ(Un)≤2-nfor all n.A point x∈R is said to pass the test if x/∈∩nUn.xis Martin-Löf randomif it pass every Martin-Löf test.

A Schnorr testis a uniformly Σ1sequence of Un⊆R such thatµ(Un)is uniformly computable on nandµ(Un)≤2-nfor all n.A point x∈R is said to pass the test if.xis Schnorr randomif it pass every Schnorr test.

Theorem3(EffectiveWeierstrassTheorem,[8])EisacompactsetonR.Afunctionfon Eis computable if and only if there is a computable sequence of rational polynomials pm(x)which converges effectively to fin uniform norm:there is a computable function d:N→N such that|pm-f|≤2-nif m≥d(n).

A function fon R is computable if and only if for any N＞0 there is a computable sequence of rational polynomials pm,N(x)which converges effectively to fon compact set[-N,N]i.e.,there is a computable function d:N×N→N such that|pm,N-f|≤2-nif m≥d(n,N).

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for all x∈E.Notice that this δdepends on θ.

Proof See[8].□

Lemma 3 ([7,9])Suppose f∈L1comp[0,1]and fkis a sequence of uniformly computable rational polynomials such that‖f-fk‖L1≤2-k.

·(Existence)The limit limk→∞fk(x)exists on all Schnorr random x.

In this section we will prove that the differentiation theorem for approximate identities holds for all Schnorr random x∈R.First,we show how it works when fis computable on a compact set.

·For any neighborhood Vof 0 we have as ε→0.

 

for all Schnorr random x

It is straightforward to see the result above also holds in L1([-N,N]).Note that if fis L1(R)-computable,then f↾[-N,N]is L1([-N,N])-computable,so actually this result also holds in L1(R).We useto denote limk→∞fk(x)when fkis the sequence as in the lemma above,which implies the value of(x)does not depend on the choice of fkwhen xis Schnorr random.

A dominated approximate identity kεis computable if kis L1-computable.Our main result is

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Theorem 4(effective version of Differentiation Theorem for Approximate Identities)Let kεbe a computable dominated approximate identity.Thenfor allif and only if xis Schnorr random.

We will use the following lemmas.

Lemma 4 ([7])Letsuch that Anis uniformly c.e.andµ(An)is uniformly computable andµ(An)≤2-n.Then Ais c.e.andµ(A)is computable.Moreover,this holds uniformly.

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Let an∈R for n∈N and limnan=a.Then we say that it has a computable speed of approximation if there is a computable function hs.t.for all k≥h(n)we have|ak-a|＜2-n.

3 Schnorr random points satisfy differentiation theorem for approximate identities

·(Uniqueness)Given another sequence of uniformly computable rational polyno

Lemma 6 ,thendyis a computable function of xon R.In particular, is computable.

According to effective Weierstrass Theorem,the‘a sequence of uniformly computable functions’in the definition of L1-computable function can be replaced by ‘a sequence of uniformly computable rational polynomials’.

Lemma7 Let kεbe a computable dominated approximate identity.Eis a compact set andFisacompactneighborhoodofE,B(E,θ)⊆F,fiscomputableonFandbounded on R,i.e.there is a M＞0 s.t.|f|＜M.Then kε∗f→funiformly as ε→0 on E.Moreover,the speed of this approximation is uniform computable on f↾F,Mand θ.

Proof Since kεis a dominated approximate identity,let c=‖kε‖L1=‖k‖L1.Let η＞0 be an arbitrary real.We know that∫kε(y)dy=1,so we have

 

where Vis the neighborhood B(0,δ).We need to find a sufficient small δ＜1.Since fis computable on F,there is a δsuch thaif x∈Eand|y|＜δ.So

 

Moreover,the pm(x)and pm,N(x)above can be obtained effectively from the index of f.

We know fhas a bound|f|＜M.Then

 

Since kεis an approximate identity,the

It remains to calculate the speed of this convergence.Obviously it is determined by the speed of this convergence∫Vc|kε|dy→0 as ε→0.For convenience we let δbe a rational.Replace t=ε-1ywe have

NLMS系统的结构如图1所示,x(n)表示远端参考信号,v(n)表示近端信号,w(n)表示实际的回声路径,d(n)表示麦克风所采集到的信号,e(n)表示误差信号,y(n)表示表示输入,x(n)通过回声信道所得的回声信号,表示自适应回声抵消滤波器的权矢量即w(n)的估计值,表示自适应滤波器模拟回声信号的估计值。单远端模式下,v(n)为0。NLMS算法利用e(n)来对进行自适应调整,使其逼近w(n)。

 

It is a computable function of εby Lemma 6.So we use θand fto compute δ,then use δand kto find the ε0to ensurefor all ε≤ε0.We have

 

The following result can be viewed as an effective version of Egorov Theorem.

Lemma 8(Lemma 3.6,[7]) Letand E=[-N,N].fkis a sequence of uniformly computable rational polynomials on Esuch that

 

then there is a Schnorr test Umon Esuch that for any m,fk→uniformly on E-Um.Moreover,the speed of this convergence is uniform on m.

ByLemma3,weknowthatlimforSchnorrrandomx.Thefollowing lemma is very useful in our proof.

Lemma 9(Lemma 3.3,[7]) Let Sbe a bounded set in R such that Sis first-order definableinthefieldofR.ThenthemeasureofSisacomputablerealnumber.Moreover,this holds uniformly on the first-order definition of S.

Theorem 5 Let kεbe a computable dominated approximate identity.Then for all f∈and Schnorr random xwe have

 

Proof Notice that the value kε∗f(x)ignores fon a null set of R,so we can assume f= without losing generality.The main idea is to prove the Lemma 8 also holds for kε∗f(x)and kε∗fk(x)for all ε.We assume x＜N.Let Ebe[-N,N]and fkbe the sequence of polynomials as in Lemma 8(fk=0 outsides E).

Our idea is to construct such a chain as ε→0 and n→∞:

 

Firstly,let us deal with the first→.Notice that

 

Since kεis a computable dominated approximate identity,it has a L1-computable radial decreasing majorant K.Since Kcan be arbitrary large we can choose such Kthat‖K‖L1(R)=ais a rational.By Lemma 1 we have

 

Define AiasBy the definition we know

 

It is easy to see that Aiis uniformly first-order definable on i,so by Lemma 9µ(Ai)is uniformly computable on i.By Lemma 2 we have

 

Let.It follows from Lemma 4 thatµ(Vs)is computable.We also have

 

So Vsis a Schnorr test.If x/∈Vs,then for all i≥2s:

 

Combining with the definition of Mand triangle inequality we can have

 

Let Gm=Vm∪Umwith Umas in Lemma 8.Vmand Umboth have measure limits,so by Lemma 5 Gmis a Schnorr test.Suppose x/∈Gm.Fix δ,search a sufficient large nsuch that|f-fn|(x)＜δ/3 and supε＞0|(kε∗(f-fn))(x)|＜δ/3.Then by Lemma 7 there is a sufficient small εsuch that|(kε∗fn-fn)(x)|＜δ/3.Then we have

 

This δcan be arbitrary small so we conclude lim.This proves the required conclusion. □

4 The points satisfy differentiation theorem for approximate identities are Schnorr

In this section we will prove the converse part of the main theorem.Actually,we will show that if xis not Schnorr random,the limit of kε∗f(x)may not exist.

Lemma 10 Given a computable dominated approximate identity kε,kε∗χA→0 asµ(A)→0.Moreover,the speed of this convergence does not depend on the specific A.

Proof Sincekεisacomputabledominatedapproximateidentity,thereisaL1-computable radial decreasing Ksuch that K≥k.Note that Kε(x)≥Kε(y)when|x|≤|y|.We have

 

Hence,kε∗χA(x)→0 asµ(A)→0.And the speed of this convergenceis uniform on ε. □

We will show that our result hold for computable xfirst.

Theorem 6 Given a computable dominated approximate identity kεand a computable real r,there is ansuch thatdiverges as ε→0.

We will use the similar technic used in Lemma 4.5 in[7]to prove our theorem.

Proof Suppose r＜Nfor some natural number N.Set E=[-N,N].We will construct a uniformly computable sequence of 0,1-valued functions fnon R such that f=limfn.First,Let fn↾Ec=0 for all n.So we only need to define fn↾E.Let δbe a small positive rational.We will also construct an computable sequence εnand an sequence of neighborhood Anof rs.t.:When nis even,kεn∗fn(r)＜δand kεn∗fn+1(r)＜δ;when nis odd,kεn∗fn(r)＞1-δand kεn∗fn+1(r)＞1-δ.

Conduct the induction on n.

 

The next stage is to find a rational neighborhood A0⊂Eof rsuch thatµ(A0)＜2-1and|kε0∗χA0|(r)＜δ.We define f1=f0+χA0.So kε0∗f1(r)＜δ.f1(x)=1 for all x∈A0,by Lemma 7 there is a ε1such that kε1∗f1(r)＞1-δ.

Stage n+1.

·Assumeniseven.ApplyLemma10toobtainarationalneighborhoodAn⊆An-1 of rsuch thatµ(An)＜2-n+1and|kεn∗fn|(r)+|kεn∗χAn|(r)＜δ.Let fn+1=fn+χAn.So we still have kεn∗fn+1(r)＜δ.Note that fn+1(x)=1 for x∈An,wecanapplyLemma7toobtainanεn+1suchthatkεn+1∗fn+1(r)＞1-δ.

· If nis odd,then we need the Ansatisfy|kεn∗fn|(r)-|kεn∗χAn|(r)＞1-δ.Let fn+1=fn-χAn.Here we have fn+1(x)=0 on An,so we can obtain an εn+1 such that kεn+1∗fn+1(r)＜δ.

So actually for every n,we define fn+1↾An=1-fnand fn+1↾Acn=fn.By this construction we have

 

which means the kεn∗fnis always close enough to fnand Anis too small to change it.

ItiseasytocheckthatfnisuniformlyL1-computable.Definefasf(x)=limn→∞fn(x).We need f∈L1compso it does not matter fis undefined on a null set.Since‖f-fn‖L1＜µ(An-1)＜2-n,we know fis a L1-computable function.This sequence kεn∗f(r)diverges since for all nit is always between kεn∗fn(r)and kεn∗fn+1(r),that is,kεn∗f(r)＜δwhen nis even,and kεn∗f(r)＞1-δwhen nis odd. □

A finite decimal is a real with the form 2-nkand a decimal interval Iis an interval on R with the form[2-nk,2-n(k+1)]where nis a natural number and kis a integer.A decimal interval can be viewed as a interval on Cantor space.Two intervals are said to be almost disjointif their intersection has at most one element.It is known that for any Σ1set U,there is a uniformly computable decimal interval Iisuch thatandare almost disjoint.

Lemma 11 Let kεbe a computable dominated approximate identity.Eis a compact set and Fis a neighborhood of E,B(E,θ)⊆F.fis constant on Fand 0,1-valued on R.Then there is a computable function S:R+×R+→R+such that for all ε≤S(θ,δ)we have

 

on E.

Proof It is a straight forward conclusion from the proof of Lemma 7.Note that this function Sonly depends on the majorant K. □

Theorem 7 Given a computable dominated approximate identity kεand a Schnorr testGm,we can construct asuch thadiverges for alwhere xis not finite decimal.

We will modify the technic used in theorem 6 to construct the desired f.

Proof SetE=[-N,N],clearlyGm∩EisaSchnorrtest.Sowithoutlosinggenerality we can assume that x∈Gm⊆Efor all m.We will construct a sequence of 0,1-valued function fnand f=limfn.Let fn↾Ec=0 for all n.Let δbe a small positive real.

Stagesuch that I0,iare almost disjoint decimal interval.fn(x)=0 for all x∈E.

Just like the proof of previous theorem we need to find an εsuch that kε∗fis close enough to fon an interval I,but this general εdoes not exist.Actually,we need to cut each Iinto infinite parts,and find a specific εfor each part.

Stagesuch that Iis a computable sequence of pairwise n,ialmost disjoint decimal intervals.Fix i,let l,k∈N and In,i=[2-lk,2-l(k+1)].We define aj∈In,ifor j∈Z as follows:

·a0is the midpoint of In,i:

 

·For j＞0,ajis the midpoint of[aj-1,2-l(k+1)]:

 

·For j＜0,ajis the midpoint of[2-lk,aj+1]:

 

Then we define Jj=[aj,aj+1].To each jwe define an εj.By the property of approximate identity,find εjs.t. for all x∈Jjandεjsatisfies εj≤S(µ(Jj)/2,δ/2)with Sdefined in Lemma 11.

Apply Lemma 10 to obtain αjsuch that|kεj∗χA|≤δ/4 ifµ(A)≤αj.Then we effectively find a large integer mjsuch that

 

Let.We construct such Hn,ifor all i∈N.Then define

Let

Due to the construction of Un+1we knowµ(Un+1)≤2-nfor all n＞0.So by the same argument of the previous theorem we conclude f=limfnis L1-computable.

It remain to show that kε∗fdiverges a.SupposeUn.Let εjbe the real as in our construction.By the definition of function SWe have.Notice that,we then have

 

Combining the definition of εjit follows that

 

which means the kεj∗fnis always close enough to fnand Un+1is too small to change it.So

Thus for all,we have

 

 

Thus,kε ∗fdiverges at xas ε→0.□

Let the function kbe,we have this result directly:

Corollary 1

 

holds for all functionsif and only if xis Schnorr random.

This effective version of the Lebesgue differentiation theorem is slightly different from[7]:

Theorem 8 ([7])

 

holds for all functionsif and only if xis Schnorr random.

Compared with this theorem,our effective version of Lebesgue differentiation theorem requires the xto be the ‘center’of Q,and also extends the theorem from[0,1]to R.

References

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