Construction Theory of Function on Local Fields

WeiyiSu

Department ofM athematics，Nanjing University，Nanjing 210093，China

Construction Theory of Function on Local Fields

WeiyiSu∗

Department ofM athematics，Nanjing University，Nanjing 210093，China

Received 12 January 2015；A ccep ted（in revised version）5 February 2015

.We establish the construction theory of function based upon a local field Kpas underlying space.By virtu re of the concep t of pseudo-differentialoperator，we introduce”fractal calcu lus”（or，p-type calcu lus，or，Gibbs-butzer calcu lus）.Then，show the Jackson Direct approximation theorem s，Berm stein inverse approximation theorem s and the equivalent approximation theorem s for com pact group D（⊂Kp）and locally com pact group K+p（=Kp），so that the foundation of construction theory of function on local fields is established.M oreover，the Jackson type，Bernstein type，and equivalent approximation theorem s on the H¨older-type space Cσ（Kp），σ＞0，are proved；then the equivalent approximation theorem on Sobolev-type space Wrσ（Kp），σ≥0，1≤r＜+∞，is show n.

Construction theory of function，local field，fractalcalcu lus，approximation theorem，H¨older-type space.

AM SSub ject Classifications：41A 65，28A 20

1 Concept and notation

A local field Kpis a locally com pact，non-trivial，totally d isconnected，non-Archim edean norm valued，T2-type，com p lete topological field［1］.It can be a p-series field，or its finite algebraic extension field（with add ition+，m u ltip lication×，term by term，m od p，and no carrying）；or can be a p-ad ic filed，or its finite algebraic extension field（with+，×，term by term，m od p，carrying from left to right），with p≥2 p rim e.

This kind of fields has im portan t theoretical and app lied Meaning，for example，the dyad ic system in the com puter science，and sw itch functions in physics science，they are special casesof local fieldsat p=2.

We concern the casesof p-series field and p-ad ic field，denoted by Kp≡（Kp，+，×），and call them local fields.For the algebraic extension of Kp，denoted by Kq，q=pc，c∈N，we refer to［1］.

1.1 Haarm easu re and Haar in tegral on a local field Kp

1.2 Non-Archim edean valued norm|x|on Kp

There existsan elem entβ∈Kpwith|β|=p-1in Kp，called primeelem ent.∀x∈Kpcan be expressed as

Each x∈Kpin p-series field or in p-aidc field can be exp ressed as the form in（1.1），the difference is：the operations in p-series field are term by term，m od p，no carrying；whereas，the operations in p-aidc field are term by term，m od p，carrying from left to right.

The range ofnon-Archim edean valued norm is|x|∈｛p-k：k∈Z｝.

1.3 Im portan tsubsets in Kp

（1）Com pactgroup in Kp（ring of integers）：D=｛x∈Kp：|x|≤1｝，it isa uniquem axim al com pact subring in Kp，and is an open，closed，com pact subsetwith Haarm easu re |D|=1.

（2）Unitopen ball in Kp（p rim e ideal）：B=｛x∈Kp：|x|＜1｝，it isa uniquem axim alideal in D，also principle ideal，prime ideal；and is open，closed，com pact subset with Haarm easu re|B|=p-1.

（3）Ball in Kp（fractional ideal）：Bk=｛x∈Kp：|x|≤p-k｝，k∈Z，it is a ball in Kpwith center0∈Kpand rad ius p-k；and isopen，closed，com pactsubsetwith Haarm easu re |Bk|=p-k，k∈Z.

（4）Base for neighborhood system ofsatisfies character group of Kp，it is a locally com pactgroup，andthe set of all p-coset representatives of B1in D isit is isomorphic with the finite Galois field

（5）Character group of Kp：is the

The annihilators inΓpareΓk=｛χ∈Γp：∀x∈Bk⇒χ（x）=1｝，k∈Z，with|Γk|=pk.

The base for neighborhood system of unit I∈Γpis，satisfies

Theorem 1.1（see［1，2］）.（i）Let S，T be twoballs in localfield Kp，then eitherSand T aredisjoint，or one ball contains the other one；（ii）Any ball in Kphasmulti-centers；（iii）Any ball in Kpis open，closed，and compact.

2 Test function class and d istribution space on local fields

We in troduce the test function class（Schw artz type space）and the d istribution space of Schw artz type space on a local field Kp.

2.1 Test function class（Schw artz type space）S（Kp）

The space

is said to be a test function class，or Schw artz type space，where

is the characteristic function of Bkj；τhjΦBkj（x）is translation（for hj）ofΦBkj（x），j=1，2，···，n.

2.1.1 The topology of S（Kp）

1.∀φnexists the same index pair（the index pair ofaφ∈S（Kp）：∃（k，l）∈Z×Z，st.（i）φis constanton the cosetof Bk；（ii）suppφ=Bl；denoted by the indexφ=（k，l））.

2.limn→+∞φn（x）=0，x∈Kp，uniform ly.

with theabove topology，S（Kp）becom esa com p lete，separated，T2-type，sem i-norm ed topological linear space.

2.1.2 The im portan tp roperties of S（Kp）

1.S（Kp）is an algebra of con tinuous functions with com pact support，consisted of finite linear com binations of translations of characteristic fucntions of balls，which distinguishes points，and

2.The Fou rier transform ation

is topological isom orphic from S（Kp）on to S（Γp）.The inverse Fou rier transform ation

is topological isom orphic from S（Γp）on to S（Kp）.And（φ∧）∨（x）=φ（x），∀φ∈S（Kp），x∈Kpholds；aswellas（ψ∨）∧（ξ）=ψ（ξ），∀ψ∈S（Γp），ξ∈Γpholds.

2.2 Schw artz d istribution space S∗（Kp）

The space S∗（Kp）=｛u：continuous linear functional on S（Kp）｝，endowed with w∗-topology，itbecom es a com p lete，separated，T2-type，topological linear space.

The Fou rier transform ation of T∈S∗（Kp）（in d istribution sense）is defined as a d istribution T∧∈S∗（Γp）satisfies

The inverse Fou rier transform ation of S∈S∗（Γp）（in d istribution sense）is defined as a d istribution S∨∈S∗（Kp）satisfies

Thus，the Fou rier transform ation in the d istribution sense∧：T→T∧is a bounded linear operator；and it is isom orphic from S∗（Kp）onto S∗（Γp）.And hold（T∧）∨=T，∀T∈S∗（Kp）；（S∨）∧=S，∀S∈S∗（Γp）.

3 Fractal calculus on local fields

By virtue of the pseudo-differentialoperator，we define a kind of new calcu lus on local fields，called fractal calcu lus（or p-type calcu lus，since the pof”pseudo”；or Gibbs-butzer calcu lus）［2-6］.

3.2 Pseudo-differen tial operator Tσ（x，D）with sym bolon Kp

Let f：Kp→C be a Haarm easu rable function on Kp.Denote by

we call Tσ（x，D）a pseudo-differen tialoperator on Kpwith the sym bol

For a functionφ∈S（Kp）in Schw artz-type space S（Kp），the action of pseudodifferentialoperator Tσ（x，D）on it can be denoted by

since the Fou rier transform ation）is isom orphic from S（Kp）on to S（Γp）.

3.3 Fractal calcu lus on Kp

The pseudo-differentialoperatorwith sym bolσ=〈ξ〉α∈Sαρδ（Kp）acting on Haarm easu rable function f：Kp→C is denoted by

Fractalderivative forα＞0，if the integral

exists，then T〈·〉αf（x）is said to be anα-order point-w ise fractal derivative of f（x）at x，denoted by f〈α〉（x）=T〈·〉αf（x）.

Fractal integral forα＞0，if the in tegral

exists，then T〈·〉-αf（x）is said to be anα-order point-w ise fractal integral of f（x）at x，denoted by f〈α〉（x）=T〈·〉-αf（x）.

Sim ilarly，theα-order Lr-strong fractal derivative of f（x）and theα-order Lr-strong fractal integralof f（x）can be defined，i.e.，the strong lim its of T〈·〉αf（x）and T〈·〉-αf（x）in Lr（Kp）sense，denoted by D〈α〉f（x）and I〈α〉f（x），respectively.

3.4 Properties of fractal calcu lus on S（Kp）

Forφ∈S（Kp）andα∈R，the derivative operator（α＞0）and integraloperator（α＜0）

has the follow ing p roperties.

Theorem 3.1.For Schwartz-type functionφ∈S（Kp），it holds in the fractal calcu lussense

（i）φisany orderpoint-w isefractalderivable，and any orderpoint-wise fractal integrable；and

i.e.，point-w ise fractal derivative operation and point-w ise fractal integral operation are closed on S（Kp）.

M oreover，φis any order Lr-strong fractal derivable，and any order Lr-strong fractal integrable；and i.e.，Lr-strong fractal derivative operation and Lr-strong fractal integral operation are closed on S（Kp）.

（ii）Forα＞0，holds D〈α〉φ（x）=φ〈α〉（x），x∈Kp；i.e.，thepoint-w isefractalderivative equals the Lr-strong fractal derivative；And so does thefractal integral，I〈α〉φ（x）=φ〈α〉（x），x∈Kp.

Thus，it is no necessary to distinguish between”point-w ise derivability”and”Lr-strong derivability”；Also neither for”point-w ise integrability”and”Lr-strong integrability”.

（iii）The fractal derivative operator and fractal integral operator are isomorphic linearmappingsfrom S（Kp）onto S（Kp）（linear，one-one，continuous）.

（iv）Thefractal derivative operatorand fractal integraloperatorare inverseeach other，i.e.，for α＞0，holds

3.5 Defin itions and p roperties of fractal calcu lus on S∗（Kp）

By Theorem 3.1，wem ay generalize the fractal calcu lus to the d istribution space S∗（Kp）.

Let T∈S∗（Kp）be a Schw artz-type d istribution.The fractal derivative and fractal in tegralof T are defined as follows.

Fractalderivative forα＞0，anα-order fractalderivative T〈α〉of T∈S∗（Kp）isdefined asa Schw artz-type d istribution satisfying

Fractal in tegral forα＞0，anα-order fractal in tegral T〈α〉of T∈S∗（Kp）is defined as a Schw artz-type d istribution satisfying

Correspond ing to Theorem 3.1，we have

Theorem 3.2.For Schwartz-type distribution T∈S∗（Kp），hold

（i）T isany orderfractalderivable，and any order fractal integrable，and T〈α〉，T〈α〉∈S∗（Kp），α≥0；i.e.，the fractalderivative operation and fractal integral operation are closed on S∗（Kp）.

（ii）Thefractalderivativeoperatorand fractalintegraloperatorareisomorphic linearmappings from S∗（Kp）onto S∗（Kp）（linear，one-one，continuous）.

（iii）Thefractal derivative operatorand fractal integraloperatorare inverse each other，i.e.，for α＞0，holds（T〈α〉）〈α〉=T=（T〈α〉）〈α〉.

3.6 Princip le for estab lish new calcu lus

The p rincip le to establish Somenew calcu lus issuggested in［5］，we now verify the fractal calcu lus on local fields satisfies the p rincip le.

（1）Fractal derivative operator and fractal in tegral operator are inverse each other（in the point of view ofm athem aticalanalysis and operator theory）

（2）Fou rier transform ation form u las（in the point of view of spectrum theory）

（3）Equivalen t theorem s（in the point of view of construction theory of function）On locally com pact group Kp，for∀s≥0，

holds

f〈s〉∈Lip（X（Kp），α），α＞0⇐⇒Epn（X（Kp），f）=O（p-n（s+α）），α＞0，n∈N，with Lip class Lip（X（Kp），α），α＞0；the bestapproximation

and Sn（Kp）=｛φ∈S（Kp）：indexφ=（n，l），l∈Z｝for a fixed n∈Z.

（4）Relationship between charactersof local fieldswith eigen-functionsof New tonm echanics（in the point of view sof group theory and physics science）.

For locally com pactgroup Kp，the character group isΓKp=｛χξ（x）：ξ∈Kp｝iso.←→∈Kp.

Character function y（k）=χk（x），x∈Kp←→eigen-function in New tonm echanics，

Character equation y〈1〉=λy←→eigen-equation in New tonm echanics，

Character valueλ=〈ξ〉，ξ∈Kp←→eigen-value in New tonm echanics.

The character values are the num bers for which the character equations have non-zero solu tions.

4 The construction theory of function in H¨older-type spaces

The classical approximation theory of functions，also called construction theory of function，hashad itsbrightera in the 40's to 90'sof last cen tu ry.Starting from theWeierstrass approximation theory of trigonom etric functionsand polynom ial functions，aswellas the Fou rier series theory，ithas created and developed successfu lly the idea and m entality of construction theory of function，and kep t back lots of valuable wealth form athem atical science.

It isw orth tom ention the tw o im portan tcon tributionsof classicalconstruction theory of function［7］

1.lots of approximation iden tity kernels and approximation identity operatorswith theoreticaland app lied sensesare constructed.

2.the Direct（Jackson）and inverse（Bernstein）approximation theorem s，and equivalent theorem on function spaces，such as，on C（［a，b］）and Lp（［a，b］），1≤p＜+∞，are proved.These theorem s revealan essentialp roperty of functions：the sm oother the functions，the faster to zero the bestapproximations；and vice versa.

In this section，we summ arize the foundation of construction theory of function on the spaces X（D）and X（Kp）；and prove the Jackson type，Bernstein type approximation theorem s，and equivalent theorem son the H¨older-type space and Sobolev-type space.

4.1 Foundation of construction theory of function on Kp

Since the 70's of last centu ry，m athem aticians in allof thew orld have contributed lots of excellentw ork for studying construction theory of function over local fields，such as，the jobs of Chinesem athem aticians for com pact group D，locally com pactgroup Kp，see［8-25］，or the citations in［2］.Resultsof foreignm athem aticianshave listed in references［25-30］，or the citations in［29，30］.

We now show the fundam en tal theorem s of construction theory of function on the function spaces X（D）and X（Kp）over local fields：Jackson theorem s，Bernstein theorem s，equivalent theorem s.

4.1.1 approximation theorem s on function spaces X（D）over com pact group D

Let D=｛x∈Kp：|x|≤1｝⊂Kpbe the com pact group of a local field Kp，and X（D）be the function spaces

with norms

Modu lus of con tinuity，Lipschitz function class，the bestapproximation on X（D）Modu lus of con tinuity

Lipschitz class

The best approximation

Epn（X（D），f，where Pnis the set of all k-degree character polynom ials···+a1χ1（x）+a0，aj∈C，0≤j≤k.

Rem ark 4.1.（i）each k-degree character polynom ial qk（x），is any s（s＞0）-order fractal derivable and（qk）〈s〉∈Pn；and is in finitely order fractal integrable；（ii）the best approximation character polynom ial q∗n∈Pnexists and unique，

Partial sun o f Fou rier series

with the Dirich let kernel

Direct（Jackson）and inverse（Bernstein）approximation theorem s on X（D）

The Direct theorem on X（D）

Lemma 4.1（Jackson type theorem）.Forfunction spaces X（D），ifs≥0，then

Specially，

The inverse theorem on X（D）

Lemma 4.2（Bernstein type theoem）.Forfunction spaces X（D），ifs≥0，α＞0，then

Specially，

Equivalen t theorem on X（D）

Theorem 4.1.Forfunction spaces X（D），ifs≥0，then the follow ing statementsareequivalent

4.1.2 Approximation theorems on function spaces X（Kp）over locally compact group Kp

Let K+p=Kpbe the locally com pactgroup ofa local field Kp，and X（Kp）be function spaces

with norm s

Modu lus of con tinuity，Lipschitz function class，the best approximation on X（Kp）Modu lus of continuity

Lipschitz class

The best approximation

Epn（X（Kp），f）=where the set S n（Kp）is a subset of S（Kp），in which∀φ∈Sn（Kp）=｛φ∈S（Kp）：indexφ=（n，l），l∈Z｝is constant on each cosetof Bnfor the sam e n∈Z，and suppφ=Bl.

also

Direct（Jackson）and inverse（Bernstein）approximation theorem s on X（Kp）The Direct theorem on X（Kp）

Lemma 4.3（Jackson type theorem）.Forfunction spaces X（Kp），ifs≥0，then

Specially，

The inverse type theorem s on X（Kp）

Lemma 4.4（Bernstein type theorem）.Forfunction spaces X（Kp），ifs≥0，α＞0，then

Specially，

Equivalen t theorem on X（Kp）

Theorem 4.2.Forfunction spaces X（Kp），ifs≥0，then thefollow ing statementsare equivalent

4.2 H¨older-type space Cσ（Kp），σ∈R

4.2.1 The definition of H¨older-type space

is the B-type space of Teribelwith norm

4.2.2 The im portan t p roperties of H¨older-type space

Theorem 4.3（Equivalent definition，see［32，33］）.For the H¨older-type space，hold

Theorem 4.4（see［32］）.For the H¨older-type space Cσ（Kp），ifσ∈［0，+∞），then

（i）f∈Cσ（Kp），∀λ∈［0，σ］⇒f has anyλ-order fractal derivatives f〈λ〉≡T〈·〉λf，and f〈λ〉∈Cσ-λ（Kp）；specially，f∈Cσ（Kp）⇒f〈σ〉∈C（Kp）.

（ii）f〈σ〉=T〈·〉σf∈C（Kp），∀λ∈［0，σ］⇒f has anyλ-order fractal derivatives f〈λ〉≡T〈·〉λf，and f〈λ〉∈Cσ-λ（Kp）；specially，f〈σ〉∈C（Kp）⇒f∈Cσ（Kp）.

Thus，f∈Cσ（Kp）⇔f〈σ〉∈C（Kp），∀σ∈［0，+∞），i.e.，the H¨older-type space Cσ（Kp）is the space in which fractals live.

As we know，the New ton k-order continuous derivable function space is Ck（Rn），k∈N；and for the H¨older space Cσ（Rn），σ∈R+N，the param eterσhas a”gap”at naturalnum bers，σ/∈N.However，in the case of local field，the param eterσ∈R has”no gap”of the H¨older type space Cσ（Kp）.On the other hand，the Lipschitz function classes Lip（C（［a，b］），α），Lip（C（R），α）on Euclidean spaces exist just for 0＜α≤1，of param eterα，but the Lip（C（D），α），Lip（C（Kp），α）on local fields exist forα＞0，withou t restriction of 0＜α≤1.These differences show that the sm oothnessof functions defined on R and on Kpare quite different.

4.3 approximation theorem s on H¨older-type space Cσ（Kp）

We study approximation theorem s on the H¨older-type Cσ（Kp），σ＞0，over a local field Kp.

4.3.1 Modu lus of con tinuity，Lipschitz function class，the bestapproximation on Cσ（Kp）

Modu lusof con tinuity

Lipschitz class

The bestapproximation

Epn（Cσ（Kp），f），n∈N，defined as

where Sn=｛φ∈S（Kp）：indexφ=（n，l），l∈Z｝，n∈N.

4.3.2 Direct（Jackson）and inverse（Bernstein）approximation theorem s on Cσ（Kp），σ＞0

The Direct type theorem on Cσ（Kp）

Lemma 4.5（Jackson type theorem）.ForH¨older-type space Cσ（Kp），σ＞0，ifs≥0，then

Proof.To prove the assertion（4.1），we only need prove the follow ing（4.2）forα＞0

By［2，Theorem 3.3.1］，φ∈S（Kp）⇒φ〈s〉∈S（Kp），s∈Z，and

whereκs∈S∗（Kp）isa bounded linear functionalon S（Kp）.Weestim ateAnd also only need to estim ate for，it holds for s≥0

Thus，

By

then for s≥0

This implies

（generally，itholdsω（Cσ（Kp），φ，δ）=O（δsω（Cσ（Kp），φ〈s〉，δ）），δ→0，s≥0）.

Since Epn（Cσ（Kp），φ）≤ω（Cσ（Kp），φ，p-n），com bining（4.3），itholds for s≥0

this implies（4.2）.M oreover，by S（Kp）⊂Cσ（Kp），it follows for s≥0

Thus，（4.1）is proved. For the special case，

since（4.2）holds for f∈Cσ（Kp），thus

then

The Lemma 4.5 is proved.

The inverse type theorem on Cσ（Kp）

Lemma 4.6（Bernstein type theorem）.For H¨older-type space Cσ（Kp），σ＞0，ifs≥0，α＞0，then

Specially，

Proof.We prove Bernstein type inequality：

Since∀qn∈Sn（Kp）⊂S（Kp），index

Theorem 3.2.3，Theorem 3.1.7，and），we have

Then by

we get Bernstein type inequality.

We prove theorem for s=0：

By theassum p tion，Epn（Cσ（Kp），f）=O（p-αn），n→+∞，then

Specially，this implies f∈Lip（Cσ（Kp），α）.

Next，wewill prove theorem for s＞0：

⇒the Bernstein inequality implies

Specially，this implies f〈s〉∈Lip（Cσ（Kp），α）.The Lemma 4.6 is proved.

Equivalen t theorem on Cσ（Kp）

Theorem 4.5.For H¨older-type space Cσ（Kp），σ＞0，ifs≥0，then the follow ing statements are equivalent

Proof.（i）⇒（iii）By the Jackson type theorem，

（iii）⇒（ii）By the Bernstein type theorem，

（ii）⇒（i）By the definition，

Then Theorem 4.5 is proved.

For the app lications of approximation theorem s on H¨older-type space Cσ（Kp），σ＞0，we refer to［5］.

5 approximation theorem s on Sobolev-type space

5.1 Definiitons of Lebesgue-type spaces

The classical Sobolev-type space，is defined as

It is clear that

5.2 Modu lus of con tinu ity，Lipschitz class，the best approximation on

5.3 Equivalen t theorem on Sobolev-type spaces

Theorem 5.1.For Sobolev-type spacesWrσ（Kp），σ≥0，1≤r＜+∞，ifs≥0，then the follow ing statementsareequivalent

Acknowledgments

The au thor is supported by NSFC 11271327.

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∗Correspond ing author.Emailaddress：suqiu@nju.edu.cn（W.Y.Su）