Research article Special Issues

New reproducing kernel functions in the reproducing kernel Sobolev spaces

  • Received: 04 October 2019 Accepted: 29 November 2019 Published: 06 December 2019
  • MSC : 46E35, 46E22

  • In this paper we construct some new reproducing kernel functions in the reproducing kernel Sobolev space. These functions are new in the literature. We can solve many problems by these functions in the reproducing kernel Sobolev spaces.

    Citation: Ali Akgül, Esra Karatas Akgül, Sahin Korhan. New reproducing kernel functions in the reproducing kernel Sobolev spaces[J]. AIMS Mathematics, 2020, 5(1): 482-496. doi: 10.3934/math.2020032

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  • In this paper we construct some new reproducing kernel functions in the reproducing kernel Sobolev space. These functions are new in the literature. We can solve many problems by these functions in the reproducing kernel Sobolev spaces.


    A finite collection of strictly contractive maps on the real line is called an iterated function system (IFS). Let Φ={φa}aΛ be an IFS and p=(pa)aΛ be a probability vector. It is well-known that there exists a unique Borel probability measure ν, called the invariant measure, such that

    ν=aΛpaφaν,

    where φaν is the push-forward of ν under the map φa:RR.

    When the construction does not involve complicated overlaps (say, under the strong separation condition), the invariant measures are relatively easy to understand. For example, if the strong separation condition holds, then the invariant measure ν is supported on a Cantor set and is singular, and the dimension of ν is given by

    dimν=hχ,

    where h=h(p) is the entropy and χ=χ(Φ,p) is the Lyapunov exponent.

    In this paper we consider IFS with inverses (i.e., IFS that contain inverse maps). IFS with inverses were first introduced by the author in [5], motivated by the Furstenberg measure. See also [6]. We show that the invariant measures for IFS with inverses agree with the invariant measures for associated graph-directed IFS under the suitable choice of weight. The main results of [5] and [6] follow directly from our result.

    The paper is organized as follows: In section 2, we recall IFS with inverses and state the main result. Section 3 is devoted to preliminary lemmas. In section 4 we prove the main result.

    Let G be the free group of rank r2, and let W be a free generating set of G. Let Λ be a set that satisfies

    WΛWW1,

    where W1={a1}aW. Let E=n1Λn and E=ΛN. For ω=ω0ω1 we denote ω|n=ω0ωn1. For ω,ξEE we denote by ωξ their common initial segment. For ωE and ξEE, we say that ω precedes ξ if ωξ=ω.

    Let p=(pa)aΛ be a non-degenerate probability vector, and let μ be the associated Bernoulli measure on E. We say that a (finite or infinite) sequence ωEE is reduced if ωiωi+1aa1 for all i0 and aΛ. Let Γ (resp. Γ) be the set of all finite (resp. infinite) reduced sequences. For ωΓ we denote the associated cylinder set in Γ by [ω]. Define the map

    red:EΓ

    in the obvious way, i.e., red(ω) is the sequence derived from ω by deleting all occurrences of consecutive pairs aa1 (aΛ). Let ¯EE be the set of all ω such that the limit

    limnred(ω|n) (1)

    exists. For example, for any aΛ we have aaa¯E and aa1aa1¯E. By abuse of notation, for ω¯E we denote the limit (1) by red(ω). The following is well-known (see, e.g., chapter 14 in [3]):

    Lemma 2.1. There exists 0<1 (drift or speed) such that

    limn1n|red(ω|n)|=

    for μ-a.e. ωE. In particular, ¯E has full measure.

    Denote

    Λ={(a,b)Λ2:ab1}.

    For aΛ, write Ra=R×{a}. We freely identify Ra with R below. Let XaRa (aΛ) be open intervals and write X=aΛXa. Assume that there exist 0<γ<1 and 0<θ1 such that for all (a,b)Λ, the map φab:XbXa is C1+θ and satisfies

    (ⅰ) ¯φab(Xb)Xa;

    (ⅱ) 0<|φab(x)|<γ for all xXb;

    (ⅲ) φ1ab:φab(Xb)Xb is C1+θ.

    We say that Φ={φab}(a,b)Λ is an IFS with inverses. For ω=ω0ωnΓ, we denote

    φω=φω0ω1φωn1ωn.

    Let Π:ΓX be the natural projection map, i.e.,

    Π(ω)=n1φω|n+1(¯Xωn).

    Define ΠE:¯EX by ΠE=Πred. Define the measure ν=ν(Φ,p) by ν=ΠEμ (i.e., the push-forward of the measure μ under the map ΠE:¯EX). We call ν an invariant measure. It is easy to see that if Λ=W, then the measure ν is an invariant measure of an IFS. Let χ=χ(Φ,p) be the Lyapunov exponent, and hRW=hRW(p) be the random walk entropy. See section 3 in [5] for the precise definition. Fix xaXa for each aΛ. For ω¯E and nN we denote xω,n=xj, where j=j(ω,n)Λ is the last letter of red(ω|n).

    Proposition 2.1 (Proposition 3.1 in [5]). We have

    χ=limn1nlog|φred(ω|n)(xω,n)|

    for μ-a.e. ω.

    Notice that an IFS with inverses Φ={φab}(a,b)Λ does not have any explicit inverse map. The next example illustrates why we call Φ an IFS with inverses. For more detail, see Example 2.1 and Appendix in [5].

    Example 2.1. Let r=2, W={0,1} and Λ={0,1,11}. For 0<k,l<1, define

    f0(x)=kx, f1(x)=(1+l)x+1l(1l)x+1+l.

    Let f11=f11. It is easy to see that we have f0(0)=0, f1(1)=1, f1(1)=1 and f0(0)=k, f1(1)=l. It is well-known that there exists a unique Borel probability measure ν that satisfies

    ν=aΛpafaν.

    The above measure is called a Fustenberg measure. See., e.g., [2]. Let

    Y0=(k,k), Y1=(f1(k),1) and Y1=(1,f1(k)).

    Then we have

    fa(YYa1)Ya,

    for all aΛ, where Y=aΛYa and Y01=. Notice that the sets {Ya}aΛ are not mutually disjoint if and only if k>f1(k), which is equivalent to

    l>1k1+k.

    It is easy to see that there exist open intervals X0,X1,X11R such that

    YaXa and ¯fa(XXa1)Xa

    for all aΛ, where X=aΛXa and X01=. Then {fa|Xb}(a,b)Λ is an IFS with inverses, and the associated invariant measure agrees with ν. For the proof, see the Appendix in [5].

    Denote

    dimν=inf{dimHY:ν(RY)=0}.

    Proposition 2.2 (Proposition 3.3 in [5]). Assume that for all aΛ, the sets {¯φab(Xb)}bΛa are mutually disjoint, where

    Λa={bΛ:(a,b)Λ}.

    Then we have

    dimνa=hRWχ

    for all aΛ.

    Given an IFS with inverses Φ={φab}(a,b)Λ, one can naturally associate a graph-directed IFS by restricting transitions from a to a1 for all aW. For the precise definitions of graph-directed IFS, see section 1.7 in [1].

    Let ˜P=(˜pab) be a |Λ|×|Λ| stochastic matrix that satisfies ˜pab>0 (ab1) and ˜pab=0 (a=b1). Let ˜p=(˜p1,,˜pN) be the unique row vector satisfying ˜p˜P=˜p. Let ˜μ be the probability measure on Γ associated with ˜P and ˜p. Define the measure ˜ν=˜ν(Φ,˜P) by ˜ν=Π˜μ. For aΛ, denote ˜νa=˜ν|Xa. It is easy to see that

    ˜νa=(a,b)Λ˜pabφab˜νb.

    Let ˜h=˜h(˜P) be the entropy and ˜χ=˜χ(Φ,˜P) be the Lyapunov exponent, i.e.,

    ˜h=(a,b)Λ˜pa˜pablog˜pab,

    and

    ˜χ=limn1nlog|φω|n(xωn)|

    for ˜μ-a.e. ω. Under the separation condition, we obtain the following. The argument is classical, so we omit the proof. See, e.g., the proof of (2.6) in [4].

    Proposition 2.3. For every aΛ, assume that the sets {¯φab(Xb)}bΛa are mutually disjoint. Then we have

    dim˜νa=˜h˜χ

    for all aΛ.

    Our main result is the following:

    Theorem 2.1. Let Φ={φab}(a,b)Λ be an IFS with inverses. Then there exists a stochastic matrix ˜P such that

    ν=˜ν, hRW=˜h and χ=˜χ.

    Since the graph directed IFS has essentially the same structure as IFS, by the above theorem most of the results of IFS can be immediately extended to IFS with inverses. For example, the main results of [5] and [6] follow directly from the above result.

    Define μred by redμ, i.e., the push-forward of the measure μ under the map red:EΓ. From below, for n>0, which is not necessarily an integer, we interpret ω|n to be ω|n. The following lemma is immediate.

    Lemma 3.1. We have

    χ=limn1nlog|φω|n(xω,n)|

    for μred-a.e. ωΓ.

    Proof. Let ωΓ, and let ηE be such that ω=red(η). We can assume that η satisfies

    limn1n|red(η|n)|=

    and

    χ=limn1nlog|φred(η|n)(xη,n)|.

    Let ϵ>0, and let nN be sufficiently large. Then, since ω|(ϵ)n precedes red(η|n) and |red(η|n)|<(+ϵ)n, we have

    log|φω|(ϵ)n(xω(ϵ)n)|<nχ<log|φω|(ϵ)n(xω(ϵ)n)|2ϵnlogλmin,

    where

    λmin=min{|φab(x)|:(a,b)Λ,xXb}.

    The result follows from this.

    For ωΓ, we denote

    Eω={υ¯E:ω precedes red(υ)}

    and

    ˆEω={υ¯E:there exists nN s.t. red(υ|n)=ω}.

    Notice that EωˆEω. For aΛ, write

    qa={1μ(Ea1)(a1Λ)1(a1Λ)

    and pa=μ(ˆEa). We next prove the following crucial lemma.

    Lemma 3.2. Let ωΓ and aΛ be such that ωaΓ. Then we have

    pa=μ(ˆEωa)μ(ˆEω).

    Proof. Fix such ωΓ and aΛ. Notice that

    ˆEω=i=|ω|ˆE(i)ω,

    where

    ˆE(i)ω={υ¯E:red(υ|i)=ω, red(υ|k)ω (|ω|k<i)}.

    Then, since

    ˆEωaˆE(i)ω={υ¯E:σi(υ)ˆEa}ˆE(i)ω,

    we have

    μ(ˆEωaˆE(i)ω)=paμ(ˆE(i)ω).

    Therefore,

    μ(ˆEωa)=μ(i=|ω|ˆEωaˆE(i)ω)=i=|ω|paμ(ˆE(i)ω)=paμ(ˆEω).

    Similarly, we have the following:

    Lemma 3.3. For aΛ and ω=ω0ωnΓ with ωn=a, we have

    μ(Eω)=qaμ(ˆEω).

    Proof. Fix such aΛ and ωΓ. Notice that

    ˆEω=i=|ω|ˆE(i)ω,

    where

    ˆE(i)ω={υ¯E:red(υ|i)=ω, red(υ|k)ω (|ω|k<i)}.

    Then, since

    EωˆE(i)ω={υ¯E:σi(υ)bΛ{a1}Eb}ˆE(i)ω,

    we have

    μ(EωˆE(i)ω)=qaμ(ˆE(i)ω).

    Therefore,

    μ(Eω)=μ(i=|ω|EωˆE(i)ω)=i=|ω|qaμ(ˆE(i)ω)=qaμ(ˆEω).

    In this section we prove Theorem 2.1. Denote

    ˜pa=paqa and ˜pab=pbqbqa.

    Let ˜P=(˜pab). By Lemma 3.2 and Lemma 3.3, we obtain the following.

    Proposition 4.1. For all ω=ω0ωnΓ, we have

    μred([ω])=˜pω0˜pω0ω1˜pωn1ωn.

    The above proposition implies that μred=˜μ. Therefore, we obtain ν=˜ν. By Lemma 3.1, we have χ=˜χ. It remains to show the following lemma. Notice that hRW, and ˜h all depend only on Λ and p.

    Lemma 4.1. We have

    hRW=˜h.

    Proof. Let Φ={ϕab}(a,b)Λ be an IFS with inverses that satisfies the separation condition. Then, by Proposition 2.2 and Proposition 2.3 we obtain hRW=˜h.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The author would like to thank the anonymous referees for the careful reading and all the helpful suggestions and remarks. Y. T. was supported by JSPS KAKENHI grant 2020L0116.

    The author declares no conflict of interest.



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