# Multiresolution Expansion and Approximation Order of Generalized Tempered Distributions

Authors: Byung Keun Sohn
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

Let𝒦Mr′(ℝ) be the generalized tempered distributions of eM(x)-growth with restricted order r∈ℕ0, where the function M(x) grows faster than any linear functions as |x|→∞. We show the convergence of multiresolution expansions of 𝒦Mr′(ℝ) in the test function space 𝒦Mr(ℝ) of 𝒦Mr′(ℝ). In addition, we show that the kernel of an integral operator K:𝒦Mr′(ℝ)→𝒦Mr′(ℝ) provides approximation order in 𝒦Mr′(ℝ) in the context of shift-invariant spaces.


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## Body

## 1. Introduction

Multiresolution analysis was shown to be very useful in extending the expansions in orthogonal wavelets fromL2(ℝ) to a certain class of tempered distributions. Some interactions between wavelets and tempered distributions have been presented by Walter’s work in [1–3]. Walter has found the analytic representation of tempered distributions of polynomial growth with restricted order, 𝒮r′(ℝ),r∈ℕ0, by wavelets [1] and the multiresolution expansions’ pointwise convergence of 𝒮r′(ℝ) [3]. Pilipović and Teofanov have showed the uniform convergence on compact sets of the derivatives of multiresolution expansions of 𝒮r′(ℝ) and the convergence of multiresolution expansions of 𝒮r′(ℝ) in the test function space 𝒮r(ℝ) of 𝒮r′(ℝ). As an application, Pilipović and Teofanov have shown that the kernel of an integral operator K:𝒮r′(ℝ)→𝒮r′(ℝ) provides approximation order in 𝒮r′(ℝ) in the context of shift-invariant spaces [4].In the meantime, the tempered distributions of polynomial growth were extended to tempered distributions ofe|x|-growth, 𝒦1′(ℝ), in [5, 6] and e|x|p-growth, 𝒦p′(ℝ), in [7, 8] or eM(x)-growth, 𝒦M′(ℝ), in [9, 10], where the function M(x) grows faster than any linear functions as |x|→∞. We have considered the analytic representation of tempered distributions of eM(x)-growth with restricted order, 𝒦Mr′(ℝ), by wavelets [11]. Also, we have shown that the multiresolution expansions of 𝒦Mr′(ℝ) converges pointwise to the value of the distribution where it exists [12].In this paper, we will show the uniform convergence on compact sets of the derivatives of multiresolution expansions of𝒦Mr′(ℝ) and convergence of multiresolution expansions of 𝒦Mr′(ℝ) in the test function space 𝒦Mr(ℝ) of 𝒦Mr′(ℝ). In addition, we will show that the kernel of an integral operator K:𝒦Mr′(ℝ)→𝒦Mr′(ℝ) provides approximation order in 𝒦Mr′(ℝ). This is an extension of the works of Pilipović and Teofanov [4] in the context of generalized tempered distributions, 𝒦M′(ℝ).


## 2. The Generalized Tempered Distribution Spaces𝒦M′(ℝ)

Throughout this paper, we will useC or Ci to denote the positive constants, which are independent parameters and may be different at each occurrence.Letμ(ξ)(0≤ξ≤∞) denote a continuous increasing function such that μ(0)=0 and μ(∞)=∞. For x≥0, we define
(1)M(x)=∫0x‍μ(ξ)dξ.The functionM(x) is an increasing, convex, and continuous function with M(0)=0,M(∞)=∞ and satisfies the fundamental convexity inequality M(x1)+M(x2)≤M(x1+x2). Further, we define M(x) for negative x by means of the equality M(x)=M(-x). Note that since the derivative μ(x) of M(x) is unbounded in ℝ, the function M(x) will grow faster than any linear function as |x|→∞. Now we list some properties of M(x) which will be frequently used later. Consider the following:
(2)M(x)+M(y)≤M(x+y)∀x,y≥0,M(x+y)≤M(2x)+M(2y)∀x,y≥0.Using the functionM(x), we define the space 𝒦M(ℝ) as the space of all functions φ∈C∞(ℝ) such that
(3)∥φ∥𝒦M=supx∈ℝ,α≤keM(kx)|dαdxαφ(x)|<∞,k=1,2,….The topology in𝒦M(ℝ) is defined by the family of the seminorms ∥·∥𝒦M. Then 𝒦M(ℝ) become a Fréchet space and 𝒟(ℝ)↪𝒦M(ℝ)↪𝒮(ℝ)↪ℰ(ℝ) are continuous and dense inclusions; here 𝒟(ℝ) denotes the spaces of all C∞(ℝ) functions with compact supports, 𝒮(ℝ) the spaces of polynomially decreasing functions (Schwartz functions), and ℰ(ℝ) the space of all C∞(ℝ) functions. By 𝒦M′(ℝ), we mean the space of continuous linear functionals on 𝒦M(ℝ).Definition 1.
We say that the elements of𝒦M′(ℝ) are generalized tempered distributions.Clearly, whenM(x)=log(1+|x|), 𝒦M′(ℝ) are tempered distributions (Schwarz distributions), 𝒮(ℝ). When M(x)=|x|, 𝒦M′(ℝ) are tempered distributions, 𝒦1′(ℝ), which are introduced and characterized by Yoshinaga [6] and Hasumi [5], independently. When M(x)=|x|p,p>1, 𝒦M′(ℝ) are tempered distributions, 𝒦p′(ℝ), which are introduced and characterized by Sznajder and Zielezny [7, 8]. For details about 𝒦M′(ℝ), we refer to [9, 10].For a natural numberr, we define by 𝒦Mr(ℝ) the space of all ϕ∈Cr(ℝ) such that
(4)∥ϕ∥𝒦Mr=supx∈ℝ,α≤reM(rx)|dαdxαφ(x)|<∞,lim|x|→∞sup0≤α≤reM(rx)|dαdxαφ(x)|=0.The topology of𝒦Mr(ℝ) is defined by the family of ∥·∥𝒦Mr and the dual of 𝒦Mr(ℝ) is denoted by 𝒦Mr′(ℝ). Clearly, 𝒦M(ℝ) is the projective limit of 𝒦Mr(ℝ) when r→∞ and 𝒦M′(ℝ) = ⋃r∈ℕ𝒦Mr′(ℝ). Also, we have continuous and dense inclusion mapping as following:
(5)𝒦M(ℝ)↪⋯↪𝒦Mr+1(ℝ)↪𝒦Mr(ℝ)↪⋯↪𝒦Mr′(ℝ)↪𝒦Mr+1′(ℝ)↪⋯↪𝒦M′(ℝ).Definition 2.
We say that the elements of𝒦Mr′(ℝ) are generalized tempered distributions of order r.We define by𝒦~Mr(ℝ) the space of all ψ∈Cr(ℝ) such that
(6)∥ψ∥𝒦~Mr=supx∈ℝ,0≤α≤reM(lx)|dαdxαψ(x)|<∞,l=1,2,….
The topology of 𝒦~Mr(ℝ) is defined by the family of ∥·∥𝒦~Mr and the dual of 𝒦~Mr(ℝ) is denoted by 𝒦~Mr′(ℝ). Obviously, 𝒦~Mr(ℝ)⊂𝒦Mr(ℝ).Now, we give a theorem that will be used later.Theorem 3.
Letϕ and sequence {ϕn}n∈ℕ be given in 𝒦Mr+1(ℝ) such that {(dα/dxα)ϕn}n∈ℕ converges uniformly to (dα/dxα)ϕ on every compact set K⊂ℝ and for α=0,1,…,r. If {ϕn}n∈ℕ is bounded in 𝒦Mr+1(ℝ), then the sequence {ϕn}n∈ℕ converges to ϕ∈𝒦Mr+1(ℝ) in 𝒦Mr(ℝ).Proof.
Letϵ>0 be given and let α∈{0,1,…,r}. Then there exist N such that
(7)supx∈KeM(rx)|dαdxα(ϕn-ϕ)(x)|<ϵ,n≥N,
for arbitrary K⊂ℝ. Also, since the sequence {ϕn}n∈ℕ is bounded in 𝒦Mr+1(ℝ), we can take a positive number A>0 and a compact set K such that |x|>A when x∉K and
(8)supx∉KeM(rx)|dαdxα(ϕn-ϕ)(x)|=supx∉KeM(rx)e-M((r+1)x)eM((r+1)x)|dαdxα(ϕn-ϕ)(x)|≤supx∉Ke-M(x)eM((r+1)x)|dαdxα(ϕn-ϕ)(x)|≤e-M(A)supx∈ℝeM((r+1)x)(|dαdxαϕn(x)|+|dαdxαϕ(x)|)<Ce-M(A)<ϵ.
From (7) and (8), we have
(9)limn→∞supx∈ℝeM(rx)|dαdxα(ϕn-ϕ)(x)|=0,0≤α≤r.


## 3. Multiresolution Expansion of𝒦M′(ℝ)

Definition 4.
A multiresolution analysis (shortly MRA) consists of a sequence of closed subspacesVn,n∈ℤ, of L2(ℝ) satisfying the following:(i)
{ψ(t-n)}n∈ℤ is an orthonormal basis of V0,(ii)
⋯⊂V-1⊂V0⊂V1⊂⋯⊂L2(ℝ),(iii)
f∈Vn⇔f(2·)∈Vn+1,(iv)
⋂nVn={0},∪nVn¯=L2(ℝ). The function ψ whose existence is asserted in (i) is called a scaling function of the given MRA.Definition 5.
We say that a multiresolution analysisVn,n∈ℤ, is (M,r-) regular MRA of L2(ℝ) if the scaling function ψ is in 𝒦~Mr(ℝ).Example 6.
It is impossible that the scaling functionψ has exponential decay and ψ∈C∞(ℝ), with all derivatives bounded, unless ψ=0. Refer to [13, Corollary  5.5.3]. So we will restrict our attention to 𝒦Mr(ℝ) or 𝒦~Mr(ℝ). From the remark in [13] or, page 152 [2, Example  4, page 48], Battle-Lemarié’s wavelets are in 𝒦Mr(ℝ) for some r∈ℕ when M(x)=|x|, but not in 𝒦~Mr(ℝ) even if they have exponential decay and smoothness. In [13], Daubechies shown that for an arbitrary nonnegative integer r, there exists an (M,r-) regular MRA of L2(ℝ) such that the scaling function ψ has compact supports.
LetVj be an (M,r-) regular MRA of L2(ℝ) and let ψ be a scaling function. The reproducing kernel of V0 is given by
(10)q0(x,y)=∑n∈ℤψ(x-n)ψ(y-n)¯.
The series and its derivatives with respect tox or y of order ≤r converge uniformly on ℝ because of the regularity of ψ∈𝒦~Mr(ℝ). The reproducing kernel of the projection operator onto Vj is
(11)qj(x,y)=2jq0(2jx,2jy),x,y∈ℝ,
and the projection of f∈L2(ℝ) onto Vj is given by
(12)qjf(x)=〈f(y),qj(x,y)〉=∫f(y)qj(x,y)dy,x∈ℝ.
The sequence{qj}j∈ℤ, given in (12), is called the multiresolution expansion of f∈L2(ℝ).Definition 7.
For a givenf∈𝒦Mr′(ℝ), the sequence {qj}j∈ℤ defined by
(13)〈qjf,ϕ〉=〈f,qjϕ〉,ϕ∈𝒦Mr(ℝ)
is called the multiresolution expansion of f∈𝒦Mr′(ℝ).We deduce the following properties of the reproducing kernelq0 with scaling function ψ∈𝒦~Mr(ℝ):(a)
q0(x,y)=q0(y,x) and q0(x+k,y+k)=q0(x,y) for all k∈ℤ.(b)
For everyl∈ℕ and 0≤α,β≤r, there exist Cl′>0 such that
(14)|∂α∂xα∂β∂yβq0(x,y)|≤∑j|dαdxαψ(x-j)||dβdxβψ(y-j)|≤∑jCle-M((2l+1)(x-j))e-M((2l+1)(y-j))≤∑jCle-M(2l(x-j))e-M(x-j)×e-M(2l(j-y))e-M(j-y)≤Cle-M(l(x-y))∑jCle-M(x-j)e-M(j-y)≤Cl′e-M(l(x-y)),where we used the properties (2).(c)
∫-∞∞‍q0(x,y)yαdy=xα,y∈ℝ,0≤α≤r.LetVj be an (M,r-) regular MRA of L2(ℝ). We fix a function g∈𝒟(ℝ) with ∫‍g(x)dx=1. We let gj denote the function 2jg(2jx) and let Gj denote the operation of convolution by gj. For each fixed x, we consider the function ∂xαq0(x,y) of the variable y. From (c), we have
(15)∫‍∂xαq0(x,y)yβdy=0,
for 0≤β<α, whereas
(16)∫‍∂xαq0(x,y)yαdy=α!.Now, it follows from integration by parts that the kernalg(x-y) of the operator G shares these properties (15) and (16) with q0(x,y).Let(17)Rα(x,y)=∂xαq0(x,y)-∂xαg(x-y).
From (b) and the fact that g∈𝒟(ℝ)⊂𝒦Mr(ℝ), we have
(18)|Rα(x,y)|≤cke-M(k(x-y)),x,y∈ℝ,k∈ℕ,
and these functions also satisfy
(19)∫‍Rα(x,y)dy=0
identically in x for every α=1,2,…,r. They, for every j∈ℤ and f∈Cr(ℝ) with at most eM(x)-growth, define operator Rjα by
(20)Rjαf(x)=2j∫Rα(2jx,2jy)f(y)dy
which are such that
(21)qjdαdxαf(x)=Gjdαdyαf(y)+Rjαdαdyαf(y),
that is,
(22)∫qj(x,y)dαdyαf(y)dy=2j∫g(2j(x-y))dαdyαf(y)dy+2j∫Rα(2jx,2jy)dαdyαf(y)dy.
From Theorem  1.1 in [14], we have
(23)limj→∞Gjdαdyαf(y)dy=dαdxαf(x),x∈ℝ,α≥0,
uniformly on compact sets. Now we will show the uniform convergence on compact sets of the derivatives of multiresolution expansions of 𝒦Mr′(ℝ).Theorem 8.
Letf∈Cr(ℝ) such that the corresponding derivatives (dα/dxα)f are bounded by a eM(k0x) when |x|→∞, for every α=0,1,…,r and some k0∈ℕ. If qjf, given by (12), be the projection of f onto an (M,r-) regular MRA of L2(ℝ), then the sequence {(dα/dxα)qjf}j∈ℤ converges uniformly on compact sets to (dα/dxα)f as j→∞, for every α=0,1,…,r.Proof.
If|y-x|≤c, we have
(24)|dαdyαf(y)-dαdyαf(y)|y=x|≤eαM(y-x),
where eαM(x) is a continuous function with eM(k0x) growth and eαM(0)=0. From (18), given a compact set K, we have
(25)2j|∫‍Rα(2jx,2jy)dαdyαf(y)dy|≤2j∫‍|Rα(2jx,2jy)(dαdyαf(y)-dαdyαf(y)|y=x)|dy≤2j∫cke-M(2jk(x-y))|eαM(y-x)|dy,
for large enough j and x∈K. Since k can be chosen arbitrary, we obtain by dominated convergence theorem,
(26)limj→∞2j|∫Rα(2jx,2jy)dαdyαf(y)dy|≤∫limj→∞2jcke-M(2jk(x-y))|eαM(y-x)|dy=0
uniformly for x∈K. From (21) and (23), we have the conclusion.We now ready to show the main theorem.Theorem 9.
Letϕ∈𝒦~Mr(ℝ) and let qjϕ(x), given by (7), be a projection of ϕ onto an (M,r-) regular MRA of L2(ℝ). If ϕ∈𝒦M2(r+1)(ℝ), then the sequence {qjϕ(x)} converges to ϕ(x) in 𝒦Mr(ℝ) as j→∞.Proof.
Letg and Rα be given in (21) such that g∈𝒟(ℝ) and ∫‍g(x)dx=1. From Theorems 3 and 8 and (21), it suffices to show that
(27)supx∈ℝeM((r+1)x)1h|∫q0(xh,yh)ϕ(y)dy|=supx∈ℝeM((r+1)x)1h|∫g(x-yh)dαdyαϕ(y)dy+∫Rα(xh,yh)dαdyαϕ(y)dy|
is bounded for every α∈{0,1,…,r} and h>0. Since g has a compact support, then
(28)supx∈ℝeM((r+1)x)1h|∫g(x-yh)dαdyαϕ(y)dy|≤supx∈ℝ1h|∫g(x-yh)eM((r+1)x)e-M(2(r+1)y)dy|≤supx∈ℝ1h|∫g(x-yh)eM(2(r+1)(x-y))dy|≤C.
Hence we have only to show that
(29)K=supx∈ℝeM((r+1)x)1h|∫Rα(xh,yh)dαdyαϕ(y)dy|≤C,x∈ℝ,
for  every  α∈{0,1,…,r}  and  h>0.  Let  S1={y:|x-y|≤1},S2={y:|x-y|>1and(1/2)|x|≤|y|}andS3={y:|x-y|>1and(1/2)|x|>|y|}. Then, by (18), we have
(30)I=supx∈ℝeM((r+1)x)1h|∫Rα(xh,yh)dαdyαϕ(y)dy|≤clsupx∈ℝeM((r+1)x)1h∫e-M(l((x-y)/h))e-M(2(r+1)y)dy=clsupx∈ℝeM((r+1)x)1h(∫S1‍+∫S2‍+∫S3‍)×e-M(l((x-y)/h))e-M(2(r+1)y)dy=cl(I1+I2+I3).
By a simple change of variable, we have
(31)I1=supx∈ℝeM((r+1)x)1h∫S1e-M(l((x-y)/h))e-M(2(r+1)y)dy≤supx∈ℝ1h∫S1e-M(l((x-y)/h))eM(2(r+1)(x-y))dy≤2eM(2(r+1))∫011he-M(l(t/h))dt=2eM(2(r+1))×∫01/he-M(lu)du≤C1.
Since (1/2)|x|<|y| and (1/2)|y|≤|x-y| on S2, then
(32)I2=supx∈ℝeM((r+1)x)1h∫S2e-M(l((x-y)/h))e-M(2(r+1)y)dy≤supx∈ℝeM((r+1)x)1h∫S2e-M(l((x-y)/h))e-M((r+1)x)dy≤1h∫S2e-M((l/2h)y)dy≤C2,
for sufficiently large l. Since (1/2)|x|>|x-y| on S3, then
(33)I3=supx∈ℝeM((r+1)x)1h∫S3e-M(l((x-y)/h))e-M(2(r+1)y)dy≤supx∈ℝeM((r+1)x)1he-M((l/2)(x/h))∫S3e-M((r+1)y)dy≤C3′1h∫S3e-M(2(r+1)y)dy≤C3,
for sufficiently large l.


## 4. Approximation Order of𝒦M′(ℝ)

A space of functionsS is called shift invariant if it is invariant under all integer translate, that is,
(34)f∈S⟺f(·+k)∈S∀k∈ℤ.The principal shift-invariant subspacesS=S(ϕ) are generated by the closure of the linear span of the shifts of ϕ. The stationary ladder of spaces {Sh(ϕ):h>0} is given by
(35)Sh(ϕ)={f(·h):f∈S}.To rate the efficiency for approximation of such spaces, the concept of approximation order is widely used. We say that the scale of the spaceSh(ϕ) provides approximation order k in F if for every sufficiently smooth f,
(36)infg∈Sh(ϕ)∥f-g∥F≤Chk,h>0,
where C=C(f)>0. For further details about the theory on the approximation order provided by shift-invariant spaces, we refer to [15, 16]. We will focus our attention to the so-called approximation order of an integral operator.LetK be an integral operator of the following form
(37)(Kf)(x)=∫K(x,y)f(y)dy,x∈ℝ.We assume thatK(x-k,y)=K(x,y+k),h∈ℤ,x,y∈ℝ. For h>0, we define
(38)Kh=ϱhKϱ1/h,
where ϱ is the scaling operator ϱhf=f(·/h). We say that the integral operator K defined by (37) provides approximation order k in F if for every sufficiently smooth f,
(39)∥Khf-f∥F≤Chk,h>0,
where C=C(f)>0. For further details about the theory on the approximation order provided by integral or kernel operator, we refer to [17, 18].Definition 10 (see [4]).
Letf∈𝒦Mr′(ℝ). Let K(x,y),x,y,∈ℝ, be the kernel of an integral operator K:𝒦Mr′(ℝ)→𝒦Mr′(ℝ). Kf is given by 〈Kf,ϕ〉=〈f,Kϕ〉. We say that the operator K provides approximation order k in 𝒦Mr′(ℝ) if
(40)∥Khf-f∥𝒦Mr+k′=sup∥ϕ∥𝒦Mr+k=1|〈Khf,ϕ〉-〈f,ϕ〉|≤Chk,h>0,
where the constant C=C(f)>0.We will now show that the kernel of an integral operatorK:𝒦Mr′(ℝ)→𝒦Mr′(ℝ) provides approximation order in 𝒦Mr′(ℝ).Theorem 11.
Letϕ∈𝒦~Mr+k(ℝ) with compact support such that the integer shifts of ϕ form an orthogonal basis of S(ϕ) with respect to the inner product in L2(ℝ). Assume that ϕ(x)=∑n∈ℕ‍cnϕ(2x-k) for some sequence {cn}n∈ℕ. Let
(41)K(x,y)=∑l∈ℤϕ(x-l)ϕ-(y-l),x,y∈ℝ
be the kernel of the integral operator given by (37). Then K provides approximation order k in 𝒦Mr′(ℝ).Proof.
Firstly, we will show that(42)J=∥Khϕ-ϕ∥𝒦Mr=supx∈ℝeM(rx)|dαdxα(〈Kh(x,y),ϕ(y)〉-ϕ(x))|≤C∥ϕ∥𝒦Mr+khk,
where ϕ∈𝒦Mr+k,k∈ℕ. If we accept the result (42) for a moment, it follows that for f∈𝒦Mr′(ℝ)⊂𝒦Mr+k′(ℝ), we have
(43)|〈Khf-f,ϕ〉|=|〈f,Khϕ-ϕ〉|≤∥f∥𝒦Mr′∥Khϕ-ϕ∥𝒦Mr≤Chk∥ϕ∥𝒦Mr+k′
hence
(44)∥Khf-f∥𝒦Mr+k′=sup∥ϕ∥𝒦Mr+k=1|〈Khf,ϕ〉-〈f,ϕ〉|≤Chk,h>0,
which implies the conclusion.
Since{S2-j(ϕ):j∈ℤ} satisfy the conditions of (M,r-) regular MRA of L2(ℝ) with S(ϕ)=V0, we can apply (21) to the operator K, that is,
(45)dαdxαKhf(x)=1h∫g(x-yh)dαdyαf(y)dy+1h∫Rα(xh,yh)dαdyαf(y)dy,
where g and Rα are given in (21). For 0≤α≤r,
(46)J=supx∈ℝeM(rx)|dαdxα(〈Kh(x,y),ϕ(y)〉-ϕ(x))|≤supx∈ℝeM(rx)|(dαdyαϕ(y)-dαdyαϕ(y)|y=x)1h∫g(x-yh)×(dαdyαϕ(y)-dαdyαϕ(y)|y=x)dy|+supx∈ℝeM(rx)|1h∫Rα(x-yh)×(dαdyαϕ(y)-dαdyαϕ(y)|y=x)dy|=J1+J2.
In order to estimateJ1, we consider g∈𝒟(ℝ) with ∫‍g(x)dx=1 and ∫‍g(x)xαdx=0,0<|α|<max{r,k-1}. Let c be a constant such that supg⊂[-c,c]. If we assume h∈(0,1), the smoothness of ϕ∈𝒦Mr+k(ℝ)⊂Cr+k(ℝ) implies
(47)J1=suphx∈ℝeM(rhx)|∫|x-y|≤c‍g(x-y)×(dαdyαϕ(y)|y=hy-dαdyαϕ(y)|y=hx)dy|=suphx∈ℝeM(rhx)|∫|x-y|≤c‍g(x-y)×(×hkdα+kdyα+kϕ(y)|y=ξ(y)(y-x)×hdα+1dyα+1ϕ(y)|y=hx+⋯+(y-x)k-1(k-1)!×hk-1dα+k-1dyα+k-1ϕ(y)|y=hx+(y-x)kk!×hkdα+kdyα+kϕ(y)|y=ξ(y))dy|≤suphx∈ℝeM(rhx)|∫|x-y|≤c‍g(x-y)(y-x)kk!×hkdα+kdyα+kϕ(y)|y=ξ(y)dy|≤suphx∈ℝeM(rhx)Cckk!hksupξ(y)∈[hx-hc,hx+hc]|dα+kdyα+kϕ(y)|=Cckk!hksups∈ℝeM(rs)supt∈[s-c,s+c]|dα+kdtα+kϕ(t)|=Cckk!hksups∈ℝeM(rs)supt∈[s-c,s+c](e-M(rt)eM(rt)|dα+kdtα+kϕ(t)|)≤Cckk!hksups∈ℝeM(rs)e-M(r(|s|-c))supt∈ℝ(eM(rt)|dα+kdtα+kϕ(t)|)=C1∥ϕ∥𝒦Mr+k,
where ξ(y)=hx+θh(y-x)∈[hx-hc,hx+hc] for some θ∈(0,1) and C1=C(ck/k!)hksups∈ℝeM(rs)e-M(r(|s|-c))<∞. To show the finiteness of C1 in the last statement, we use
(48)sups∈ℝeM(rs)e-M(r(|s|-c))≤sup|s|≤ceM(rs)e-M(r(|s|-c))+sup|s|>ceM(rs)e-M(r(|s|-c))≤sup|s|≤ceM(rs)e-M(r(|s|-c))+sup|s|>ceM(rs)e-M(r(|s|-c))≤eM(rc)+e-M(rc).
We will estimateJ2 by using the following facts. Since ϕ has a compact support, there exists M>0 such that K(x,y)=0 for |x-y|>M. Also, by the choice of g and property (c) of the reproducing kernel q0, we have
(49)∫‍Rα(x,y)ysdy=dαdxα∫‍K(x,y)ysdy-dαdxα∫‍g(x-y)ysdy=dαdxαxs-dαdxα∫‍g(t)xsdt=dαdxαxs-dαdxαxs=0,0≤s≤r+k-1.
Hence
(50)J2=suphx∈ℝeM(rhx)|{(dαdyαϕ(y)|y=hy-dαdyαϕ(y)|y=hx)}∫‍Rα(x,y)×(dαdyαϕ(y)|y=hy-dαdyαϕ(y)|y=hx)dy|=suphx∈ℝeM(rhx)|{(dαdyαϕ(y)|y=hy-dαdyαϕ(y)|y=hx)}∫‍Rα(x,y)×(hkdα+kdyα+k(y-x)×hdα+1dyα+1ϕ(y)|y=hx+⋯+(y-x)k-1(k-1)!×hk-1dα+k-1dyα+k-1ϕ(y)|y=hx+(y-x)kk!×hkdα+kdyα+kϕ(y)|y=ξ(y))dy|=hkk!suphx∈ℝeM(rhx)|{(dαdyαϕ(y)|y=hy-dαdyαϕ(y)|y=hx)}∫|x-y|≤M‍Rα(x,y)(y-x)k×dα+kdyα+kϕ(y)|y=ξ(y)dy|≤Mkhkk!suphx∈ℝsuphx<ξ(y)<hyeM(rhx)|dα+kdyα+kϕ(y)|y=ξ(y)|×|∫|x-y|≤M‍Rα(x,y)(y-x)kdy|≤C′ckk!hksups∈ℝeM(rs)e-M(r(|s|-c))supt∈ℝ(eM(rt)|dα+kdtα+kϕ(t)|)=C2∥ϕ∥𝒦Mr+k.



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*Source: 190981-2013-02-05.xml*