Multiresolution Expansion and Approximation Order of Generalized Tempered Distributions

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Volume 2013, Article ID 190981, 8 pages http://dx.doi.org/10.1155/2013/190981

Research Article

Multiresolution Expansion and Approximation Order of

Generalized Tempered Distributions

Byung Keun Sohn

Department of Mathematics, Inje University, Kimhae 621-749, Republic of Korea Correspondence should be addressed to Byung Keun Sohn; [email protected] Received 22 October 2012; Accepted 25 November 2012

Academic Editor: Palle E. Jorgensen

Copyright © 2013 Byung Keun Sohn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. LetK𝑟𝑟_𝑀𝑀󸀠󸀠(R)be the generalized tempered distributions of𝑒𝑒𝑀𝑀𝑀𝑀𝑀𝑀-growth with restricted order𝑟𝑟 𝑟 N₀, where the function𝑀𝑀(𝑀𝑀)

grows faster than any linear functions as|𝑀𝑀| 𝑥 𝑥. We show the convergence of multiresolution expansions ofK𝑟𝑟_𝑀𝑀󸀠󸀠(R)in the test

function spaceK𝑟𝑟_𝑀𝑀(R)ofK𝑟𝑟_𝑀𝑀󸀠󸀠(R). In addition, we show that the kernel of an integral operator𝐾𝐾 𝐾K𝑟𝑟_𝑀𝑀󸀠󸀠(R) 𝑥 K𝑟𝑟_𝑀𝑀󸀠󸀠(R)provides approximation order inK𝑟𝑟_𝑀𝑀󸀠󸀠(R)in the context of shift-invariant spaces.

1. Introduction

Multiresolution analysis was shown to be very useful in

extending the expansions in orthogonal wavelets from𝐿𝐿2₍_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 ana-lytic representation of tempered distributions of polynomial

growth with restricted order,S󸀠󸀠_𝑟𝑟(R), 𝑟𝑟 𝑟 N₀, by wavelets [1]

and the multiresolution expansions’ pointwise convergence

of S󸀠󸀠_𝑟𝑟(R) [3]. Pilipovi´c and Teofanov have showed the

uniform convergence on compact sets of the derivatives of

multiresolution expansions of S󸀠󸀠_𝑟𝑟(R) and the convergence

of multiresolution expansions ofS󸀠󸀠_𝑟𝑟(R)in the test function

space S_𝑟𝑟(R) of S󸀠󸀠_𝑟𝑟(R). As an application, Pilipovi´c and

Teofanov have shown that the kernel of an integral operator 𝐾𝐾 𝐾S󸀠󸀠_𝑟𝑟(R) 𝑥 S󸀠󸀠_𝑟𝑟(R)provides approximation order inS󸀠󸀠_𝑟𝑟(R) in the context of shift-invariant spaces [4].

In the meantime, the tempered distributions of poly-nomial growth were extended to tempered distributions of 𝑒𝑒|𝑀𝑀|_-growth, _K󸀠󸀠

1(R), in [5, 6] and 𝑒𝑒|𝑀𝑀| 𝑝𝑝

-growth, K󸀠󸀠_𝑝𝑝(R), in

[7, 8] or𝑒𝑒𝑀𝑀𝑀𝑀𝑀𝑀-growth,K󸀠󸀠_𝑀𝑀(R), in [9, 10], where the function

𝑀𝑀(𝑀𝑀)grows faster than any linear functions as|𝑀𝑀| 𝑥 𝑥. We have considered the analytic representation of tempered

distributions of𝑒𝑒𝑀𝑀𝑀𝑀𝑀𝑀_{-growth with restricted order,}_K𝑟𝑟

𝑀𝑀󸀠󸀠(R), by wavelets [11]. Also, we have shown that the multiresolution

expansions ofK𝑟𝑟_𝑀𝑀󸀠󸀠(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

ofK𝑟𝑟_𝑀𝑀󸀠󸀠(R)and convergence of multiresolution expansions

ofK𝑟𝑟_𝑀𝑀󸀠󸀠(R)in the test function spaceK_𝑀𝑀𝑟𝑟 (R)ofK𝑟𝑟_𝑀𝑀󸀠󸀠(R). In addition, we will show that the kernel of an integral operator

𝐾𝐾 𝐾 K𝑟𝑟_𝑀𝑀󸀠󸀠(R) 𝑥 K𝑟𝑟_𝑀𝑀󸀠󸀠(R) provides approximation order

inK𝑟𝑟_𝑀𝑀󸀠󸀠(R). This is an extension of the works of Pilipovi´c and Teofanov [4] in the context of generalized tempered distributions,K󸀠󸀠_𝑀𝑀(R).

2. The Generalized Tempered Distribution

Spaces

K

󸀠󸀠_𝑀𝑀

(

R

)

Throughout this paper, we will use 𝐶𝐶 or𝐶𝐶𝑖𝑖 to denote the

positive constants, which are independent parameters and may be different at each occurrence.

Let𝜇𝜇(𝜇𝜇) (𝜇 𝜇 𝜇𝜇 𝜇 𝑥)denote a continuous increasing

function such that𝜇𝜇(𝜇) 𝜇 𝜇and𝜇𝜇(𝑥) 𝜇 𝑥. For𝑀𝑀 𝑥 𝜇, we

define

𝑀𝑀 (𝑀𝑀) 𝜇 ∫𝑀𝑀

0 𝜇𝜇 (𝜇𝜇) 𝑑𝑑𝜇𝜇𝑑 (1)

The function𝑀𝑀(𝑀𝑀)is an increasing, convex, and

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fundamental convexity inequality𝑀𝑀𝑀𝑀𝑀1) + 𝑀𝑀𝑀𝑀𝑀2) ≤ 𝑀𝑀𝑀𝑀𝑀1+

𝑀𝑀2). Further, we define𝑀𝑀𝑀𝑀𝑀)for negative𝑀𝑀by means of the

equality𝑀𝑀𝑀𝑀𝑀) 𝑀 𝑀𝑀𝑀𝑀𝑀𝑀). Note that since the derivative𝜇𝜇𝑀𝑀𝑀)

of𝑀𝑀𝑀𝑀𝑀) is unbounded inR, the function 𝑀𝑀𝑀𝑀𝑀) will grow

faster than any linear function as|𝑀𝑀| 𝑥 𝑥. Now we list

some properties of𝑀𝑀𝑀𝑀𝑀)which will be frequently used later.

Consider the following:

𝑀𝑀 𝑀𝑀𝑀) + 𝑀𝑀 𝑀𝑀𝑀𝑀 ≤ 𝑀𝑀 𝑀𝑀𝑀 + 𝑀𝑀𝑀 𝑀𝑀𝑀𝑀 𝑀𝑀 𝑀 𝑀𝑀

𝑀𝑀 𝑀𝑀𝑀 + 𝑀𝑀𝑀 ≤ 𝑀𝑀 𝑀2𝑀𝑀) + 𝑀𝑀 𝑀2𝑀𝑀𝑀 𝑀𝑀𝑀𝑀 𝑀𝑀 𝑀 𝑀𝑀 (2)

Using the function𝑀𝑀𝑀𝑀𝑀), we define the spaceK_𝑀𝑀𝑀R)as

the space of all functions𝜑𝜑 𝜑 𝜑𝜑∞_𝑀_R₎_{such that}

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜑𝜑󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑀𝑀𝑀_𝑥𝑥𝑥sup

R,𝛼𝛼𝛼𝛼𝛼𝑒𝑒

𝑀𝑀𝑀𝛼𝛼𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑀𝑀𝑑𝑑𝛼𝛼𝛼𝛼𝜑𝜑 𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < 𝑥𝑀 𝑘𝑘 𝑀 𝑘𝑀2𝑀𝑀𝑀𝑀𝑀

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The topology inK_𝑀𝑀𝑀R)is defined by the family of the

seminorms‖ ⋅ ‖K_𝑀𝑀. Then K_𝑀𝑀𝑀R)become a Fr´echet space

andD𝑀R) 󳨅󳨅𝑥 K_𝑀𝑀𝑀R) 󳨅󳨅𝑥 S𝑀R) 󳨅󳨅𝑥 E𝑀R)are continuous

and dense inclusions; hereD𝑀R) denotes the spaces of all

𝜑𝜑∞𝑀R)functions with compact supports,S𝑀R)the spaces of

polynomially decreasing functions (Schwartz functions), and

E𝑀R)the space of all𝜑𝜑∞𝑀R)functions. ByK󸀠󸀠_𝑀𝑀𝑀R), we mean

the space of continuous linear functionals onK_𝑀𝑀𝑀R).

Definition 1. We say that the elements ofK󸀠󸀠_𝑀𝑀𝑀R)are

gener-alized tempered distributions.

Clearly, when𝑀𝑀𝑀𝑀𝑀) 𝑀log𝑀𝑘 + |𝑀𝑀|),K󸀠󸀠_𝑀𝑀𝑀R)are tempered

distributions (Schwarz distributions),S𝑀R). When𝑀𝑀𝑀𝑀𝑀) 𝑀

|𝑀𝑀|, K󸀠󸀠_𝑀𝑀𝑀R) are tempered distributions, K󸀠󸀠₁𝑀R), which are

introduced and characterized by Yoshinaga [6] and Hasumi

[5], independently. When𝑀𝑀𝑀𝑀𝑀) 𝑀 |𝑀𝑀|𝑝𝑝_{𝑀 𝑝𝑝 𝑝 𝑘}_,_K󸀠󸀠

𝑀𝑀𝑀R)are

tempered distributions, K󸀠󸀠_𝑝𝑝𝑀R), which are introduced and

characterized by Sznajder and Zielezny [7, 8]. For details aboutK󸀠󸀠_𝑀𝑀𝑀R), we refer to [9, 10].

For a natural number𝑟𝑟, we define byK𝑟𝑟_𝑀𝑀𝑀R)the space of

all𝜙𝜙 𝜑 𝜑𝜑𝑟𝑟_𝑀_R₎_{such that}

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜙𝜙󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟_𝑀𝑀𝑀 sup

𝑥𝑥𝑥R,𝛼𝛼𝛼𝑟𝑟𝑒𝑒

𝑀𝑀𝑀𝑟𝑟𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑀𝑀𝑑𝑑𝛼𝛼𝛼𝛼𝜑𝜑 𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < 𝑥𝑀

lim

|𝑥𝑥| → ∞0𝛼𝛼𝛼𝛼𝑟𝑟sup𝑒𝑒

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑀𝑀𝑑𝑑𝛼𝛼𝛼𝛼𝜑𝜑 𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑀 𝑀𝑀

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The topology ofK𝑟𝑟_𝑀𝑀𝑀R)is defined by the family of‖ ⋅ ‖K𝑟𝑟 𝑀𝑀

and the dual of K𝑟𝑟_𝑀𝑀𝑀R) is denoted by K𝑟𝑟_𝑀𝑀󸀠󸀠𝑀R). Clearly,

K_𝑀𝑀𝑀R) is the projective limit of K𝑟𝑟_𝑀𝑀𝑀R) when𝑟𝑟 𝑥 𝑥

andK󸀠󸀠_𝑀𝑀𝑀R)=⋃_𝑟𝑟𝑥_N K𝑟𝑟_𝑀𝑀󸀠󸀠𝑀R). Also, we have continuous and

dense inclusion mapping as following:

K_𝑀𝑀𝑀R) 󳨅󳨅𝑥 ⋅ ⋅ ⋅ 󳨅󳨅𝑥K𝑟𝑟𝑟1_𝑀𝑀 𝑀R) 󳨅󳨅𝑥K𝑟𝑟_𝑀𝑀𝑀R) 󳨅󳨅𝑥 ⋅ ⋅ ⋅ 󳨅󳨅𝑥K𝑟𝑟_𝑀𝑀󸀠󸀠𝑀R) 󳨅󳨅𝑥K_𝑀𝑀𝑟𝑟𝑟1󸀠󸀠𝑀R) 󳨅󳨅𝑥 ⋅ ⋅ ⋅ 󳨅󳨅𝑥K󸀠󸀠_𝑀𝑀𝑀R) 𝑀

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Definition 2. We say that the elements of K𝑟𝑟_𝑀𝑀󸀠󸀠𝑀R) are

generalized tempered distributions of order𝑟𝑟.

We define byK̃𝑟𝑟_𝑀𝑀𝑀R)the space of all𝜓𝜓 𝜑 𝜑𝜑𝑟𝑟𝑀R)such that 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜓𝜓󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩̃K𝑟𝑟_𝑀𝑀𝑀 sup

𝑥𝑥𝑥R,0𝛼𝛼𝛼𝛼𝑟𝑟𝑒𝑒

𝑀𝑀𝑀𝑀𝑀𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑀𝑀𝑑𝑑𝛼𝛼𝛼𝛼𝜓𝜓 𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < 𝑥𝑀 𝑙𝑙 𝑀 𝑘𝑀2𝑀𝑀𝑀𝑀𝑀

(6) The topology ofK̃𝑟𝑟_𝑀𝑀𝑀R)is defined by the family of‖ ⋅ ‖K̃𝑟𝑟_𝑀𝑀

and the dual ofK̃𝑟𝑟_𝑀𝑀𝑀R)is denoted byK̃𝑟𝑟_𝑀𝑀󸀠󸀠𝑀R). Obviously, ̃

K𝑟𝑟_𝑀𝑀𝑀R) ⊂K𝑟𝑟_𝑀𝑀𝑀R).

Now, we give a theorem that will be used later.

Theorem 3. Let𝜙𝜙and sequence{𝜙𝜙𝑛𝑛}𝑛𝑛𝑥Nbe given inK𝑟𝑟𝑟1𝑀𝑀𝑀R)

such that{𝑀𝑑𝑑𝛼𝛼/𝑑𝑑𝑀𝑀𝛼𝛼)𝜙𝜙𝑛𝑛}𝑛𝑛𝑥Nconverges uniformly to𝑀𝑑𝑑𝛼𝛼/𝑑𝑑𝑀𝑀𝛼𝛼)𝜙𝜙 on every compact set𝐾𝐾 ⊂Rand for𝛼𝛼 𝑀 𝑀𝑀 𝑘𝑀 𝑀 𝑀 𝑀 𝑀 𝑟𝑟. If{𝜙𝜙_𝑛𝑛}_𝑛𝑛𝑥N is bounded inK𝑟𝑟𝑟1_𝑀𝑀 𝑀R), then the sequence{𝜙𝜙_𝑛𝑛}_𝑛𝑛𝑥_Nconverges to 𝜙𝜙 𝜑K𝑟𝑟𝑟1_𝑀𝑀𝑀R)inK𝑟𝑟_𝑀𝑀𝑀R).

Proof. Let𝜖𝜖 𝑝 𝑀be given and let𝛼𝛼 𝜑 {𝑀𝑀 𝑘𝑀 𝑀 𝑀 𝑀 𝑀 𝑟𝑟}. Then there

exist𝑁𝑁such that

sup

𝑥𝑥𝑥𝑥𝑥𝑒𝑒

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑀𝑀𝑑𝑑𝛼𝛼𝛼𝛼𝑀𝜙𝜙𝑛𝑛𝑀 𝜙𝜙𝑀 𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < 𝜖𝜖𝑀 𝜖𝜖 𝑀 𝑁𝑁𝑀 (7)

for arbitrary 𝐾𝐾 ⊂ R. Also, since the sequence {𝜙𝜙_𝑛𝑛}_𝑛𝑛𝑥_N is

bounded inK𝑟𝑟𝑟1_𝑀𝑀𝑀R), we can take a positive number𝐴𝐴 𝑝 𝑀

and a compact set𝐾𝐾such that|𝑀𝑀| 𝑝 𝐴𝐴when𝑀𝑀 𝑥 𝐾𝐾and

sup

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑀𝑀𝑑𝑑𝛼𝛼𝛼𝛼𝑀𝜙𝜙𝑛𝑛𝑀 𝜙𝜙𝑀 𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑀sup

𝑀𝑀𝑀𝑟𝑟𝑥𝑥𝑀_𝑒𝑒−𝑀𝑀𝑀𝑀𝑟𝑟𝑟1𝑀𝑥𝑥𝑀_𝑒𝑒𝑀𝑀𝑀𝑀𝑟𝑟𝑟1𝑀𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤sup

−𝑀𝑀𝑀𝑥𝑥𝑀_𝑒𝑒𝑀𝑀𝑀𝑀𝑟𝑟𝑟1𝑀𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤ 𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀sup

𝑥𝑥𝑥R𝑒𝑒

𝑀𝑀𝑀𝑀𝑟𝑟𝑟1𝑀𝑥𝑥𝑀₍󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑀𝑀𝑑𝑑𝛼𝛼𝛼𝛼𝜙𝜙𝑛𝑛𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑑𝑑 𝛼𝛼

𝑑𝑑𝑀𝑀𝛼𝛼𝜙𝜙 𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨)

< 𝜑𝜑𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀< 𝜖𝜖𝑀

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From (7) and (8), we have

lim

𝑛𝑛 → ∞sup_𝑥𝑥𝑥_R𝑒𝑒

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑀𝑀𝑑𝑑𝛼𝛼𝛼𝛼𝑀𝜙𝜙𝑛𝑛𝑀 𝜙𝜙𝑀 𝑀𝑀𝑀)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑀 𝑀𝑀 𝑀 ≤ 𝛼𝛼 ≤ 𝑟𝑟𝑀 (9)

3. Multiresolution Expansion of

K

󸀠󸀠_𝑀𝑀

𝑀

R

)

Definition 4. A multiresolution analysis (shortly MRA) con-sists of a sequence of closed subspaces𝑉𝑉𝑛𝑛𝑀 𝜖𝜖 𝜑 Z, of𝐿𝐿2𝑀R)

satisfying the following:

(i){𝜓𝜓𝑀𝜓𝜓 𝑀 𝜖𝜖)}𝑛𝑛𝑥Zis an orthonormal basis of𝑉𝑉0,

(ii)⋅ ⋅ ⋅ ⊂ 𝑉𝑉−1 ⊂ 𝑉𝑉0⊂ 𝑉𝑉1⊂ ⋅ ⋅ ⋅ ⊂ 𝐿𝐿2𝑀R),

(iii)𝑓𝑓 𝜑 𝑉𝑉𝑛𝑛⇔ 𝑓𝑓𝑀2⋅) 𝜑 𝑉𝑉𝑛𝑛𝑟1,

(iv)⋂_𝑛𝑛𝑉𝑉𝑛𝑛 𝑀 {𝑀}𝑀 ∪𝑛𝑛𝑉𝑉𝑛𝑛𝑀 𝐿𝐿2𝑀R).

The function𝜓𝜓whose existence is asserted in (i) is

(3)

Definition 5. We say that a multiresolution analysis𝑉𝑉𝑛𝑛, 𝑛𝑛 𝑛

Z, is(𝑀𝑀, 𝑀𝑀-)regular MRA of𝐿𝐿2(R)if the scaling function𝜓𝜓

is inK̃𝑟𝑟_𝑀𝑀(R).

Example 6. It is impossible that the scaling function𝜓𝜓has

exponential decay and 𝜓𝜓 𝑛 𝜓𝜓∞₍_R₎_{, with all derivatives}

bounded, unless𝜓𝜓 𝜓 𝜓. Refer to [13, Corollary 5.5.3]. So we will restrict our attention toK𝑟𝑟_𝑀𝑀(R)orK̃𝑟𝑟_𝑀𝑀(R). From the

remark in [13] or, page 152 [2, Example 4, page 48], Battle-Lemari´e’s wavelets are in K𝑟𝑟_𝑀𝑀(R) for some 𝑀𝑀 𝑛 N when

𝑀𝑀(𝑀𝑀) 𝜓 𝑀𝑀𝑀𝑀, but not inK̃𝑟𝑟_𝑀𝑀(R)even if they have exponential

decay and smoothness. In [13], Daubechies shown that for an arbitrary nonnegative integer𝑀𝑀, there exists an(𝑀𝑀, 𝑀𝑀-)regular MRA of𝐿𝐿2₍_R₎_{such that the scaling function}_𝜓𝜓_{has compact}

supports.

Let𝑉𝑉𝑗𝑗be an(𝑀𝑀, 𝑀𝑀-)regular MRA of𝐿𝐿2(R)and let𝜓𝜓be a

scaling function. The reproducing kernel of𝑉𝑉0is given by

𝑞𝑞0(𝑀𝑀, 𝑥𝑥𝑥 𝜓 ∑

𝑛𝑛𝑛Z

𝜓𝜓 (𝑀𝑀 𝑥 𝑛𝑛) 𝜓𝜓 (𝑥𝑥 𝑥 𝑛𝑛𝑥𝜓 ₍₁₀₎

The series and its derivatives with respect to𝑀𝑀 or𝑥𝑥of order≤ 𝑀𝑀converge uniformly onRbecause of the regularity

of𝜓𝜓 𝑛 ̃K𝑟𝑟_𝑀𝑀(R). The reproducing kernel of the projection

operator onto𝑉𝑉𝑗𝑗is

𝑞𝑞𝑗𝑗(𝑀𝑀, 𝑥𝑥𝑥 𝜓 𝑥𝑗𝑗𝑞𝑞0(𝑥𝑗𝑗𝑀𝑀, 𝑥𝑗𝑗𝑥𝑥𝑦 , 𝑀𝑀, 𝑥𝑥 𝑛R, (11)

and the projection of𝑓𝑓 𝑛 𝐿𝐿2₍_R₎_onto_𝑉𝑉

𝑗𝑗is given by

𝑞𝑞𝑗𝑗𝑓𝑓 (𝑀𝑀) 𝜓 ⟨𝑓𝑓 (𝑥𝑥𝑥 , 𝑞𝑞𝑗𝑗(𝑀𝑀, 𝑥𝑥𝑥𝑥

𝜓 ∫ 𝑓𝑓 (𝑥𝑥𝑥 𝑞𝑞𝑗𝑗(𝑀𝑀, 𝑥𝑥𝑥 𝑥𝑥𝑥𝑥, 𝑀𝑀 𝑛R𝜓

(12)

The sequence{𝑞𝑞𝑗𝑗}𝑗𝑗𝑛Z, given in (12), is called the

multires-olution expansion of𝑓𝑓 𝑛 𝐿𝐿2₍_R₎_.

Definition 7. For a given𝑓𝑓 𝑛 K𝑟𝑟_𝑀𝑀󸀠󸀠(R), the sequence{𝑞𝑞_𝑗𝑗}_𝑗𝑗𝑛_Z

defined by

⟨𝑞𝑞𝑗𝑗𝑓𝑓, 𝑓𝑓𝑥 𝜓 ⟨𝑓𝑓, 𝑞𝑞𝑗𝑗𝑓𝑓𝑥 , 𝑓𝑓 𝑛K𝑟𝑟𝑀𝑀(R) (13)

is called the multiresolution expansion of𝑓𝑓 𝑛 K𝑟𝑟_𝑀𝑀󸀠󸀠(R).

We deduce the following properties of the reproducing kernel𝑞𝑞0with scaling function𝜓𝜓 𝑛 ̃K𝑟𝑟_𝑀𝑀(R):

(a) 𝑞𝑞0(𝑀𝑀, 𝑥𝑥) 𝜓 𝑞𝑞0(𝑥𝑥, 𝑀𝑀)and𝑞𝑞0(𝑀𝑀 𝑥 𝑥𝑥, 𝑥𝑥 𝑥 𝑥𝑥) 𝜓 𝑞𝑞0(𝑀𝑀, 𝑥𝑥)for

all𝑥𝑥 𝑛Z.

(b) For every𝑙𝑙 𝑛Nand𝜓 ≤ 𝛼𝛼, 𝛼𝛼 ≤ 𝑀𝑀, there exist𝜓𝜓󸀠󸀠_𝑙𝑙 > 𝜓

such that

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨

𝜕𝜕𝛼𝛼

𝜕𝜕𝑀𝑀𝛼𝛼 𝜕𝜕

𝛽𝛽

𝜕𝜕𝑥𝑥𝛽𝛽𝑞𝑞0(𝑀𝑀, 𝑥𝑥𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨≤ ∑

𝑗𝑗 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑥𝑥𝑀𝑀𝑥𝑥𝛼𝛼𝛼𝛼𝜓𝜓 (𝑀𝑀 𝑥 𝜓𝜓𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨

𝑥𝑥𝛽𝛽

𝑥𝑥𝑀𝑀𝛽𝛽𝜓𝜓 (𝑥𝑥 𝑥 𝜓𝜓𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨

≤ ∑

𝑗𝑗𝜓𝜓𝑙𝑙𝑒𝑒

−𝑀𝑀𝑀𝑀2𝑙𝑙𝑀𝑀𝑀𝑀𝑀𝑀−𝑗𝑗𝑀𝑀_𝑒𝑒−𝑀𝑀𝑀𝑀2𝑙𝑙𝑀𝑀𝑀𝑀𝑀𝑀−𝑗𝑗𝑀𝑀

≤ ∑

𝑗𝑗𝜓𝜓𝑙𝑙𝑒𝑒

−𝑀𝑀𝑀2𝑙𝑙𝑀𝑀𝑀−𝑗𝑗𝑀𝑀_𝑒𝑒−𝑀𝑀𝑀𝑀𝑀−𝑗𝑗𝑀

× 𝑒𝑒−𝑀𝑀𝑀2𝑙𝑙𝑀𝑗𝑗−𝑀𝑀𝑀𝑀𝑒𝑒−𝑀𝑀𝑀𝑗𝑗−𝑀𝑀𝑀

≤ 𝜓𝜓𝑙𝑙𝑒𝑒−𝑀𝑀𝑀𝑙𝑙𝑀𝑀𝑀−𝑀𝑀𝑀𝑀∑

𝑗𝑗 𝜓𝜓𝑙𝑙𝑒𝑒

−𝑀𝑀𝑀𝑀𝑀−𝑗𝑗𝑀_𝑒𝑒−𝑀𝑀𝑀𝑗𝑗−𝑀𝑀𝑀

≤ 𝜓𝜓󸀠󸀠_𝑙𝑙𝑒𝑒−𝑀𝑀𝑀𝑙𝑙𝑀𝑀𝑀−𝑀𝑀𝑀𝑀,

(14)

where we used the properties (2).

(c) ∫_−∞∞ 𝑞𝑞0(𝑀𝑀, 𝑥𝑥)𝑥𝑥𝛼𝛼𝑥𝑥𝑥𝑥 𝜓 𝑀𝑀𝛼𝛼, 𝑥𝑥 𝑛R, 𝜓 ≤ 𝛼𝛼 ≤ 𝑀𝑀.

Let 𝑉𝑉𝑗𝑗 be an (𝑀𝑀, 𝑀𝑀-) regular MRA of 𝐿𝐿2(R). We fix a

function𝑔𝑔 𝑛 D(R)with∫ 𝑔𝑔(𝑀𝑀)𝑥𝑥𝑀𝑀 𝜓 𝑔. We let𝑔𝑔𝑗𝑗 denote

the function 𝑥𝑗𝑗𝑔𝑔(𝑥𝑗𝑗𝑀𝑀) and let 𝐺𝐺𝑗𝑗 denote the operation of

convolution by𝑔𝑔𝑗𝑗. For each fixed𝑀𝑀, we consider the function

𝜕𝜕𝛼𝛼

𝑀𝑀𝑞𝑞0(𝑀𝑀, 𝑥𝑥)of the variable𝑥𝑥. From (c), we have

∫ 𝜕𝜕_𝑀𝑀𝛼𝛼𝑞𝑞0(𝑀𝑀, 𝑥𝑥𝑥 𝑥𝑥𝛽𝛽𝑥𝑥𝑥𝑥 𝜓 𝜓, (15)

for𝜓 ≤ 𝛼𝛼 𝛽 𝛼𝛼, whereas

∫ 𝜕𝜕𝛼𝛼_𝑀𝑀𝑞𝑞0(𝑀𝑀, 𝑥𝑥𝑥 𝑥𝑥𝛼𝛼𝑥𝑥𝑥𝑥 𝜓 𝛼𝛼𝑑𝜓 (16)

Now, it follows from integration by parts that the kernal

𝑔𝑔(𝑀𝑀 𝑥 𝑥𝑥)of the operator𝐺𝐺shares these properties (15) and

(16) with𝑞𝑞0(𝑀𝑀, 𝑥𝑥). Let

𝑅𝑅𝛼𝛼(𝑀𝑀, 𝑥𝑥𝑥 𝜓 𝜕𝜕𝛼𝛼𝑀𝑀𝑞𝑞0(𝑀𝑀, 𝑥𝑥𝑥 𝑥 𝜕𝜕𝑀𝑀𝛼𝛼𝑔𝑔 (𝑀𝑀 𝑥 𝑥𝑥𝑥 𝜓 (17)

From (b) and the fact that𝑔𝑔 𝑛D(R) ⊂ K𝑟𝑟_𝑀𝑀(R), we have 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑅𝑅𝛼𝛼

(𝑀𝑀, 𝑥𝑥𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 𝑐𝑐𝑘𝑘𝑒𝑒−𝑀𝑀𝑀𝑘𝑘𝑀𝑀𝑀−𝑀𝑀𝑀𝑀, 𝑀𝑀, 𝑥𝑥 𝑛R, 𝑥𝑥 𝑛N, (18)

and these functions also satisfy

∫ 𝑅𝑅𝛼𝛼(𝑀𝑀, 𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝜓 𝜓 (19)

identically in𝑀𝑀for every𝛼𝛼 𝜓 𝑔, 𝑥, 𝜓 𝜓 𝜓 , 𝑀𝑀. They, for every𝜓𝜓 𝑛Z

and𝑓𝑓 𝑛 𝜓𝜓𝑟𝑟₍_R₎_{with at most}_𝑒𝑒𝑀𝑀𝑀𝑀𝑀𝑀_{-growth, define operator}_𝑅𝑅𝛼𝛼

𝑗𝑗

by

(4)

which are such that

𝑞𝑞𝑗𝑗 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 (𝑑𝑑) = 𝐺𝐺𝑗𝑗 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑓 𝑓𝑓𝛼𝛼𝑗𝑗 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑓 (21)

that is,

∫ 𝑞𝑞𝑗𝑗𝑓𝑑𝑑𝑓 𝑑𝑑𝑓 𝑑𝑑_{𝑑𝑑𝑑𝑑}𝛼𝛼_𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑

= 2𝑗𝑗∫ 𝑔𝑔 𝑔2𝑗𝑗𝑓𝑑𝑑 𝑥 𝑑𝑑𝑓𝑥 𝑑𝑑_{𝑑𝑑𝑑𝑑}𝛼𝛼_𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑

𝑓 2𝑗𝑗∫ 𝑓𝑓𝛼𝛼𝑔2𝑗𝑗_{𝑑𝑑𝑓 2}𝑗𝑗_{𝑑𝑑𝑥 𝑑𝑑}𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑𝑓

(22)

From Theorem 1.1 in [14], we have

lim

𝑗𝑗 𝑗 𝑗𝐺𝐺𝑗𝑗

𝑑𝑑𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑 = 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 (𝑑𝑑) 𝑓 𝑑𝑑 𝑥R𝑓 𝛼𝛼 𝛼 𝛼𝑓 (23)

uniformly on compact sets. Now we will show the uniform convergence on compact sets of the derivatives of

multireso-lution expansions ofK𝑟𝑟_𝑀𝑀󸀠󸀠(R).

Theorem 8. Let 𝑓𝑓 𝑥 𝑓𝑓𝑟𝑟(R) such that the corresponding

derivatives(𝑑𝑑𝛼𝛼_{/𝑑𝑑𝑑𝑑}𝛼𝛼_)𝑓𝑓_{are bounded by a}_𝑒𝑒𝑀𝑀𝑀𝑀𝑀0𝑥𝑥𝑥when|𝑑𝑑| 𝑥

∞, for every 𝛼𝛼 = 𝛼𝑓 𝛼𝑓 𝑓 𝑓 𝑓 𝑓 𝛼𝛼 and some 𝑘𝑘0 𝑥 N. If 𝑞𝑞𝑗𝑗𝑓𝑓,

given by(12), be the projection of𝑓𝑓onto an(𝑀𝑀𝑓 𝛼𝛼-) regular

MRA of𝐿𝐿2(R), then the sequence{(𝑑𝑑𝛼𝛼/𝑑𝑑𝑑𝑑𝛼𝛼)𝑞𝑞_𝑗𝑗𝑓𝑓𝑓_𝑗𝑗𝑗Zconverges

uniformly on compact sets to(𝑑𝑑𝛼𝛼_{/𝑑𝑑𝑑𝑑}𝛼𝛼_)𝑓𝑓_as_{𝑗𝑗 𝑥 ∞, for every}

𝛼𝛼 = 𝛼𝑓 𝛼𝑓 𝑓 𝑓 𝑓 𝑓 𝛼𝛼.

Proof. If|𝑑𝑑 𝑥 𝑑𝑑| 𝑦 𝑦𝑦, we have

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝑑𝑑𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑥 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑦𝑦𝑦𝑥𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑦 𝑒𝑒

𝑀𝑀

𝛼𝛼 𝑓𝑑𝑑 𝑥 𝑑𝑑𝑓 𝑓 (24)

where𝑒𝑒𝑀𝑀𝛼𝛼(𝑑𝑑)is a continuous function with𝑒𝑒𝑀𝑀𝑀𝑀𝑀0𝑥𝑥𝑥growth and

𝑒𝑒𝑀𝑀

𝛼𝛼(𝛼) = 𝛼. From (18), given a compact set𝐾𝐾, we have

2𝑗𝑗󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑓𝑓}𝛼𝛼𝑔2𝑗𝑗𝑑𝑑𝑓 2𝑗𝑗𝑑𝑑𝑥 𝑑𝑑𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑦 2𝑗𝑗∫󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑓𝑓

𝛼𝛼_𝑔2𝑗𝑗_{𝑑𝑑𝑓 2}𝑗𝑗_{𝑑𝑑𝑥 𝑦 𝑑𝑑}𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑥 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑦𝑦𝑦𝑥𝑥)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑑𝑑𝑑𝑑

𝑦 2𝑗𝑗∫ 𝑦𝑦𝑀𝑀𝑒𝑒−𝑀𝑀𝑀2

𝑗𝑗_{𝑀𝑀𝑀𝑥𝑥−𝑦𝑦𝑥𝑥}

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨𝑒𝑒𝑀𝑀𝛼𝛼 𝑓𝑑𝑑 𝑥 𝑑𝑑𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑑𝑑𝑓

(25)

for large enough𝑗𝑗and𝑑𝑑 𝑥 𝐾𝐾. Since𝑘𝑘can be chosen arbitrary,

we obtain by dominated convergence theorem,

lim

𝑗𝑗 𝑗 𝑗2

𝑗𝑗󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑓𝑓𝛼𝛼𝑔2𝑗𝑗𝑑𝑑𝑓 2𝑗𝑗𝑑𝑑𝑥 𝑑𝑑_{𝑑𝑑𝑑𝑑}𝛼𝛼_𝛼𝛼𝑓𝑓 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨}

𝑦 ∫ lim

𝑗𝑗 𝑗 𝑗2

𝑗𝑗_𝑦𝑦

𝑀𝑀𝑒𝑒−𝑀𝑀𝑀2

𝑗𝑗_{𝑀𝑀𝑀𝑥𝑥−𝑦𝑦𝑥𝑥}

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨𝑒𝑒𝑀𝑀𝛼𝛼 𝑓𝑑𝑑 𝑥 𝑑𝑑𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑑𝑑 = 𝛼

(26)

uniformly for 𝑑𝑑 𝑥 𝐾𝐾. From (21) and (23), we have the

conclusion.

We now ready to show the main theorem.

Theorem 9. Let𝜙𝜙 𝑥 ̃K𝑟𝑟_𝑀𝑀(R)and let𝑞𝑞𝑗𝑗𝜙𝜙(𝑑𝑑), given by(7), be

a projection of𝜙𝜙onto an(𝑀𝑀𝑓 𝛼𝛼-)regular MRA of𝐿𝐿2(R). If𝜙𝜙 𝑥

K2𝑀𝑟𝑟𝑟𝑟𝑥_𝑀𝑀 (R), then the sequence{𝑞𝑞_𝑗𝑗𝜙𝜙(𝑑𝑑)𝑓converges to𝜙𝜙(𝑑𝑑)in

K𝑟𝑟_𝑀𝑀(R)as𝑗𝑗 𝑥 ∞.

Proof. Let𝑔𝑔and𝑓𝑓𝛼𝛼_{be given in (21) such that}_{𝑔𝑔 𝑥}_D₍_R₎_and

∫ 𝑔𝑔(𝑑𝑑)𝑑𝑑𝑑𝑑 = 𝛼. From Theorems 3 and 8 and (21), it suffices to show that

sup

𝑥𝑥𝑗R𝑒𝑒

𝑀𝑀𝑀𝑀𝑟𝑟𝑟𝑟𝑥𝑥𝑥𝑥𝛼

ℎ󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑞𝑞0(𝑑𝑑ℎ𝑓

𝑑𝑑

ℎ) 𝜙𝜙 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

=sup

ℎ󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑔𝑔(𝑑𝑑 𝑥 𝑑𝑑ℎ ) 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝜙𝜙 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑

𝑓 ∫ 𝑓𝑓𝛼𝛼_(𝑑𝑑

ℎ𝑓

𝑑𝑑

ℎ) 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝜙𝜙 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

(27)

is bounded for every𝛼𝛼 𝑥 {𝛼𝑓 𝛼𝑓 𝑓 𝑓 𝑓 𝑓 𝛼𝛼𝑓andℎ > 𝛼. Since𝑔𝑔has a

compact support, then

sup

ℎ󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑔𝑔(

𝑑𝑑 𝑥 𝑑𝑑

ℎ ) 𝑑𝑑

𝛼𝛼

𝑦sup

𝑥𝑥𝑗R

𝛼

ℎ󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑔𝑔(

𝑑𝑑 𝑥 𝑑𝑑

ℎ ) 𝑒𝑒𝑀𝑀𝑀𝑀𝑟𝑟𝑟𝑟𝑥𝑥𝑥𝑥𝑒𝑒−𝑀𝑀𝑀2𝑀𝑟𝑟𝑟𝑟𝑥𝑦𝑦𝑥𝑑𝑑𝑑𝑑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑦sup

𝑥𝑥𝑗R

𝛼

ℎ󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑔𝑔(𝑑𝑑 𝑥 𝑑𝑑ℎ ) 𝑒𝑒𝑀𝑀𝑀2𝑀𝑟𝑟𝑟𝑟𝑥𝑀𝑥𝑥−𝑦𝑦𝑥𝑥𝑑𝑑𝑑𝑑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑦 𝑓𝑓𝑓

(28)

Hence we have only to show that

𝐾𝐾 =sup

ℎ󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑓𝑓𝛼𝛼(𝑑𝑑ℎ𝑓

𝑑𝑑

ℎ) 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝜙𝜙 𝑓𝑑𝑑𝑓 𝑑𝑑𝑑𝑑󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑦 𝑓𝑓𝑓

𝑑𝑑 𝑥R𝑓

(29)

for every𝛼𝛼 𝑥 {𝛼𝑓 𝛼𝑓 𝑓 𝑓 𝑓 𝑓 𝛼𝛼𝑓and ℎ > 𝛼. Let𝑆𝑆𝑟= {𝑑𝑑 𝑦 |𝑑𝑑 𝑥 𝑑𝑑| 𝑦

𝛼𝑓𝑓 𝑆𝑆2 = {𝑑𝑑 𝑦 |𝑑𝑑 𝑥 𝑑𝑑| > 𝛼and(𝛼/2)|𝑑𝑑| 𝑦 |𝑑𝑑|𝑓and 𝑆𝑆3 = {𝑑𝑑 𝑦

|𝑑𝑑 𝑥 𝑑𝑑| > 𝛼and (𝛼/2)|𝑑𝑑| > |𝑑𝑑|𝑓. Then, by (18), we have

𝐼𝐼 = sup

ℎ󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫𝑓𝑓𝛼𝛼(𝑑𝑑ℎ𝑓

𝑑𝑑

ℎ) 𝑑𝑑

𝛼𝛼

𝑦 𝑦𝑦𝑙𝑙sup

ℎ∫ 𝑒𝑒−𝑀𝑀𝑀𝑙𝑙𝑀𝑀𝑥𝑥−𝑦𝑦𝑥𝑀𝑀𝑥𝑥𝑒𝑒−𝑀𝑀𝑀2𝑀𝑟𝑟𝑟𝑟𝑥𝑦𝑦𝑥𝑑𝑑𝑑𝑑

= 𝑦𝑦𝑙𝑙sup

ℎ(∫𝑆𝑆1

𝑓 ∫

𝑆𝑆2

𝑓 ∫

𝑆𝑆3

)

× 𝑒𝑒−𝑀𝑀𝑀𝑙𝑙𝑀𝑀𝑥𝑥−𝑦𝑦𝑥𝑀𝑀𝑥𝑥𝑒𝑒−𝑀𝑀𝑀2𝑀𝑟𝑟𝑟𝑟𝑥𝑦𝑦𝑥𝑑𝑑𝑑𝑑

= 𝑦𝑦𝑙𝑙𝑓𝐼𝐼𝑟𝑓 𝐼𝐼2𝑓 𝐼𝐼3𝑓 𝑓

(5)

By a simple change of variable, we have

𝐼𝐼1=sup 𝑥𝑥𝑥R𝑒𝑒

𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑥𝑥𝑀1

ℎ∫𝑆𝑆1

𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑥𝑥−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑀𝑀𝑀𝑑𝑑𝑑𝑑

≤sup

𝑥𝑥𝑥R

1 ℎ∫𝑆𝑆1

𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑥𝑥−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑒𝑒𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑀𝑥𝑥−𝑀𝑀𝑀𝑀𝑑𝑑𝑑𝑑

≤ 2𝑒𝑒𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑀∫1

0

1

ℎ𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑑𝑑𝑑𝑑 = 2𝑒𝑒𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑀

× ∫1𝑀𝑀

0 𝑒𝑒

−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀_{𝑑𝑑𝑑𝑑 ≤ 𝑑𝑑} 1.

(31)

Since(1/2)|𝑥𝑥| 𝑥 |𝑑𝑑|and(1/2)|𝑑𝑑| ≤ |𝑥𝑥 𝑦 𝑑𝑑|on𝑆𝑆𝑀, then

𝐼𝐼𝑀=sup 𝑥𝑥𝑥R𝑒𝑒

ℎ∫𝑆𝑆2

𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑥𝑥−𝑀𝑀𝑀𝑀𝑀𝑀𝑀_𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑀𝑀𝑀_{𝑑𝑑𝑑𝑑}

≤sup

ℎ∫𝑆𝑆2

𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑥𝑥−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑥𝑥𝑀𝑑𝑑𝑑𝑑

≤ 1 ℎ∫𝑆𝑆2

𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑑𝑑𝑑𝑑 ≤ 𝑑𝑑𝑀,

(32)

for sufficiently large𝑙𝑙. Since(1/2)|𝑥𝑥| 𝑥 |𝑥𝑥 𝑦 𝑑𝑑|on𝑆𝑆3, then

𝐼𝐼3=sup 𝑥𝑥𝑥R𝑒𝑒

ℎ∫𝑆𝑆3

𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑥𝑥−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑀𝑀𝑀𝑑𝑑𝑑𝑑

≤sup

ℎ𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑥𝑥𝑀𝑀𝑀𝑀∫𝑆𝑆3

𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑀𝑀𝑀𝑑𝑑𝑑𝑑

≤ 𝑑𝑑󸀠󸀠31_ℎ∫ 𝑆𝑆3

𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀1𝑀𝑀𝑀𝑀𝑑𝑑𝑑𝑑 ≤ 𝑑𝑑3,

(33)

for sufficiently large𝑙𝑙.

4. Approximation Order of

K

󸀠󸀠_𝑀𝑀

(

R

)

A space of functions𝑆𝑆is called shift invariant if it is invariant under all integer translate, that is,

𝑓𝑓 𝑓 𝑆𝑆 𝑓𝑓 𝑓𝑓 (⋅ + 𝑘𝑘) 𝑓 𝑆𝑆 𝑆𝑘𝑘 𝑓Z. (34)

The principal shift-invariant subspaces 𝑆𝑆 = 𝑆𝑆(𝑆𝑆) are

generated by the closure of the linear span of the shifts of𝑆𝑆. The stationary ladder of spaces{𝑆𝑆𝑀_{(𝑆𝑆) 𝜙 ℎ 𝑥 𝜙𝜙}_{is given by}

𝑆𝑆𝑀(𝑆𝑆𝜙 = 𝜙𝑓𝑓 𝜙 ⋅

ℎ) 𝜙 𝑓𝑓 𝑓 𝑆𝑆𝑓 . (35)

To rate the efficiency for approximation of such spaces, the concept of approximation order is widely used. We say that the scale of the space𝑆𝑆𝑀_(𝑆𝑆)_{provides approximation order}

𝑘𝑘in𝐹𝐹if for every sufficiently smooth𝑓𝑓,

inf

𝑔𝑔𝑥𝑆𝑆ℎ₍_𝜙𝜙₎󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑓𝑓 𝑦 𝑓𝑓󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐹𝐹≤ 𝑑𝑑ℎ

𝑘𝑘_{, ℎ 𝑥 𝜙,}

(36)

where𝑑𝑑 = 𝑑𝑑(𝑓𝑓) 𝑥 𝜙. 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.

Let𝐾𝐾be an integral operator of the following form

(𝐾𝐾𝑓𝑓𝜙 (𝑥𝑥) = ∫ 𝐾𝐾 (𝑥𝑥, 𝑑𝑑𝜙 𝑓𝑓 (𝑑𝑑𝜙 𝑑𝑑𝑑𝑑, 𝑥𝑥 𝑓R. (37)

We assume that𝐾𝐾(𝑥𝑥 𝑦 𝑘𝑘, 𝑑𝑑) = 𝐾𝐾(𝑥𝑥, 𝑑𝑑 + 𝑘𝑘), ℎ 𝑓Z, 𝑥𝑥, 𝑑𝑑 𝑓 R. Forℎ 𝑥 𝜙, we define

𝐾𝐾𝑀= 󰜚󰜚𝑀𝐾𝐾󰜚󰜚1𝑀𝑀, (38)

where󰜚󰜚is the scaling operator󰜚󰜚𝑀𝑓𝑓 = 𝑓𝑓(⋅/ℎ). We say that the

integral operator𝐾𝐾defined by (37) provides approximation

order𝑘𝑘in𝐹𝐹if for every sufficiently smooth𝑓𝑓,

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐾𝐾𝑀𝑓𝑓 𝑦 𝑓𝑓󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐹𝐹≤ 𝑑𝑑ℎ𝑘𝑘, ℎ 𝑥 𝜙, (39)

where𝑑𝑑 = 𝑑𝑑(𝑓𝑓) 𝑥 𝜙. 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]). Let𝑓𝑓 𝑓K𝑀𝑀_𝑀𝑀󸀠󸀠(R). Let𝐾𝐾(𝑥𝑥, 𝑑𝑑), 𝑥𝑥, 𝑑𝑑, 𝑓 R, be the kernel of an integral operator𝐾𝐾 𝜙 K𝑀𝑀_𝑀𝑀󸀠󸀠(R) →

K𝑀𝑀_𝑀𝑀󸀠󸀠(R).𝐾𝐾𝑓𝑓is given by⟨𝐾𝐾𝑓𝑓, 𝑆𝑆𝐾 = ⟨𝑓𝑓, 𝐾𝐾𝑆𝑆𝐾. We say that the

operator𝐾𝐾provides approximation order𝑘𝑘inK𝑀𝑀_𝑀𝑀󸀠󸀠(R)if

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐾𝐾𝑀𝑓𝑓 𝑦 𝑓𝑓󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟𝑟𝑟𝑟_𝑀𝑀 󸀠󸀠 = sup ‖𝜙𝜙‖K𝑟𝑟𝑟𝑟𝑟_{𝑀𝑀 =1}

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨⟨𝐾𝐾𝑀𝑓𝑓, 𝑆𝑆𝑓 𝑦 ⟨𝑓𝑓, 𝑆𝑆𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 𝑑𝑑ℎ𝑘𝑘,

ℎ 𝑥 𝜙,

(40)

where the constant𝑑𝑑 = 𝑑𝑑(𝑓𝑓) 𝑥 𝜙.

We will now show that the kernel of an integral operator

𝐾𝐾 𝜙 K𝑀𝑀_𝑀𝑀󸀠󸀠(R) → K𝑀𝑀_𝑀𝑀󸀠󸀠(R)provides approximation order in

K𝑀𝑀_𝑀𝑀󸀠󸀠(R).

Theorem 11. Let 𝑆𝑆 𝑓 ̃K𝑀𝑀𝑀𝑘𝑘_𝑀𝑀 (R) with compact support such

that the integer shifts of𝑆𝑆 form an orthogonal basis of𝑆𝑆(𝑆𝑆) with respect to the inner product in𝐿𝐿𝑀₍_R₎_{. Assume that}_{𝑆𝑆(𝑥𝑥) =}

∑𝑛𝑛𝑥N𝑐𝑐𝑛𝑛𝑆𝑆(2𝑥𝑥 𝑦 𝑘𝑘)for some sequence{𝑐𝑐𝑛𝑛𝜙𝑛𝑛𝑥N. Let

𝐾𝐾 (𝑥𝑥, 𝑑𝑑𝜙 = ∑

𝑀𝑀𝑥Z

𝑆𝑆 (𝑥𝑥 𝑦 𝑙𝑙) 𝑆𝑆 (𝑑𝑑 𝑦 𝑙𝑙𝜙 , 𝑥𝑥, 𝑑𝑑 𝑓R ₍₄₁₎

be the kernel of the integral operator given by(37). Then 𝐾𝐾 provides approximation order𝑘𝑘inK𝑀𝑀_𝑀𝑀󸀠󸀠(R).

Proof. Firstly, we will show that

𝐽𝐽 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐾𝐾𝑀𝑆𝑆 𝑦 𝑆𝑆󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟_𝑀𝑀

=sup

𝑥𝑥𝑥R𝑒𝑒 𝑀𝑀𝑀𝑀𝑀𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑥𝑥𝑑𝑑𝛼𝛼𝛼𝛼(⟨𝐾𝐾𝑀(𝑥𝑥, 𝑑𝑑𝜙 , 𝑆𝑆 (𝑑𝑑𝜙𝑓 𝑦 𝑆𝑆 (𝑥𝑥)𝜙󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤ 𝑑𝑑󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑆𝑆󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟𝑟𝑟𝑟_𝑀𝑀ℎ

𝑘𝑘_,

(6)

where𝜙𝜙 𝜙 K𝑟𝑟𝑟𝑟𝑟_𝑀𝑀, 𝑘𝑘 𝜙 N. If we accept the result (42) for a moment, it follows that for𝑓𝑓 𝜙 K𝑟𝑟_𝑀𝑀󸀠󸀠(R) ⊂K𝑟𝑟𝑟𝑟𝑟 󸀠󸀠_𝑀𝑀 (R), we have

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨⟨𝐾𝐾ℎ𝑓𝑓 𝑓 𝑓𝑓, 𝜙𝜙𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨⟨𝑓𝑓,𝐾𝐾ℎ𝜙𝜙 𝑓 𝜙𝜙𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤ 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑓𝑓󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟_𝑀𝑀󸀠󸀠󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐾𝐾ℎ𝜙𝜙 𝑓 𝜙𝜙󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟_𝑀𝑀 ≤ 𝐶𝐶𝐶

𝑟𝑟󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜙𝜙󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

K𝑟𝑟𝑟𝑟𝑟󸀠󸀠_𝑀𝑀

(43)

hence

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐾𝐾ℎ𝑓𝑓 𝑓 𝑓𝑓󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟𝑟𝑟𝑟󸀠󸀠_𝑀𝑀 = sup

‖𝜙𝜙‖K𝑟𝑟𝑟𝑟𝑟_{𝑀𝑀 =1}

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨⟨𝐾𝐾ℎ𝑓𝑓, 𝜙𝜙𝑓 𝑓 ⟨𝑓𝑓, 𝜙𝜙𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 𝐶𝐶𝐶𝑟𝑟,

𝐶 > 0,

(44)

which implies the conclusion. Since{𝑆𝑆2−𝑗𝑗

(𝜙𝜙) 𝜙 𝜙𝜙 𝜙 Z}satisfy the conditions of(𝑀𝑀, 𝑀𝑀-) regular MRA of𝐿𝐿2₍_R₎_with_{𝑆𝑆(𝜙𝜙) = 𝑆𝑆}

0, we can apply (21) to

the operator𝐾𝐾, that is,

𝑑𝑑𝛼𝛼

𝑑𝑑𝑑𝑑𝛼𝛼𝐾𝐾ℎ𝑓𝑓 (𝑑𝑑) = 1_𝐶∫ 𝑔𝑔 𝑔𝑑𝑑 𝑓 𝑥𝑥_𝐶 ) 𝑑𝑑 𝛼𝛼

𝑑𝑑𝑥𝑥𝛼𝛼𝑓𝑓 𝑓𝑥𝑥𝑓 𝑑𝑑𝑥𝑥

+ 1_𝐶∫ 𝑅𝑅𝛼𝛼𝑔𝑑𝑑_𝐶,𝑥𝑥_𝐶) 𝑑𝑑_{𝑑𝑑𝑥𝑥}𝛼𝛼_𝛼𝛼𝑓𝑓 𝑓𝑥𝑥𝑓 𝑑𝑑𝑥𝑥,

(45)

where𝑔𝑔and𝑅𝑅𝛼𝛼_{are given in (21). For}_{0 ≤ 𝛼𝛼 ≤ 𝑀𝑀}_,

𝐽𝐽 =sup

𝑥𝑥𝑥R𝑒𝑒 𝑀𝑀𝑀𝑟𝑟𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑𝑑𝑑𝑑𝑑𝑑𝛼𝛼𝛼𝛼 𝑓⟨𝐾𝐾ℎ𝑓𝑑𝑑, 𝑥𝑥𝑓 , 𝜙𝜙 𝑓𝑥𝑥𝑓𝑓 𝑓 𝜙𝜙 (𝑑𝑑)𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤sup

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 1

𝐶∫ 𝑔𝑔 𝑔𝑑𝑑 𝑓 𝑥𝑥𝐶 )

× ( 𝑑𝑑𝛼𝛼

𝑑𝑑𝑥𝑥𝛼𝛼𝜙𝜙 𝑓𝑥𝑥𝑓 𝑓 𝑑𝑑 𝛼𝛼

𝑑𝑑𝑥𝑥𝛼𝛼𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑦𝑦𝑦𝑥𝑥) 𝑑𝑑𝑥𝑥

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 +sup

󵄨󵄨󵄨󵄨󵄨󵄨1𝐶∫ 𝑅𝑅𝛼𝛼𝑔 𝑑𝑑 𝑓 𝑥𝑥

𝐶 )

× ( 𝑑𝑑𝛼𝛼

𝑑𝑑𝑥𝑥𝛼𝛼𝜙𝜙 𝑓𝑥𝑥𝑓 𝑓 𝑑𝑑 𝛼𝛼

𝑑𝑑𝑥𝑥𝛼𝛼𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑦𝑦𝑦𝑥𝑥) 𝑑𝑑𝑥𝑥

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 = 𝐽𝐽1+ 𝐽𝐽2.

(46)

In order to estimate 𝐽𝐽1, we consider 𝑔𝑔 𝜙 D(R) with

∫ 𝑔𝑔(𝑑𝑑)𝑑𝑑𝑑𝑑 = 1and∫ 𝑔𝑔(𝑑𝑑)𝑑𝑑𝛼𝛼𝑑𝑑𝑑𝑑 = 0, 0 𝑑 𝑑𝛼𝛼𝑑 𝑑max{𝑀𝑀, 𝑘𝑘 𝑓 1}.

Let𝑐𝑐be a constant such that sup𝑔𝑔 ⊂ 𝑔𝑓𝑐𝑐, 𝑐𝑐𝑔. If we assume

𝐶 𝜙 (0, 1), the smoothness of𝜙𝜙 𝜙K𝑟𝑟𝑟𝑟𝑟_𝑀𝑀 (R) ⊂ 𝐶𝐶𝑟𝑟𝑟𝑟𝑟(R)implies

𝐽𝐽1= sup ℎ𝑥𝑥𝑥R

𝑒𝑒𝑀𝑀𝑀𝑟𝑟ℎ𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨}

󵄨󵄨∫|𝑥𝑥𝑥𝑦𝑦|𝑥𝑥𝑥𝑔𝑔 𝑓𝑑𝑑 𝑓 𝑥𝑥𝑓

× ( 𝑑𝑑𝛼𝛼 𝑑𝑑𝑥𝑥𝛼𝛼𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑦𝑦𝑦ℎ𝑦𝑦𝑓 𝑑𝑑

𝛼𝛼

𝑑𝑑𝑥𝑥𝛼𝛼𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑦𝑦𝑦ℎ𝑥𝑥) 𝑑𝑑𝑥𝑥

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

= sup

ℎ𝑥𝑥𝑥R𝑒𝑒 𝑀𝑀𝑀𝑟𝑟ℎ𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨∫|𝑥𝑥𝑥𝑦𝑦|𝑥𝑥𝑥𝑔𝑔 𝑓𝑑𝑑 𝑓 𝑥𝑥𝑓

× ( 𝑓𝑥𝑥 𝑓 𝑑𝑑𝑓 ×𝐶 𝑑𝑑𝛼𝛼𝑟1 𝑑𝑑𝑥𝑥𝛼𝛼𝑟1𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨𝑦𝑦𝑦ℎ𝑥𝑥

+ ⋅ ⋅ ⋅ + 𝑓𝑥𝑥 𝑓 𝑑𝑑𝑓

𝑟𝑟𝑥1

(𝑘𝑘 𝑓 1)! ×𝐶𝑟𝑟𝑥1 𝑑𝑑𝛼𝛼𝑟𝑟𝑟𝑥1

𝑑𝑑𝑥𝑥𝛼𝛼𝑟𝑟𝑟𝑥1𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨 𝑦𝑦𝑦ℎ𝑥𝑥

+𝑓𝑥𝑥 𝑓 𝑑𝑑𝑓

𝑟𝑟

𝑘𝑘! ×𝐶𝑟𝑟 𝑑𝑑

𝛼𝛼𝑟𝑟𝑟

𝑑𝑑𝑥𝑥𝛼𝛼𝑟𝑟𝑟𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨

𝑦𝑦𝑦𝑦𝑦𝑀𝑦𝑦𝑀) 𝑑𝑑𝑥𝑥

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

≤ sup

ℎ𝑥𝑥𝑥R𝑒𝑒 𝑀𝑀𝑀𝑟𝑟ℎ𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨∫|𝑥𝑥𝑥𝑦𝑦|𝑥𝑥𝑥𝑔𝑔 𝑓𝑑𝑑 𝑓 𝑥𝑥𝑓

𝑓𝑥𝑥 𝑓 𝑑𝑑𝑓𝑟𝑟 𝑘𝑘!

×𝐶𝑟𝑟 𝑑𝑑𝛼𝛼𝑟𝑟𝑟 𝑑𝑑𝑥𝑥𝛼𝛼𝑟𝑟𝑟𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨

𝑦𝑦𝑦𝑦𝑦𝑀𝑦𝑦𝑀𝑑𝑑𝑥𝑥

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

≤ sup

ℎ𝑥𝑥𝑥R𝑒𝑒

𝑀𝑀𝑀𝑟𝑟ℎ𝑥𝑥𝑀_{𝐶𝐶𝑐𝑐}𝑟𝑟

𝑘𝑘!𝐶𝑟𝑟_𝑦𝑦₍_𝑦𝑦₎_{𝑥[ℎ𝑥𝑥𝑥ℎ𝑥𝑥𝑥ℎ𝑥𝑥𝑟ℎ𝑥𝑥]}sup 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨 𝑑𝑑

𝛼𝛼𝑟𝑟𝑟

𝑑𝑑𝑥𝑥𝛼𝛼𝑟𝑟𝑟𝜙𝜙 𝑓𝑥𝑥𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨

= 𝐶𝐶𝑐𝑐_𝑘𝑘!𝑟𝑟𝐶𝑟𝑟sup

𝑠𝑠𝑥R𝑒𝑒

𝑀𝑀𝑀𝑟𝑟𝑠𝑠𝑀 _sup 𝑡𝑡𝑥[𝑠𝑠𝑥𝑥𝑥𝑥𝑠𝑠𝑟𝑥𝑥] 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨 𝑑𝑑𝛼𝛼𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝛼𝛼𝑟𝑟𝑟𝜙𝜙 (𝑑𝑑)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨

= 𝐶𝐶𝑐𝑐𝑟𝑟 𝑘𝑘!𝐶𝑟𝑟sup𝑠𝑠𝑥R𝑒𝑒

𝑀𝑀𝑀𝑟𝑟𝑠𝑠𝑀 _sup

𝑡𝑡𝑥[𝑠𝑠𝑥𝑥𝑥𝑥𝑠𝑠𝑟𝑥𝑥](𝑒𝑒

𝑥𝑀𝑀𝑀𝑟𝑟𝑡𝑡𝑀_𝑒𝑒𝑀𝑀𝑀𝑟𝑟𝑡𝑡𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨

𝑑𝑑𝛼𝛼𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝛼𝛼𝑟𝑟𝑟𝜙𝜙 (𝑑𝑑)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨)

≤ 𝐶𝐶𝑐𝑐𝑟𝑟 𝑘𝑘!𝐶𝑟𝑟sup𝑠𝑠𝑥R𝑒𝑒

𝑀𝑀𝑀𝑟𝑟𝑠𝑠𝑀_𝑒𝑒𝑥𝑀𝑀𝑀𝑟𝑟𝑀|𝑠𝑠|𝑥𝑥𝑥𝑀𝑀_sup 𝑡𝑡𝑥R(𝑒𝑒

𝑀𝑀𝑀𝑟𝑟𝑡𝑡𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨

𝑑𝑑𝛼𝛼𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝛼𝛼𝑟𝑟𝑟𝜙𝜙 (𝑑𝑑)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨)

= 𝐶𝐶1󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜙𝜙󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟𝑟𝑟𝑟_𝑀𝑀 ,

(47)

where𝜉𝜉(𝑥𝑥) = 𝐶𝑑𝑑 + 𝜉𝜉𝐶(𝑥𝑥 𝑓 𝑑𝑑) 𝜙 𝑔𝐶𝑑𝑑 𝑓 𝐶𝑐𝑐, 𝐶𝑑𝑑 + 𝐶𝑐𝑐𝑔for some

𝜉𝜉 𝜙 (0, 1)and𝐶𝐶1 = 𝐶𝐶(𝑐𝑐𝑟𝑟/𝑘𝑘!)𝐶𝑟𝑟sup𝑠𝑠𝑥R𝑒𝑒𝑀𝑀𝑀𝑟𝑟𝑠𝑠𝑀𝑒𝑒𝑥𝑀𝑀𝑀𝑟𝑟𝑀|𝑠𝑠|𝑥𝑥𝑥𝑀𝑀 𝑑 ∞. To show the finiteness of𝐶𝐶1in the last statement, we use

sup

𝑀𝑀𝑀𝑟𝑟𝑠𝑠𝑀_𝑒𝑒𝑥𝑀𝑀𝑀𝑟𝑟𝑀|𝑠𝑠|𝑥𝑥𝑥𝑀𝑀

≤sup

|𝑠𝑠|𝑥𝑥𝑥𝑒𝑒

(7)

+sup

|𝑠𝑠|>𝑐𝑐𝑒𝑒

𝑀𝑀𝑀𝑀𝑀𝑠𝑠𝑀_𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀|𝑠𝑠|−𝑐𝑐𝑀𝑀

≤sup

|𝑠𝑠|≤𝑐𝑐𝑒𝑒

+sup

|𝑠𝑠|>𝑐𝑐𝑒𝑒

≤ 𝑒𝑒𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑀+ 𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑀.

(48)

We will estimate𝐽𝐽2by using the following facts. Since𝜙𝜙

has a compact support, there exists𝑀𝑀 𝑀 𝑀such that𝐾𝐾𝐾𝐾𝐾𝐾 𝐾𝐾𝐾 𝐾 𝑀for|𝐾𝐾 𝑥 𝐾𝐾| 𝑀 𝑀𝑀. Also, by the choice of𝑔𝑔and property (c) of the reproducing kernel𝑞𝑞0, we have

∫ 𝑅𝑅𝛼𝛼(𝐾𝐾𝐾 𝐾𝐾𝑥 𝐾𝐾𝑠𝑠𝑑𝑑𝐾𝐾 𝐾 𝑑𝑑𝛼𝛼

𝑑𝑑𝐾𝐾𝛼𝛼∫ 𝐾𝐾 (𝐾𝐾𝐾 𝐾𝐾𝑥 𝐾𝐾𝑠𝑠𝑑𝑑𝐾𝐾

𝑥 𝑑𝑑𝛼𝛼

𝑑𝑑𝐾𝐾𝛼𝛼∫ 𝑔𝑔 (𝐾𝐾 𝑥 𝐾𝐾𝑥 𝐾𝐾𝑠𝑠𝑑𝑑𝐾𝐾

𝐾 𝑑𝑑𝛼𝛼 𝑑𝑑𝐾𝐾𝛼𝛼𝐾𝐾𝑠𝑠𝑥 𝑑𝑑

𝛼𝛼

𝑑𝑑𝐾𝐾𝛼𝛼∫ 𝑔𝑔 𝐾𝑡𝑡𝐾 𝐾𝐾𝑠𝑠𝑑𝑑𝑡𝑡

𝐾 𝑑𝑑_{𝑑𝑑𝐾𝐾}𝛼𝛼_𝛼𝛼𝐾𝐾𝑠𝑠𝑥 𝑑𝑑_{𝑑𝑑𝐾𝐾}𝛼𝛼_𝛼𝛼𝐾𝐾𝑠𝑠𝐾 𝑀𝐾 𝑀 ≤ 𝑠𝑠 ≤ 𝑠𝑠 + 𝑠𝑠 𝑥 𝑠.

(49)

Hence

𝐽𝐽2𝐾 sup ℎ𝑥𝑥𝑥R𝑒𝑒

𝑀𝑀𝑀𝑀𝑀ℎ𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨∫ 𝑅𝑅

𝛼𝛼_{(𝐾𝐾𝐾 𝐾𝐾𝑥}

× ( 𝑑𝑑𝛼𝛼 𝑑𝑑𝐾𝐾𝛼𝛼𝜙𝜙 (𝐾𝐾𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑦𝑦𝑦ℎ𝑦𝑦𝑥 𝑑𝑑 𝛼𝛼

𝑑𝑑𝐾𝐾𝛼𝛼𝜙𝜙 (𝐾𝐾𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

𝑦𝑦𝑦ℎ𝑥𝑥) 𝑑𝑑𝐾𝐾

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝐾 sup

ℎ𝑥𝑥𝑥R

𝑒𝑒𝑀𝑀𝑀𝑀𝑀ℎ𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨∫ 𝑅𝑅

𝛼𝛼_{(𝐾𝐾𝐾 𝐾𝐾𝑥}

× ( (𝐾𝐾 𝑥 𝐾𝐾𝑥 ×𝑦 𝑑𝑑𝛼𝛼𝛼𝛼 𝑑𝑑𝐾𝐾𝛼𝛼𝛼𝛼𝜙𝜙 (𝐾𝐾𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨𝑦𝑦𝑦ℎ𝑥𝑥

+ ⋅ ⋅ ⋅ + (𝐾𝐾 𝑥 𝐾𝐾𝑥

𝑘𝑘−𝛼

𝐾𝑠𝑠 𝑥 𝑠𝐾! ×𝑦𝑘𝑘−𝛼 𝑑𝑑

𝛼𝛼𝛼𝑘𝑘−𝛼

𝑑𝑑𝐾𝐾𝛼𝛼𝛼𝑘𝑘−𝛼𝜙𝜙 (𝐾𝐾𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨 𝑦𝑦𝑦ℎ𝑥𝑥

+(𝐾𝐾 𝑥 𝐾𝐾𝑥

𝑘𝑘

𝑠𝑠! ×𝑦𝑘𝑘 𝑑𝑑

𝛼𝛼𝛼𝑘𝑘

𝑑𝑑𝐾𝐾𝛼𝛼𝛼𝑘𝑘𝜙𝜙 (𝐾𝐾𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨𝑦𝑦𝑦𝑦𝑦𝑀𝑦𝑦𝑀) 𝑑𝑑𝐾𝐾

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝐾 𝑦𝑘𝑘 𝑠𝑠!ℎ𝑥𝑥𝑥supR

𝑒𝑒𝑀𝑀𝑀𝑀𝑀ℎ𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨∫|𝑥𝑥−𝑦𝑦|≤𝑀𝑀𝑅𝑅

𝛼𝛼_{(𝐾𝐾𝐾 𝐾𝐾𝑥 (𝐾𝐾 𝑥 𝐾𝐾𝑥}𝑘𝑘

× 𝑑𝑑𝛼𝛼𝛼𝑘𝑘 𝑑𝑑𝐾𝐾𝛼𝛼𝛼𝑘𝑘𝜙𝜙 (𝐾𝐾𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨𝑦𝑦𝑦𝑦𝑦𝑀𝑦𝑦𝑀𝑑𝑑𝐾𝐾

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

≤ 𝑀𝑀_𝑠𝑠!𝑘𝑘𝑦𝑘𝑘 sup

ℎ𝑥𝑥𝑥R sup

ℎ𝑥𝑥𝑥𝑦𝑦𝐾𝑦𝑦𝐾𝑥ℎ𝑦𝑦𝑒𝑒 𝑀𝑀𝑀𝑀𝑀ℎ𝑥𝑥𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝑑𝑑𝛼𝛼𝛼𝑘𝑘

𝑑𝑑𝐾𝐾𝛼𝛼𝛼𝑘𝑘𝜙𝜙 (𝐾𝐾𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨𝑦𝑦𝑦𝑦𝑦𝑀𝑦𝑦𝑀

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

×󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨}

󵄨󵄨∫|𝑥𝑥−𝑦𝑦|≤𝑀𝑀𝑅𝑅

𝛼𝛼_{(𝐾𝐾𝐾 𝐾𝐾𝑥 (𝐾𝐾 𝑥 𝐾𝐾𝑥}𝑘𝑘_{𝑑𝑑𝐾𝐾}󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨

≤ 𝐶𝐶󸀠󸀠𝑐𝑐_𝑠𝑠!𝑘𝑘𝑦𝑘𝑘sup

𝑀𝑀𝑀𝑀𝑀𝑠𝑠𝑀_𝑒𝑒−𝑀𝑀𝑀𝑀𝑀𝑀|𝑠𝑠|−𝑐𝑐𝑀𝑀_sup 𝑡𝑡𝑥R(𝑒𝑒

𝑀𝑀𝑀𝑀𝑀𝑡𝑡𝑀󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨

𝑑𝑑𝛼𝛼𝛼𝑘𝑘

𝑑𝑑𝑡𝑡𝛼𝛼𝛼𝑘𝑘𝜙𝜙 𝐾𝑡𝑡𝐾󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨) 𝐾 𝐶𝐶2󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜙𝜙󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩K𝑟𝑟𝑟𝑟𝑟_𝑀𝑀 .

(50)

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(8)

[17] G. C. Kyriazis, “Approximation of distribution spaces by means of kernel operators,”The Journal of Fourier Analysis and Appli-cations, vol. 2, no. 3, pp. 261–286, 1996.

(9)

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Abstract and Applied Analysis Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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