Multiresolution Expansion and Approximation Order of Generalized Tempered Distributions

í µí±í µí± í µí±í µí± í® í° í® í° (R) be the generalized tempered distributions of í µí±í µí± í µí±í µí±í µí±í µí±í µí±í µí±-growth with restricted order í µí±í µí± í µí± N 0 , where the function í µí±í µí±(í µí±í µí±) grows faster than any linear functions as |í µí±í µí±| í µí±¥ í µí±¥. We show the convergence of multiresolution expansions of K í µí±í µí± í µí±í µí± í® í° í® í° (R) in the test function space K í µí±í µí± í µí±í µí± (R) of K í µí±í µí± í µí±í µí± í® í° í® í° (R). In addition, we show that the kernel of an integral operator í µí°¾í µí°¾ í µí°¾ K í µí±í µí± í µí±í µí± í® í° í® í° (R) í µí±¥ K í µí±í µí± í µí±í µí± í® í° í® í° (R) provides approximation order in K í µí±í µí± í µí±í µí± í® í° í® í° (R) in the context of shift-invariant spaces.


Introduction
Multiresolution analysis was shown to be very useful in extending the expansions in orthogonal wavelets from  2 (R) to a certain class of tempered distributions.Some interactions between wavelets and tempered distributions have been presented by Walter's work in [1][2][3].Walter has found the analytic representation of tempered distributions of polynomial growth with restricted order, S   (R),   N 0 , by wavelets [1] and the multiresolution expansions' pointwise convergence of S   (R) [3].Pilipović and Teofanov have showed the uniform convergence on compact sets of the derivatives of multiresolution expansions of S   (R) and the convergence of multiresolution expansions of S   (R) in the test function space S  (R) of S   (R).As an application, Pilipović and Teofanov have shown that the kernel of an integral operator   S   (R)  S   (R) provides approximation order in S   (R) in the context of shift-invariant spaces [4].
In the meantime, the tempered distributions of polynomial growth were extended to tempered distributions of  || -growth, K  1 (R), in [5,6] and  ||  -growth, K   (R), in [7,8] or   -growth, K   (R), in [9,10], where the function () grows faster than any linear functions as ||  .We have considered the analytic representation of tempered distributions of   -growth with restricted order, K    (R), by wavelets [11].Also, we have shown that the multiresolution expansions of K    (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 of K    (R) and convergence of multiresolution expansions of K    (R) in the test function space K   (R) of K    (R).In addition, we will show that the kernel of an integral operator   K    (R)  K    (R) provides approximation order in K    (R).This is an extension of the works of Pilipović and Teofanov [4] in the context of generalized tempered distributions, K   (R).

The Generalized Tempered Distribution Spaces K 󸀠󸀠 𝑀𝑀 (R)
Throughout this paper, we will use  or   to denote the positive constants, which are independent parameters and may be different at each occurrence.
Let () (    ) denote a continuous increasing function such that ()   and ()  .For   , we define The function () is an increasing, convex, and continuous function with ()  , ()   and satisfies the fundamental convexity inequality  1 ) +  2 ) ≤  1 +  2 ).Further, we define ) for negative  by means of the equality )  ).Note that since the derivative ) of ) is unbounded in R, the function ) will grow faster than any linear function as ||  .Now we list some properties of ) which will be frequently used later.Consider the following: Using the function ), we define the space K  R) as the space of all functions The topology in K  R) is defined by the family of the seminorms ‖ ⋅ ‖ K  .Then K  R) become a Fréchet space and DR)   K  R)   SR)   ER) are continuous and dense inclusions; here DR) denotes the spaces of all  ∞ R) functions with compact supports, SR) the spaces of polynomially decreasing functions (Schwartz functions), and ER) the space of all  ∞ R) functions.By K   R), we mean the space of continuous linear functionals on K  R).Definition 1.We say that the elements of K   R) are generalized tempered distributions.
Clearly, when )  log + ||), K   R) are tempered distributions (Schwarz distributions), SR).When )  ||, K   R) are tempered distributions, K  1 R), which are introduced and characterized by Yoshinaga [6] and Hasumi [5], independently.When )  ||     , K   R) are tempered distributions, K   R), which are introduced and characterized by Sznajder and Zielezny [7,8].For details about K   R), we refer to [9,10].For a natural number , we define by K   R) the space of all     R) such that The topology of K   R) is defined by the family of ‖ ⋅ ‖ K   and the dual of K   R) is denoted by K    R).Clearly, K  R) is the projective limit of K   R) when    and K   R) = ⋃ N K    R).Also, we have continuous and dense inclusion mapping as following: From ( 7) and ( 8), we have lim

Multiresolution Expansion of K 󸀠󸀠 𝑀𝑀 𝑀R)
Definition 4. A multiresolution analysis (shortly MRA) consists of a sequence of closed subspaces      Z, of  2 R) satisfying the following: The function  whose existence is asserted in (i) is called a scaling function of the given MRA.
Definition 5. We say that a multiresolution analysis Example 6.It is impossible that the scaling function  has exponential decay and    ∞ (R), with all derivatives bounded, unless   .Refer to [13,Corollary 5.5.3].So we will restrict our attention to K   (R) or K  (R).From the remark in [13] or, page 152 [2, Example 4, page 48], Battle-Lemarié's wavelets are in K   (R) for some   N when ()  , but not in K  (R) even if they have exponential decay and smoothness.In [13], Daubechies shown that for an arbitrary nonnegative integer , there exists an (, -) regular MRA of  2 (R) such that the scaling function  has compact supports.
Let   be an (, -) regular MRA of  2 (R) and let  be a scaling function.The reproducing kernel of  0 is given by The series and its derivatives with respect to  or  of order ≤  converge uniformly on R because of the regularity of   K  (R).The reproducing kernel of the projection operator onto   is   (,      0 (  ,    , ,   R, (11) and the projection of    2 (R) onto   is given by The sequence {  } Z , given in (12), is called the multiresolution expansion of    2 (R).
is called the multiresolution expansion of   K    (R).
We deduce the following properties of the reproducing kernel  0 with scaling function   K  (R): (a)  0 (, )   0 (, ) and  0 (  ,   )   0 (, ) for all   Z. ( where we used the properties (2). (c) Let   be an (, -) regular MRA of  2 (R).We fix a function   D(R) with ∫ ()  .We let   denote the function   (  ) and let   denote the operation of convolution by   .For each fixed , we consider the function     0 (, ) of the variable .From (c), we have for  ≤   , whereas Now, it follows from integration by parts that the kernal (  ) of the operator  shares these properties (15) and (16) with  0 (, ). Let From (b) and the fact that and these functions also satisfy By a simple change of variable, we have for sufficiently large .Since (1/2)||  |  | on  3 , then for sufficiently large .

Approximation Order of K 󸀠󸀠 𝑀𝑀 (R)
A space of functions  is called shift invariant if it is invariant under all integer translate, that is, The principal shift-invariant subspaces  = () are generated by the closure of the linear span of the shifts of .The stationary ladder of spaces {  ()  ℎ   is given by To rate the efficiency for approximation of such spaces, the concept of approximation order is widely used.We say that the scale of the space   () provides approximation order  in  if for every sufficiently smooth , where  = ()  .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.
Let  be an integral operator of the following form ( () = ∫  (,   ( ,   R. We assume that (  , ) = (,  + ), ℎ  Z, ,   R. For ℎ  , we define where  is the scaling operator    = (⋅/ℎ) .We say that the integral operator  defined by (37) provides approximation order  in  if for every sufficiently smooth , where  = ()  .For further details about the theory on the approximation order provided by integral or kernel operator, we refer to [17,18].
We will now show that the kernel of an integral operator

Theorem 11. Let 𝑆𝑆 𝑓 K𝑀𝑀𝑀𝑘𝑘
(R) with compact support such that the integer shifts of  form an orthogonal basis of () with respect to the inner product in   (R).Assume that () = ∑ N   (2  ) for some sequence {   N .Let be the kernel of the integral operator given by (37).Then  provides approximation order  in K    (R).
Let  be a constant such that sup  ⊂ , .If we assume   (0, 1), the smoothness of   K   (R) ⊂   (R) implies ∞ R) such that         K   sup ) Let  and sequence {  } N be given in K 1  R) such that {  /  )  } N converges uniformly to   /  ) on every compact set  ⊂ R and for         .If {  } N is bounded in K 1  R), then the sequence {  } N converges to   K 1  R) in K   R).