Research Article | Open Access
Multiresolution Expansion and Approximation Order of Generalized Tempered Distributions
Let be the generalized tempered distributions of -growth with restricted order , where the function grows faster than any linear functions as . We show the convergence of multiresolution expansions of in the test function space of . In addition, we show that the kernel of an integral operator provides approximation order in in the context of shift-invariant spaces.
Multiresolution analysis was shown to be very useful in extending the expansions in orthogonal wavelets from 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, , by wavelets  and the multiresolution expansions’ pointwise convergence of . Pilipović and Teofanov have showed the uniform convergence on compact sets of the derivatives of multiresolution expansions of and the convergence of multiresolution expansions of in the test function space of . As an application, Pilipović and Teofanov have shown that the kernel of an integral operator provides approximation order in in the context of shift-invariant spaces .
In the meantime, the tempered distributions of polynomial growth were extended to tempered distributions of -growth, , in [5, 6] and -growth, , in [7, 8] or -growth, , 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, , by wavelets . Also, we have shown that the multiresolution expansions of converges pointwise to the value of the distribution where it exists .
In this paper, we will show the uniform convergence on compact sets of the derivatives of multiresolution expansions of and convergence of multiresolution expansions of in the test function space of . In addition, we will show that the kernel of an integral operator provides approximation order in . This is an extension of the works of Pilipović and Teofanov  in the context of generalized tempered distributions, .
2. The Generalized Tempered Distribution Spaces
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 . Further, we define for negative by means of the equality . Note that since the derivative of is unbounded in , 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 as the space of all functions such that
The topology in is defined by the family of the seminorms . Then become a Fréchet space and are continuous and dense inclusions; here denotes the spaces of all functions with compact supports, the spaces of polynomially decreasing functions (Schwartz functions), and the space of all functions. By , we mean the space of continuous linear functionals on .
Definition 1. We say that the elements of are generalized tempered distributions.
Clearly, when , are tempered distributions (Schwarz distributions), . When , are tempered distributions, , which are introduced and characterized by Yoshinaga  and Hasumi , independently. When , are tempered distributions, , which are introduced and characterized by Sznajder and Zielezny [7, 8]. For details about , we refer to [9, 10].
For a natural number , we define by the space of all such that
The topology of is defined by the family of and the dual of is denoted by . Clearly, is the projective limit of when and = . Also, we have continuous and dense inclusion mapping as following:
Definition 2. We say that the elements of are generalized tempered distributions of order .
We define by the space of all such that The topology of is defined by the family of and the dual of is denoted by . Obviously, .
Now, we give a theorem that will be used later.
Theorem 3. Let and sequence be given in such that converges uniformly to on every compact set and for . If is bounded in , then the sequence converges to in .
Proof. Let be given and let . Then there exist such that for arbitrary . Also, since the sequence is bounded in , we can take a positive number and a compact set such that when and From (7) and (8), we have
3. Multiresolution Expansion of
Definition 4. A multiresolution analysis (shortly MRA) consists of a sequence of closed subspaces , of satisfying the following:(i) is an orthonormal basis of , (ii), (iii), (iv). 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 , is regular MRA of if the scaling function is in .
Example 6. It is impossible that the scaling function has exponential decay and , with all derivatives bounded, unless . Refer to [13, Corollary 5.5.3]. So we will restrict our attention to or . From the remark in  or, page 152 [2, Example 4, page 48], Battle-Lemarié’s wavelets are in for some when , but not in even if they have exponential decay and smoothness. In , Daubechies shown that for an arbitrary nonnegative integer , there exists an regular MRA of such that the scaling function has compact supports.
Let be an regular MRA of and let be a scaling function. The reproducing kernel of is given by
The series and its derivatives with respect to or of order converge uniformly on because of the regularity of . The reproducing kernel of the projection operator onto is and the projection of onto is given by
The sequence , given in (12), is called the multiresolution expansion of .
Definition 7. For a given , the sequence defined by is called the multiresolution expansion of .
We deduce the following properties of the reproducing kernel with scaling function :(a) and for all .(b) For every and , there exist such that where we used the properties (2).(c) .
Let be an regular MRA of . We fix a function with . We let denote the function and let denote the operation of convolution by . For each fixed , we consider the function of the variable . From (c), we have for , whereas
Let From (b) and the fact that , we have and these functions also satisfy identically in for every . They, for every and with at most -growth, define operator by which are such that that is, From Theorem 1.1 in , we have uniformly on compact sets. Now we will show the uniform convergence on compact sets of the derivatives of multiresolution expansions of .
Theorem 8. Let such that the corresponding derivatives are bounded by a when , for every and some . If , given by (12), be the projection of onto an regular MRA of , then the sequence converges uniformly on compact sets to as , for every .
Proof. If , we have where is a continuous function with growth and . From (18), given a compact set , we have for large enough and . Since can be chosen arbitrary, we obtain by dominated convergence theorem, uniformly for . From (21) and (23), we have the conclusion.
We now ready to show the main theorem.
Theorem 9. Let and let , given by (7), be a projection of onto an regular MRA of . If , then the sequence converges to in as .
Proof. Let and be given in (21) such that and . From Theorems 3 and 8 and (21), it suffices to show that is bounded for every and . Since has a compact support, then Hence we have only to show that for every and . Let . Then, by (18), we have By a simple change of variable, we have Since and on , then for sufficiently large . Since on , then for sufficiently large .
4. Approximation Order of
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 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
We assume that . For , we define where is the scaling operator . 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].
Definition 10 (see ). Let . Let , be the kernel of an integral operator . is given by . We say that the operator provides approximation order in if where the constant .
We will now show that the kernel of an integral operator provides approximation order in .
Theorem 11. Let with compact support such that the integer shifts of form an orthogonal basis of with respect to the inner product in . Assume that for some sequence . Let be the kernel of the integral operator given by (37). Then provides approximation order in .
Proof. Firstly, we will show that
where . If we accept the result (42) for a moment, it follows that for , we have
which implies the conclusion.
Since satisfy the conditions of regular MRA of with , we can apply (21) to the operator , that is, where and are given in (21). For ,
In order to estimate , we consider with and . Let be a constant such that . If we assume , the smoothness of implies where for some and . To show the finiteness of in the last statement, we use
We will estimate by using the following facts. Since has a compact support, there exists such that for . Also, by the choice of and property (c) of the reproducing kernel , we have Hence
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