© Hindawi Publishing Corp. RECONSTRUCTION IN TIME-WARPED WEIGHTED SHIFT-INVARIANT SPACES WITH APPLICATION TO SPLINE SUBSPACES

We discuss the reproducing kernel structure in shift-invariant 
spaces and the weighted shift-invariant spaces, and obtain the 
reconstruction formula in time-warped weighted shift-invariant 
spaces, then apply them to a spline subspace. In the spline 
subspace, we give a reconstruction formula in a time-warped spline 
subspace.


Introduction.
The problem of reconstruction of a function f has been applied widely to signal or image processing, so it is of vital importance to study the subject in the field of signal or image processing.The problem of reconstruction means that we reconstruct the function f on R d from its countable samples.It is well known that in the sampling and reconstruction problem, the function f is often assumed to belong to a shift-invariant space [1,2,5].For instance, which is the classical space of band-limited spaces, often used as a model in sampling theory.For the special shift-invariant subspaces, spline subspaces, there are many practical applications to signal or image processing.So the research of spline subspaces received much attention in [1,6,7] and so on.
In [4], Clark et al. first introduced the concept of time-warped space and obtained the reconstruction formula in time-warped band-limited signals.They generalized the classical Shannon sampling theorem and provided a method of reconstruction for a certain space of non-band-limited signal from irregular spaced samples.Since we often deal with space of non-band-limited signal and irregular spaced samples in practical applications, Clark's method is very valuable.
In this paper, we discuss the reproducing kernel structure in shift-invariant spaces V 2 (ϕ) and time-warped shift-invariant spaces V 2 γ (ϕ).Then we obtain the reconstruction formula in the time-warped weighted shift-invariant spaces V p m,γ (ϕ).As a special example or a particular application, we obtain reconstruction formula in the time-warped spline subspaces.Then we use another method to obtain the reconstruction formula in time-warped spline subspaces.Moreover, we construct reproducing kernel in spline and time-warped spline subspaces.Because time-warped spaces are more general than the original spaces themselves, we generalize the results of [2,3,5,7] with time-warped method.

Notations and definitions.
Given a space S of function f : R n → C n and an invertible continuous function γ : R n → R n , the time-warped function space and γ is called the warping function.
A reproducing kernel on a Hilbert space H is a function We also consider the weighted sequence spaces , where (m k ) is the restriction of m to Z n .Shift-invariant spaces are defined by Weighted shift-invariant spaces are defined by Time-warped shift-invariant spaces are defined by Time-warped weighted shift-invariant spaces are defined by Two sets {ϕ n } and { φn } are biorthogonal if ϕ n , φm = δ mn .If both {ϕ n } and { φn } are the frames of some space V , and for any f ∈ V , the following equalities hold: f = k f ,ϕ k φk = k f , φk ϕ k , then {ϕ n } and { φn } are mutually called dual frames.

Reproducing kernel and basic properties in
, and the time-warped space V 2 γ (ϕ) is a Hilbert space with reproducing kernel defined by k γ (w, x) = k(γ(w), γ(x)).
In fact, for any The above-mentioned inner product •, • γ exists in the space V 2 γ (ϕ).For some time-warping function γ, the following example gives an explanation of the above definition of the inner product.
Example 3.1.Let γ(w) = Aw + B, where the inner product •, • is defined as the common definition in L 2 (R n ), where A, B are n × n and n × 1 constant matrices, respectively.If we define the inner product by f γ ,g γ γ ≡ |A| f γ ,g γ , we have f γ ,g γ γ = f ,g through some simple computations.
If X is a set of sampling for V 2 (ϕ), then we can obtain that X = {γ −1 (x) : x ∈ X} is a sampling space for V 2 γ (ϕ) for the warped function γ.More exactly, we have the following proposition.
Proof.By the definition of the set of sampling, we know that Since γ −1 is differentiable and |dγ −1 /dw| is uniformly bounded, it is easy to check that Combining (3.2) with (3.1), we only need to prove that (3.4) Remark 3.3.Similar to the proof of Proposition 3.2, if X is a set of sampling for V 2 γ (ϕ), then there exists x such that y = γ(x) for every y ∈ X, and X = {x} is a set of sampling for the space V 2 (ϕ) for the time-warping function γ.Proposition 3.4.If {k x } and { kx } are dual frames for V 2 (ϕ), X is a set of sampling for V 2 (ϕ), and γ is a time-warped function such that γ −1 is differentiable and |dγ −1 /dw| is uniformly bounded, then {k y,γ } and { ky,γ } are dual frames for V 2 γ (ϕ), where k y,γ = k γ (•,y), ky,γ = kγ (•,y), and X is the same as that in Proposition 3.2.
Proof.From the definition of dual frames, we have (3.5)By Proposition 3.2, we know that X is a set of sampling for the time-warped space V 2 γ (ϕ).And we have The second equality of the above equalities depends on the definition of the dual frames.We know the third equality from the definitions of X and •, • γ .
Remark 3.5.Similar to Proposition 3.4, it is easy to see that if {k y,γ } and { ky,γ } are dual frames for V 2 γ (ϕ), and X is a set of sampling space for the timewarped space V 2 r (ϕ), then there exists x such that y = γ(x) for any y ∈ X.So, X ≡ {x} is a set of sampling, and {k x } and { kx } are dual frames for V 2 (ϕ) for the proper warping function γ.

Reconstruction in time-warped weighted shift-invariant spaces. In [5],
Gröchenig obtained the following theorem.Theorem 4.1.Assume that the generator ϕ satisfies the assumption (i), (ii), (iii), m is s-moderate, and X is a set of sampling for V 2 (ϕ) with dual frames The following theorem gives the version of the time-warped space V p m,γ (ϕ).Theorem 4.2.Suppose that the warped function γ is the same as that in Proposition 3.2 or Proposition 3.4, the generator ϕ satisfies assumption (i), (ii), (iii), m is s-moderate, and X is a set of sampling for V 2 γ (ϕ) with dual frames { ky,γ }, then for every Proof.From Propositions 3.2 and 3.4 and Remarks 3.3 and 3.5, we know that X = {x : x = γ(y), y ∈ X} is a set of sampling for V 2 (ϕ) with dual frames { kx }.
By Theorem 4.1, for any It is well known that V p (ϕ) is 1/2-band-limited signal space for p = 2 and ϕ = sin(π x)/π x.So the following example, which is the main result in [4], can be regarded as a special case or corollary of Theorem 4.2 Example 4.3.If γ is a time-warped function, and T ,Ω > 0 with 0 < 2T Ω ≤ 1, then for every g ∈ B γ , holds in L 2 (R), where τ n := γ −1 (nT ) and B γ are time-warped spaces of Ωband-limited signal spaces

Reconstruction in time-warped spline spaces.
It is well known that the spline V N is a special case of weighted shift-invariant spaces.In [7], the following regular sampling theorem in a spline space is shown. where and s is given by ( As a special case of Theorem 4.1, or using the same method, we can obtain the reconstruction formula in a time-warped spline space.

Theorem 5.2. For any
and {t n } are the same as that in Theorem 5.1, and γ is a time-warping function.
It is obvious that {t k } is a regular sampling, but for the proper warped function γ, X and V N,γ are irregular sampling and non-band-limited signal space, respectively.
In the spline space V N , we can construct the reproducing kernel k x by where In fact, from [8], we know that We will give another proof for Theorem 5.2 below.First, we show the following theorem that already has been given in [9].Theorem 5.3.Suppose S is a reproducing kernel Hilbert space (RKHS) with inner product •, • , then a sampling basis {ϕ n } of S yields a reconstruction formula of the form from a sampling set {t n } if and only if its biorthogonal basis { φn } is given by φn (x) = k(t n ,x).
Using Theorem 5.3, we can prove Theorem 5.2.
In fact, by the discussion of Section 2, we know that V N,γ is an RKHS with the reproducing kernel k x,γ given in the above part of this section and the inner product •, • γ defined in Section 2. So, we only need to prove the biorthogonal relation between s n,γ and k γ (τ n ,x).
It is easy to know that

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning shift-invariant space and time-warped shift-invariant space.If ϕ satisfies the following conditions: (i) {ϕ(•−k),k ∈ Z d } form a Riesz basis for the Hilbert space V 2 (ϕ), (ii) ϕ is continuous, (iii) ϕ satisfies the decay condition |ϕ(x)| ≤ C(1 + |x|) −d−s− for some > 0 and s ≥ 0 such that the weight m is s-moderate, then for p = 2 and m = 1, there exists reproducing kernel k .6) At the same time, from Theorem 5.1 and the necessary conditions of Theorem 5.3, we can imply the biorthogonal relation between {s n (•)} and {k(t n , •)}.Sok γ τ n , • ,s m,γ (•) γ = k t n , • ,s m (•) = δ mn , (5.7)that is, {k γ (τ n , •)} is a biorthogonal basis of {s n,γ }.Then by the sufficient condition of Theorem 5.3, we can obtain Theorem 5.2.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation