Fast Analytic Sampling Approximation from Cauchy Kernel

The paper aims at establishing a fast numerical algorithm forB k (f), wheref is any function in the Hardy spaceH(T) and k is the scale level. Here,Bk(f) is an approximation tofwe recently constructed by applying themultiscale transform to the Cauchy kernel. We establish the matrix expression ofBk(f) and find that it has the structure of a multilevel Hankel matrix. Based on the structure, a fast numerical algorithm is established to compute Bk(f). The computational complexity is given. A numerical experiment is carried out to check the efficiency of our algorithm.


Introduction
Approximation to a function (a time-continuous signal) by its samples is the heart of modern applied mathematics and engineering and has attracted much attention.Readers are referred to [1][2][3][4][5] for just a few references.For any function  ∈  2 (T  ) with T = [0, 2), it can be reconstructed by its Fourier coefficients as follows: Here,  2 (T  ) is the space of 2Z  -periodic and square integrable functions, equipped with the inner product where  fl ( 0 , . . .,  −1 ) ∈ T  ,   fl (  0 , . . .,   −1 ), and  fl  0 ⋅ ⋅ ⋅  −1 .Truncating the series in (1) leads to the classical Fourier sampling approximation.Specifically, for sufficiently large ,  can be approximated by its Fourier coefficients { f(k) : ‖k‖ ℓ 2 ≤ } as follows: where ‖ ⋅ ‖ ℓ 2 is the Euclid norm of a vector in R  .From the point of view of feature characterization, (3) can be used to characterize some features such as instantaneous phase and frequency by those of the linear Fourier atoms { k⋅ : k ∈ Z  }.Note that each linear Fourier atom  k⋅ , k ∈ Z  has the linear phase and constant frequency.Therefore,  k⋅ is time-stable.Then, the characterization from (3) implies that, whether  is time-stable or not, its features are characterized by those of the linear Fourier atoms.However, when  is not time-stable, the effectiveness of characterization by the linear atoms in ( 3) is inferior to that by nonlinear Fourier atoms [6][7][8].
Recently, an interesting sampling approximation from Takenaka-Malmquist (T-M) system, a special class of nonlinear Fourier atoms, has attracted much attention in the literature; for example, see [9,10] and the references therein.Note that the T-M system is substantially generated from the Schmidt orthogonalization of Cauchy kernel 1/(1 − ), where  ∈ D fl { ∈ C, || < 1}.In this sense, we say that the sampling approximation from the T-M system is substantially from the Cauchy kernel.Implementing the multiscale transform on the tensor product of the Cauchy kernel, Li and Qian [11] constructed the analytic sampling approximation to any function in the Hardy space  2 (T  ), where  2 (T  ), a subspace of  2 (T  ), is defined by Journal of Function Spaces with N 0 being the set of nonnegative integers.Then, using the partial Hilbert transform, [11] generalized the approximation to  2 (T  ).A subsequent unsolved problem of the approximation in [11] is that the fast numerical algorithm has not been established.The aim of this present paper is to solve the problem by the fast computation theory of multilevel circulant matrices.

Multiscale Analytic Sampling Approximation from Cauchy
Kernel.Before introducing the approximation result in [11], we give the definition of the multiscale transform on  2 (T  ).
For any  ∈ N 0 , define the -scale transform  m  :  2 (T  ) →  2 (T  ) associated with the shift parameter m ∈ Z  by As for the parameter m, we just need to focus on the case of m ∈ L  due to the periodicity of , where Suppose that {  } ∞ =0 is a sequence of 2  -nonstationary refinement functions in  2 (T  ); namely, they satisfy where {  (m)} m∈L  belongs to (2  ), the space of 2  Z periodic complex-valued sequences.From the perspective of the Fourier coefficient, it is easy to check that (7) is equivalent to where P (n) fl ∑ m∈L    (m) −22 − n⋅m .Related to   , the socalled -scale projection operator B  is defined by Defining  0 (  ) to be )), the tensor product of the Cauchy kernel on the unit disc D, Li and Qian [11] gave the expression of B  () using the analytic samples of  and estimated the error ‖ − B  ()‖ 2 concretely.Lemma 1 (see [11]).Let where   = (  0 , . . .,   −1 ),  = ( 0 , . . .,  −1 ) ∈ T  , and where Then, for any  ∈  2 (T  ), B  () defined in ( 9) can be expressed by the -scale analytic samples )} as follows: where where 0 fl (0, . . ., 0).
An  ×  matrix is referred to as a 1-level matrix of order .Recursively, matrix  n is a -level matrix of order n if it consists of  2 0 matrices of ( − 1)-level of order ( 1 , . . .,  −1 ).To point to the entries of  n , denote  n by [ i,j ] i,j∈N n .A -level matrix  n = [ i,j ] i,j∈N n of order n is referred to as a -level circulant matrix if where i fl ( 0 , . . .,  −1 ) and j fl ( 0 , . . .,  −1 ).
An  × 1 column (or 1 ×  row) vector is referred to as a 1level vector of order .Following the definition of a multilevel matrix, vector  n is a -level vector of order n if it consists of  0 vectors of ( − 1)-level of order ( 1 , . . .,  −1 ).To point to the entries of  n , denote  n by ( n (i)) i∈N n .
A multilevel circulant matrix has a nice structure as shown in the following lemma.
Lemma 2 (see [12][13][14]).Suppose that  n is a -level matrix of order n and c n is the transpose of the first row of  n , where n = ( 0 , . . .,  −1 ).Then,  n is a -level circulant matrix if and only if where and Φ * n is the transpose conjugate of Φ n .Here,   ] is the Fourier matrix of order  ] and ⊗ is the Kronecker product of matrices.
Following (15), a -level matrix  n fl [ℎ i,j ] i,j∈N n of order n is referred to as a -level Hankel matrix if Since it can be converted to a -level circulant matrix by a transform to be given in Lemma 3, the Hankel matrix with the additional property is crucial for establishing the fast algorithm in Section 2.3.

Lemma 3.
Define a -level matrix of order n by where If  n is a -level Hankel matrix of order n satisfying (18), then is a -level circulant matrix.
The proof is concluded.
Note 1.It is clear that  −1 n =  n .Then, it follows from (21) that a -level Hankel matrix  n satisfying (18) can be written as where hn is the transpose of the first row of Hn given in (21).

Lemma 4.
Using the matrix notation, B  () in ( 13) can be expressed by where  2 k is a matrix defined by P 2 k fl ( ,m  ) m  ∈L  is a column vector, and  2 k is a matrix of functions given by Moreover,  2 k and  2 k are both -level Hankel matrices with property (18).
Proof.For any m ∈ L  , it follows from (7) that Then, the column vector ( m    ) m∈L  can be expressed by By (33) and the Cauchy integral formula, for any m ∈ L  , we have Therefore, the column vector (⟨,  m    ⟩) m∈L  can be rewritten as Now, the proof of ( 30) is concluded by ( 9), (34), and (36).
The following lemma is crucial for investigating the matrix expression of a ,x .Lemma 5.For any  ∈ N 0 , define a matrix of functions where  ∈ D.Then, for any  ∈ R and  ∈ N 2  , the value of where Proof.We first prove by induction on  that Suppose (41) holds with  being replaced by  − 1.Then, Therefore, (41) holds for any  ∈ N 2  .For any ,  ∈ N 2  , we derive from (41) that The proof of (39) is concluded.
Using (39), we will give the matrix expression of a ,x defined in (29).
It follows from ( 10), ( 32), (38), and (39) that the value of  2 k at  m,x defined in (28) can be computed by It is deduced from (30) and (48) that By (49), the column vector a ,x defined in (29) can be expressed by where g2 k , b2 k , and ã2 k are the transposes of the first rows of Based on Note 2, a fast algorithm for a ,x will be established as follows.
Algorithm 7. Let  0 , { ,m } m∈L  , and { P,m } m∈L  be as in Lemma 1.According to (51), a ,x defined in (29) can be computed by the following six steps.

Theorem 6 .
Let { 2 k ,  2 k , P 2 k } and  2  be as in Lemmas 4 and 5, respectively.Define a -level matrix  2 k fl ( m  ,m ) m  ,m∈L  by Note 2. Since  2 k ,  2 k ( 0,x ), and  2 k are all -level Hankel matrices with property (18), by (27) and (45), a ,x can be factorized as Since both  2 k ( 0,x ) and  2 k are -level Hankel matrices, they are both symmetric.Hence, the proof of (45) is concluded.