The paper aims at establishing a fast numerical algorithm for Bk(f), where f is any function in the Hardy space H2(Td) and k is the scale level. Here, Bk(f) is an approximation to f we recently constructed by applying the multiscale transform to the Cauchy kernel. We establish the matrix expression of Bk(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.

1. 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 [1–5] for just a few references. For any function f∈L2(Td) with T=[0,2π), it can be reconstructed by its Fourier coefficients as follows: (1)f=∑k∈Zdf^keik·.Here, L2(Td) is the space of 2πZd-periodic and square integrable functions, equipped with the inner product (2)f,g=12πd∫Tdfeitgeit¯dt,∀f,g∈L2Td,where t≔(t0,…,td-1)∈Td, eit≔(eit0,…,eitd-1), and dt≔dt0⋯dtd-1. Truncating the series in (1) leads to the classical Fourier sampling approximation. Specifically, for sufficiently large N, f can be approximated by its Fourier coefficients {f^(k):kl2≤N} as follows:(3)f≈∑kl2≤Nf^keik·,where ·l2 is the Euclid norm of a vector in Rd. 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 {eik·:k∈Zd}. Note that each linear Fourier atom eik·,k∈Zd has the linear phase and constant frequency. Therefore, eik· is time-stable. Then, the characterization from (3) implies that, whether f is time-stable or not, its features are characterized by those of the linear Fourier atoms. However, when f is not time-stable, the effectiveness of characterization by the linear atoms in (3) is inferior to that by nonlinear Fourier atoms [6–8].

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-z), where z∈D≔{z∈C,|z|<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 H2(Td), where H2(Td), a subspace of L2(Td), is defined by (4)H2Td≔f∈L2Td:f^n=0,∀n∈Zd∖N0dwith N0 being the set of nonnegative integers. Then, using the partial Hilbert transform, [11] generalized the approximation to L2(Td). 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.

2. Fast Algorithm for Multiscale Analytic Sampling Approximation2.1. Multiscale Analytic Sampling Approximation from Cauchy Kernel

Before introducing the approximation result in [11], we give the definition of the multiscale transform on H2(Td). For any k∈N0, define the k-scale transform Tkm:H2(Td)→H2(Td) associated with the shift parameter m∈Zd by (5)Tkmf=fei·-2π2-km,∀f∈H2Td.As for the parameter m, we just need to focus on the case of m∈Lk due to the periodicity of f, where (6)Lk≔m0,…,md-1:mν∈0,1,…,2k-1,ν∈0,1,…,d-1.Suppose that {ϕk}k=0∞ is a sequence of 2Id-nonstationary refinement functions in H2(Td); namely, they satisfy(7)ϕk=∑m∈LkPkmTkmϕ0,∀k∈N0,where {Pk(m)}m∈Lk belongs to S(2k), the space of 2kZd-periodic complex-valued sequences. From the perspective of the Fourier coefficient, it is easy to check that (7) is equivalent to(8)ϕk^n=Pk^nϕ0^n,∀n∈N0d,where Pk^(n)≔∑m∈LkPk(m)e-i2π2-kn·m. Related to ϕk, the so-called k-scale projection operator Bk is defined by(9)Bkf=∑m∈Lkf,TkmϕkTkmϕk,∀f∈H2Td.Defining ϕ0(eit) to be ∏ν=0d-11/1-a¯νeitν, the tensor product of the Cauchy kernel on the unit disc D, Li and Qian [11] gave the expression of Bk(f) using the analytic samples of f and estimated the error f-Bk(f)2 concretely.

Lemma 1 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Let(10)ϕ0eit≔∏ν=0d-111-a¯νeitν,where eit=(eit0,…,eitd-1), t=(t0,…,td-1)∈Td, and (a0,…,ad-1)∈Dd. Construct ϕk by(11)ϕk^m=Pk^mϕ0^m,∀m∈N0d,k≥1,where {Pk^(m)}m∈N0d belongs to S(2k) such that(12)ϕk^m=2-k/2.Then, for any f∈H2(Td), Bk(f) defined in (9) can be expressed by the k-scale analytic samples {f(a0ei2π2-km0′′+m0,…,ad-1ei2π2-k(md-1′′+md-1))} as follows:(13)Bkf=∑m∈Lk∑m′∈Lk∑m′′∈LkPkm′P¯km′′fa0ei2π2-km0′′+m0,…,ad-1ei2π2-kmd-1′′+md-1ϕ0ei·-2π2-km+m′,where {Pk(m)}m∈Lk is the inverse Fourier transform of {Pk^(m)}m∈Lk. Moreover,(14)f-Bkf22≤2∑m∈Lk∑q∈N0d∖0f^m+2kq2+9f22+4f^02∑ν=0d-1aν2k2∏ν=0d-11-aν2k2,where 0≔(0,…,0).

2.2. Mathematical Materials on Multilevel Circulant Matrix

Following [12–14], we will define a multilevel circulant matrix. We begin with some denotations. For any n∈N and n≔(n0,…,nd-1)∈Nd, let Nn≔{0,1,…,n-1} and Nn≔Nn0×⋯×Nnd-1, where × is the Cartesian product. An n×n matrix is referred to as a 1-level matrix of order n. Recursively, matrix Cn is a d-level matrix of order n if it consists of n02 matrices of (d-1)-level of order (n1,…,nd-1). To point to the entries of Cn, denote Cn by [ci,j]i,j∈Nn. A d-level matrix Cn=[ci,j]i,j∈Nn of order n is referred to as a d-level circulant matrix if(15)ci,j=ci0-j0modn0,…,id-1-jd-1modnd-1,∀i,j∈Nn,where i≔(i0,…,id-1) and j≔(j0,…,jd-1).

An n×1 column (or 1×n row) vector is referred to as a 1-level vector of order n. Following the definition of a multilevel matrix, vector Vn is a d-level vector of order n if it consists of n0 vectors of (d-1)-level of order (n1,…,nd-1). To point to the entries of Vn, denote Vn by (Vn(i))i∈Nn.

A multilevel circulant matrix has a nice structure as shown in the following lemma.

Lemma 2 (see [<xref ref-type="bibr" rid="B12">12</xref>–<xref ref-type="bibr" rid="B14">14</xref>]).

Suppose that Cn is a d-level matrix of order n and cn is the transpose of the first row of Cn, where n=(n0,…,nd-1). Then, Cn is a d-level circulant matrix if and only if(16)Cn=1ΠnΦn∗diagΦn∗cnΦn,where Πn≔∏ν=0d-1nν, Φn≔Fn0⊗⋯⊗Fnd-1, and Φn∗ is the transpose conjugate of Φn. Here, Fnν is the Fourier matrix of order nν and ⊗ is the Kronecker product of matrices.

Following (15), a d-level matrix Hn≔[hi,j]i,j∈Nn of order n is referred to as a d-level Hankel matrix if (17)hi,j=hi0+j0,…,id-1+jd-1,∀i,j∈Nn.Since it can be converted to a d-level circulant matrix by a transform to be given in Lemma 3, the Hankel matrix with the additional property(18)hi,j=hi0+j0modn0,…,id-1+jd-1modnd-1is crucial for establishing the fast algorithm in Section 2.3.

Lemma 3.

Define a d-level matrix of order n by (19)Dn≔En0⊗En1⊗⋯⊗End-1,where (20)Env≔0⋯0010⋯0100⋯100⋮⋰⋮⋮⋮1⋯000nv×nv.If Hn is a d-level Hankel matrix of order n satisfying (18), then(21)H~n≔HnDn=h~i,ji,j∈Nnis a d-level circulant matrix.

Proof.

From (21) and (18), we arrive at(22)h~i,j=hi,n-1-j=hi0-1-j0modn0,…,id-1-1-jd-1modnd-1,where 1≔(1,…,1). By (22), if i′≔(i0′,…,id-1′) and j′≔(j0′,…,jd-1′) satisfy(23)i0-1-j0modn0=i0′-1-j0′modn0,…,id-1-1-jd-1modn0=id-1′-1-jd-1′modnd-1,then(24)h~i,j=hi,n-1-j=hi′,n-1-j′=h~i′,j′.Recall that (23) is equivalent to(25)i0-j0modn0=i0′-j0′modn0,…,id-1-jd-1modn0=id-1′-jd-1′modnd-1,which together with (24) leads to (26)h~i,j=h~i0-j0modn0,…,id-1-jd-1modnd-1.The proof is concluded.

Note 1.

It is clear that Dn-1=Dn. Then, it follows from (21) that a d-level Hankel matrix Hn satisfying (18) can be written as(27)Hn=1ΠnΦn∗diagΦn∗h~nΦnDn,where h~n is the transpose of the first row of H~n given in (21).

2.3. Fast Multiscale Analytic Sampling Approximation

This subsection aims at developing a fast algorithm to compute the numerical values of Bk(f) at eitm,x, where(28)tm,x≔x+2π2-km=x0+2π2-km0,…,xd-1+2π2-kmd-1,x≔(x0,…,xd-1) is any fixed point on Rd, and m=(m0,…,md-1)∈Lk. For convenient narration, define a column vector ak,x by(29)ak,x≔Bkfeitm,xm∈Lkand a row vector 2k by 2k≔(2k,…,2k).

Lemma 4.

Using the matrix notation, Bk(f) in (13) can be expressed by(30)Bkf=B2kP2kTA2kP¯2k,where A2k is a matrix defined by (31)A2k≔fa0ei2π2-km0+m0′,…,ad-1ei2π2-kmd-1+md-1′m,m′∈Lk,P2k≔(Pk,m′)m′∈Lk is a column vector, and B2k is a matrix of functions given by(32)B2k≔ϕ0ei·-2π2-km+m′m,m′∈Lk.Moreover, A2k and B2k are both d-level Hankel matrices with property (18).

Proof.

For any m∈Lk, it follows from (7) that(33)Tkmϕk=Tkm∑m′∈LkPk,m′Tkm′ϕ0=∑m′∈LkPk,m′Tkm+m′ϕ0=∑m′∈LkPk,m′ϕ0ei·-2π2-km+m′.Then, the column vector (Tkmϕk)m∈Lk can be expressed by(34)Tkmϕkm∈Lk=B2kP2k.By (33) and the Cauchy integral formula, for any m∈Lk, we have (35)f,Tkmϕk=f,∑m′∈LkPk,m′ϕ0ei·-2π2-km+m′=12πd∫Tdfeit∑m′∈LkPk,m′ϕ0eit-2π2-km+m′¯dt0⋯dtd-1=∑m′∈LkPk,m′¯12πid∫Tdfeit∏ν=0d-11eitν-aνei2π2-kmν+mν′deit0⋯deitd-1=∑m′∈LkPk,m′¯12πid-1∫Td-1fa0ei2π2-km0+m0′,eit1,…,eitd-1∏ν=1d-11eitν-aνei2π2-kmν+mν′deit1⋯deitd-1⋮=∑m′∈LkPk,m′¯fa0ei2π2-km0+m0′,…,ad-1ei2π2-kmd-1+md-1′.Therefore, the column vector (〈f,Tkmϕk〉)m∈Lk can be rewritten as(36)f,Tkmϕkm∈Lk=A2kP¯2k.Now, the proof of (30) is concluded by (9), (34), and (36).

For any m=(m0,m1,…,md-1) and m′=(m0′,m1′,…,md-1′)∈Lk, it is straightforward to check that (37)fa0ei2π2-km0+m0′,…,ad-1ei2π2-kmd-1+md-1′=fa0ei2π2-km0+m0′mod2k,…,ad-1ei2π2-kmd-1+md-1′mod2k,and then A2k is a d-level Hankel matrix satisfying (18). Similarly, B2k is also a d-level Hankel matrix with property (18).

The following lemma is crucial for investigating the matrix expression of ak,x.

Lemma 5.

For any k∈N0, define a matrix of functions(38)K2k,a≔11-a¯ei·-2π2-kξ+ηξ,η∈N2k,where a∈D. Then, for any x∈R and m∈N2k, the value of K2k,a at x+2π2-km satisfies(39)K2k,ax+2π2-km=K2k,axS2km,where (40)S2k≔O2k-1,1I2k-11O1,2k-1.

Proof.

We first prove by induction on m that(41)S2km=O2k-m,mI2k-mImOm,2k-m,m∈N2k.Suppose (41) holds with m being replaced by m-1. Then, (42)S2km=S2kS2km-1=O2k-1,1I2k-11O1,2k-1O2k-m+1,m-1I2k-m+1Im-1Om-1,2k-m+1=O2k-1,1I2k-11O1,2k-1O1,m-11O1,2k-mO2k-m,m-1O2k-m,1I2k-mIm-1Om-1,1Om-1,2k-m=O2k-m,m-1O2k-m,1I2k-mIm-1Om-1,1Om-1,2k-mO1,m-11O1,2k-m=O2k-m,mI2k-mImOm,2k-m.Therefore, (41) holds for any m∈N2k.

For any ξ,η∈N2k, we derive from (41) that (43)K2k,axS2kmξ,η=K2k,axξ,2k-m+η,0≤η≤m-1;K2k,axξ,η-m,m-1<η≤2k-1;=11-a¯eix+2π2-km-2π2-kξ+η=K2k,ax+2π2-kmξ,η.The proof of (39) is concluded.

Using (39), we will give the matrix expression of ak,x defined in (29).

Theorem 6.

Let {A2k,B2k,P2k} and S2k be as in Lemmas 4 and 5, respectively. Define a d-level matrix G2k≔(gm′,m)m′,m∈Lk by(44)gm′,m=S2km0⊗⋯⊗S2kmd-1P2km′,where m=(m0,…,md-1). Then, ak,x in (29) can be expressed by(45)ak,x=G2kB2kt0,xA2kP¯2k.

Proof.

Using (41) to directly compute (44) gives us that (46)gm′,m=P2km0+m0′mod2k,…,md-1+md-1′mod2k,from which we arrive at (47)gm′,m=gm0+m0′mod2k,…,md-1+md-1′mod2k,where m′=(m0′,…,md-1′). Hence, G2k is a d-level Hankel matrix with additional property (18).

It follows from (10), (32), (38), and (39) that the value of B2k at tm,x defined in (28) can be computed by(48)B2ktm,x=K2k,a0x0+2π2-km0⊗⋯⊗K2k,ad-1x0+2π2-kmd-1=K2k,a0x0S2km0⊗⋯⊗K2k,ad-1x0S2kmd-1=K2k,a0x0⊗⋯⊗K2k,ad-1x0S2km0⊗⋯⊗S2kmd-1=B2kt0,xS2km0⊗⋯⊗S2kmd-1.It is deduced from (30) and (48) that(49)Bkftm,x=B2ktm,xP2kTA2kP¯2k=B2kt0,xS2km0⊗⋯⊗S2kmd-1P2kTA2kP¯2k=S2km0⊗⋯⊗S2kmd-1P2kTB2kTt0,xA2kP¯2k.By (49), the column vector ak,x defined in (29) can be expressed by (50)ak,x=G2kTB2kTt0,xA2kP¯2k.Since both B2k(t0,x) and G2k are d-level Hankel matrices, they are both symmetric. Hence, the proof of (45) is concluded.

Note 2.

Since G2k, B2k(t0,x), and A2k are all d-level Hankel matrices with property (18), by (27) and (45), ak,x can be factorized as(51)ak,x=1Π2kΦ2k∗diagΦ2k∗g~2kΦ2kD2k1Π2kΦ2k∗diagΦ2k∗b~2kΦ2kD2k1Π2kΦ2k∗diagΦ2k∗a~2kΦ2kD2kP¯2k,where g~2k, b~2k, and a~2k are the transposes of the first rows of G~2k≔G2kD2k, B~2k(t0,x)≔B2k(t0,x)D2k, and A~2k≔A2kD2k, respectively.

Based on Note 2, a fast algorithm for ak,x will be established as follows.

Algorithm 7.

Let ϕ0, {Pk,m}m∈Lk, and {P^k,m}m∈Lk be as in Lemma 1. According to (51), ak,x defined in (29) can be computed by the following six steps.

By ϕ0^(m)=∏ν=0d-1a¯νmν, (11), and (12), compute {P^k,m}m∈Lk, where m=(m0,…,md-1).

By implementing IFFT on {P^k,m}m∈Lk, we compute P2k.

Since Φ2k=F2k⊗⋯⊗F2k and F2k is the Fourier matrix of order 2k, Φ2k∗g~2k, Φ2k∗a~2k, and Φ2k∗b~2k are computed by IFFTs.

Computational Complexity. It is easy to check that directly computing ak,x through (13) costs O(23dk) operations. In Algorithm 7, however, FFT and IFFT are used for three and seven times, respectively, which cost O(2kd·dk) operations. Meanwhile, the complexity of other operations such as the multiplication of D2k and P¯2k is O(2kd). Therefore, the computational complexity of Algorithm 7 is O(2kd·dk).

Note 3.

So far, on computing ak,x, we have not found any numerical algorithm better than Algorithm 7. On the other hand, we notice that Algorithm 7 holds for equally spaced points on Td. In another occasion, we will establish the fast algorithm for the case of nonequally spaced points.

3. Experiment

To check the efficiency of Algorithm 7, a numerical computation experiment on the function(55)feit=eit11+0.3eit01.5+0.4eit0+2t1,t=t0,t1∈T2,will be carried out. Table 1 shows the approximation error ratios (56)ratio1≔bk,0-ak,0l2bk,0l2,ratio2≔Rebk,0-ak,0l2Rebk,0l2,ratio3≔Imbk,0-ak,0l2Imbk,0l2,and the times T1 and T2 corresponding to different choices of k, a0, and a1, where bk,0=(f(eitm,0))m∈Lk and T1 and T2 are the compute running times cost by directly computing ak,0 and Algorithm 7, respectively. The data on the time cost confirms that Algorithm 7 is faster than direct computing. Therefore, the result in this experiment matches the computational complexity analysis in Section 2. On the other hand, as for the approximation accuracy, as the scale level k increases, the approximation efficiency becomes better. This coincides with the approximation error estimation given in (14).

The error ratios (i.e., ratio 1, ratio 2, and ratio 3) corresponding to different k, a0, and a1. T1 and T2 are the time cost by direct computing and Algorithm 7, respectively.

k

a0

a1

ratio 1

ratio 2

ratio 3

T1

T2

2

0.01

0.32

0.0704

0.0733

0.0676

0.107 s

0.086 s

3

0.08

0.16

0.0051

0.0051

0.0051

0.214 s

0.103 s

4

0.44

0.41

2.5618×10-5

2.5652×10-5

2.5584×10-5

4.052 s

0.116 s

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The work is supported by National Natural Science Foundation of China (Grant nos. 61561006, 11501132, and 11461002), Guangxi Natural Science Foundation (Grant no. 2013GXNSFBA019010), and Natural Science Foundation of Guangdong Province (Grant no. 2015A030313443).

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