# Approximate Hermite Interpolations for Compactly Supported Linear Canonical Transforms

Authors: I. A. Al-Abdi
Journal: Computational and Mathematical Methods
 (2022)

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## Abstract

There has been several Lagrange and Hermite type interpolations of entire functions whose linear canonical transforms have compact supports inℝ. There interpolation representations are called sampling theorems for band-limited signals in signal analysis. The truncation, amplitude, and jitter errors associated with the Lagrange type interpolations are investigated rigorously. Nevertheless, the amplitude and jitter errors arising from perturbing samples and nodes are not studied before. The aim of this work is to establish rigorous analysis of their types of perturbation errors, which is important from both practical and theoretical points of view. We derive precise estimates for both types of errors and expose various numerical examples.


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## Body

## 1. Introduction

In recent times, the interpolation representations in the linear canonical transform (LCT) domain have become one of the important areas in different theoretical and practical disciplines. For instance, it has an important role in signal and image processing [1–4], optics [5–8], filter design [9, 10], radar system analysis [11, 12], and many others (see, e.g., [13, 14]). The LCT of a function fz is defined as follows [15–17]:
(1)LfMx≔LfMfzx=1i2πb∫−∞∞fzeidz2+ax2−2zx/2bdz,b≠0,deicdz2/2fdx,b=0,where a,b,c,d are real numbers satisfying ad−cb=1. The case b=0, LfM is merely a chirp multiplication, and it is excluded. Moreover, we assume that b>0. The importance of the LCT arises from the fact that offers a time-frequency analysis of signals and that it generalizes other important othogonal transformations (see [13] for more details).Letσ>0 be fixed. The space of L2ℝ-functions with compact support in the LCT domain is
(2)Bσ2≔f∈L2ℝ:LfMx=0,x>σ.It is also called the space of band-limited signals in the LCT domain. IfPWσ2 denotes the Paley-Wiener space of L2ℝ-functions with compact support −σ,σ in the Fourier transform domain, then f∈Bσ2 iff there exists g∈PWσ/b2 such that fz=e−ia/2bz2gz. It is known in this case that gz is entire of exponential type σ/b, [18]. Moreover, both Bσ2 and PWσ2 are reproducing kernel Hilbert space, see [18].The derivation of Lagrange-type and Hermite-type interpolations for elements ofBσ2 attracts the work of many researchers because of its importance in theoretical and applied problems. If f∈Bσ2, then f has the Lagrange-type sampling representaion:
(3)fz≔∑n=−∞∞e−ia/2bz2−zn2fznSnz,z∈ℝ,where zn≔nπb/σ and
(4)Snz≔sincσ/bz−zn=sinσ/bz−znσ/bz−zn,z≠zn,1,z=zn,see, e.g., [15, 19–23]. Series (3) converges absolutely on ℂ and uniformly on compact subsets of ℂ and on ℝ (see [24]). The Hermite sampling theorem (or derivative sampling theorem) associated with the LCT is obtained for f∈Bσ2⊆B2σ2 in [15] (see also [25]) to be
(5)fz≔∑n=−∞∞e−ia/2bz2−zn21+iabznz−znfzn+z−znf′znSn2z,z∈ℂ, zn=nπb/σ, and a,b∈ℝ,b>0 are arbitrary. In [26], the authors established a convergence analysis for (5). In particular it is shown that (5) converges absolutely and uniformly on ℝ and locally uniformly on ℂ. In addition, the truncation error associated with (5) is investigated in both local (pointwise) and global (uniform). For N∈ℤ+, z∈ℝ, the truncated series of (5) is
(6)fNDz≔∑n=−NNe−ia/2bz2−zn21+iabznz−znfzn+z−znf′znSn2z,and the associated truncation error is
(7)TN,f;z≔fz−fNDz=∑n>Ne−ia/2bz2−zn21+iabznz−zn×fzn+z−znf′znSn2z,Forf∈Bσ2 and zkfz∈L2ℝ, for some k∈ℕ, (7) is estimated in [26] via
(8)TN,f;z≤sinσ/bz2N+1kξkEk3×1Nπ−σ/bz3/2+1Nπ+σ/bz3/2+ηk1Nπ−σ/bz+1Nπ+σ/bz,where
(9)ηk≔1σπaN+1k∫N∞zfz2dz+πbξkσbEk+kEk−1,Ek≔∫−∞∞zkfz2dz,ξk≔σ/bk+1/2πk+11−4−k.In this paper, we will study other types of errors associated with (5). This involves the investigation of rigorous estimates for the amplitude and the jitter errors. This is completed in Sections 2–3. Section 4 is devoted to the numerical examples with illustrative figures and numerical comparisons.


## 2. Amplitude Error Estimate

This section involves the analysis of the amplitude error associated with the Hermite sampling series with LCT (5). The amplitude error arises from using alternate samples f~zn,f′~zn instead of the exact ones fzn,f′zn in the sampling series (5). Let εn≔fzn−f~zn, εn′≔f′zn−f′~zn be uniformly bounded by ε, i.e., εn,εn′<ε, for a sufficiently small ε>0. The amplitude error is defined for z∈ℝ in this case to be
(10)Aε,f;z≔∑n=−∞∞e−ia/2bz2−zn21+iabznz−znεn+z−znεn′Sn2z,where the following decay conditions are presumed
(11)εn≤fzn,εn′≤f′zn,n∈ℤ.Forf∈Bσ2, we define f∞≔supz∈Rfz.Theorem 1.
Suppose thatf∈Bσ2 satisfies a decay condition
(12)fz≤Azα+1,z≠0,where A>0 and 1/2<α≤1 are constants, and let (11) holds. Then, for
(13)0<ε≤minπbσ,σπb,1e,we have
(14)Aε,f;z∞≤4e1/4σα+13eσ+b+ρε−10+2aσπb3ε−1/α+1+2πb2σM1ρε−10+2aπbσ3M2εlog1ε,where
(15)M1≔3σπbf0+Aσπbα+1,M2≔Aσπbα+2,ρx≔2σAπρ2x+π2+2,ρx≔γ+logx,and γ is the Euler-Mascheroni constant.Proof.
Letz∈ℝ. From the triangle inequality and using the fact that eiθ=1,θ∈ℝ, we obtain
(16)Aε,f;z=∑n=−∞∞e−ia/2bz2−zn21+iabznz−znεn+z−znεn′Sn2z≤∑n=−∞∞1+iabznz−znεn+z−znεn′Sn2z≤∑n=−∞∞εnSn2z+∑n=−∞∞abznz−znεnSn2z+∑n=−∞∞z−znεn′Sn2z.
From (4) we have
(17)Aε,f;z≤∑n=−∞∞εnSn2z+aσ∑n=−∞∞znεnsinσbz−znSnz+bσ∑n=−∞∞εn′sinσbz−znSnz.
Now letp,q>1 be such that 1/p+1/q=1. Applying Hölder’s inequality and using the fact that Snz≤1 leads to
(18)Aε,f;z≤∑n=−∞∞εnp1/p+aσ∑n=−∞∞znεnp1/p+bσ∑n=−∞∞εn′p1/p∑n=−∞∞Snzq1/q.
Substituting from the inequality (see [18]),
(19)∑n=−∞∞Snzq1/q<pin (18) yields
(20)Aε,f;z≤p∑n=−∞∞εnp1/p+aσ∑n=−∞∞znεnp1/p+bσ∑n=−∞∞εn′p1/p.
Now, we estimate the infinite sums above. Applying Minkowski’s inequality, we obtain forN≥1(21)∑n=−∞∞εnp1/p≤∑n≤Nεnp1/p+∑n>Nεnp1/palso
(22)∑n=−∞∞znεnp1/p≤∑n≤Nznεnp1/p+∑n>Nznεnp1/p.
Forp≥2 such that αp≥2, we get from (11) and (12)
(23)∑n>Nεnp1/p≤∑n>Nfnπbσp1/p≤Aσπbα+1∑n>N1npα+p1/p=Aσπbα+12∑n>N1npα+p1/p≤Aσπbα+12∫N∞1zpα+pdz1/p≤M2πbσ2pα+p−11/p1Nα+1−1/p.
Similarly,(24)∑n>Nznεnp1/p≤∑n>Nnπbσfnπbσp1/p≤M2πbσ22pα−11/p1Nα−1/p.
Moreover,(25)∑n=−NNεnp1/p≤ε2N+11/p,and
(26)∑n=−NNznεnp1/p≤21/pπbσεN1+1/p.
Combining (23) and (25), as well as (24) and (26), we get for N≥1,p≥2(27)∑n=−∞∞εnp1/p≤ε2N+11/p+M2πbσ2pα+p−11/p×1Nα+1−1/p,and
(28)∑n=−∞∞znεnp1/p≤21/pπbσεN1+1/p+M2πbσ22pα−11/p×1Nα−1/p.
Sinceft satisfies Condition (12), then from (11) and Minkowski’s inequality, we obtain for N≥1,p≥2:
(29)∑n>Nεn′p1/p<∑n>Nf′nπbσp1/p<∑n>Nf′nπbσp1/p+∑−n>Nf′nπbσp1/p.
We have as follows (see [27]):
(30)∑±n>Nf′nπbσp1/p<M1ρN2N1−1/p+M2ρ2N2+2Nα+1−1/p.
Hence,(31)∑n>Nεn′p1/p<2M1ρN2N1−1/p+2M2ρ2N2+2Nα+1−1/p.
Furthermore,(32)∑n=−NNεn′p1/p≤ε2N+11/p,
Combining (31) and (32), we get for N≥1,p≥2(33)∑n=−∞∞εn′p1/p≤ε2N+11/p+2M1ρN2N1−1/p+2M2ρ2N2+2Nα+1−1/p,
Substituting from (27), (28), and (33) into (20), we get
(34)Aε,f;z≤pε1+bσ2N+11/p+21/paπbσ2εN1+1/p+2bM1ρN2σN1−1/p+πbσ22pα−11/paM2σNα−1/p+bM2σNα+1−1/p2ρ2N2+π2pα+p−11/p+4.When σ≥πb, we choose N and p to be
(35)N=ε−1/α+1σπbα+1p/α+1p−1,p=4α+1log1ε.
Sinceε≤πb/σ,1/e, then σ/πb≤1/ε and N≥1. By simple calculations, we have
(36)2N+11/p≤3e5/4,N1+1/p≤e1/4ε−1/α+1σπb4,N1/p−1≤e1/4επbσ,N1/p−α+1≤e1/4επbσα+1,N1/p−α≤e1/4επbσ,2pα+p−11/p≤2,2pα−11/p≤2,ρN2≤γ+10log1ε.
Combining (34), (36) and noting that p=4/α+1log1/ε, we obtain (14). If 0<σ<πb, we take N=ε−1/α+1πb/σα+1p/α+1p−1, and by the same manner, we can prove (14).


## 3. Jitter Error Estimate

In this section, we will derive the jitter error estimate associated with (5) which arises when the sampling nodes nπb/σ,n∈ℤ are perturbed from the exact nodes. Let δn,δ′n,n∈ℤ denote the sets of perturbation values. For a sufficiently small δ>0 and δn,δn′≤δ, n∈ℤ, the jitter error Jδ,f;z associated with (5) is defined for z∈ℝ by
(37)Jδ,f;z≔∑n=−∞∞eia/2bzn2−z21+iabznz−zn×fzn−fzn+δn+z−znf′zn−f′zn+δn′Sn2z.In the following, we derive estimate for the jitter error according to the decay Condition (12).Theorem 2.
Letf∈Bσ2 be such that Condition (12) holds. Then for
(38)0<δ≤minπbσ,σπb,1e,we have
(39)Jδ,f;z∞≤4e1/4σα+1C1f′∞+3bef″∞+4πb2M1ρδ−10σ+22aM2πbσ3+2ρδ−10δlog1δ,where
(40)C1≔5σe3/4+2aδ−1/α+1σπb3,
M1,M2,ρx, and ρx are defined in (15).Proof.
Letp,q>1 such that 1/p+1/q=1. Appling the triangle inequality to (37) and using (4) yield
(41)Jδ,f;z≤∑n=−∞∞fzn−fzn+δnSn2z+aσ∑n=−∞∞znfzn−fzn+δn×sinσbz−znSnz+bσ∑n=−∞∞f′zn−f′zn+δn′×sinσbz−znSnz.
From Hölder’s inequality (19) and using the fact that Snz≤1,z∈ℝ, we obtain
(42)Jδ,f;z≤p∑n=−∞∞fzn−fzn+δnp1/p+aσ∑n=−∞∞znfzn−fzn+δnp1/p+bσ∑n=−∞∞f′zn−f′zn+δn′p1/p.
From Annaby and Asharabi [27], we have
(43)∑n=−∞∞fzn−fzn+δnp1/p≤5eδf′∞+22Aδe1/4,and
(44)∑n=−∞∞f′zn−f′zn+δn′p1/p≤δf″∞2N+11/p+4M1ρN2N1−1/p+4M2ρ2N2+2Nα+1−1/p.
It remains to estimate(45)∑n=−∞∞znfzn−fzn+δnp1/p.
For convenience, let(46)hzn≔znfzn−fzn+δnp.
Using the mean value theorem,N≥1 leads to
(47)∑n≤Nhzn1/p=∑n≤Nznf′snδnp1/p≤δf′∞∑n≤Nnπbσp1/p=δπbσf′∞2∑n=1Nnp1/p≤21/pδπbσf′∞N1+1/p,where sn∈nπb/σ,nπb/σ+δn, n≤N. Applying Minkowski’s inequality yields
(48)∑n>Nhzn1/p≤∑n>Nznfznp1/p+∑n>Nznfzn+δnp1/p.
Using (12), and choosing p≥2 such that αp≥2 leads to
(49)∑n>Nznfznp1/p≤Aσπbα∑n>N1npα1/p=Aσπbα2∑n>N1npα1/p≤σπbα2∫N∞1zpαdz1/p=M2πbσ22pα−11/p×1Nα−1/p.
Hence,(50)∑n>Nhzn1/p≤∑n>Nznfznp1/p+∑n>Nznfzn+δnp1/p≤2M2πbσ22pα−11/p1Nα−1/p,where we have used the same calculations as in (49) to estimate the second sum of (48). Substituting from (43)–(47) and (50) into (42) yields
(51)Jδ,f;z≤p5eδf′∞+22Aδe1/4+21/paπbδσ2f′∞N1+1/p+bδσf″∞2N+11/p+2aM2σπbσ22pα−11/p1Nα−1/p+4bM1ρN2σN1−1/p+4bM2ρ2N2+2σNα+1−1/p.
Whenσ≥πb, we choose N and p to be
(52)N=δ−1/α+1σπbα+1p/α+1p−1,p=4α+1log1δ.
Hence, as in (36), we get
(53)2N+11/p≤3e5/4,N1/p−α+1≤e1/4δπbσα+1,N1/p−1≤e1/4δπbσ,N1+1/p≤e1/4δ−1/α+1σπb4,N1/p−α≤e1/4δπbσ,ρN2≤γ+10log1δ,2pα−11/p≤2.
Substituting from (53) into (51) and noting p=4/α+1log1/δ, we obtain (39). When 0<σ<πb, take N=δ−1/α+1πb/σα+1p/α+1p−1, and proceed as in the previous case.


## 4. Numerical Examples

This section contains three examples. The first example shows that the use of the Hermite sampling theorem with LCT (5) in approximation theory may be better compared to the classical sampling theorem with LCT (3). In the other two examples, we give tables illustrating the amplitude and jitter errors for some numerical values where BN,f;z, Aε,f;z, and Jδ,f;z denote to the bound of the truncation error in (8), the bound of the amplitude error in (14), and the bound of jitter error in (39), respectively. For f∈Bσ2, z∈ℝ and N∈ℕ, we set
(54)fNCz≔∑n=−NNe−ia/2bz2−Zn2fznSnz,fz−fε,NDz=fz−∑n≤Neia/2bzn2−z2×1+iabznz−znf~zn+z−znf~′znSnz≤BN,f;z+Aε,f;z,fz−fNDδ,z=fz−∑n≤Neia/2bzn2−z2×1+ia/bznz−znfzn+δn+z−znf′zn+δn′Snz≤BN,f;z+Jδ,f;z.Example 3.
The function(55)fz=e−iz2/2sin2πzπ22z−z3.is a Bπ2⊆B2π2 function where a=b=d=1/2. Table 1 demonstrates the comparison between the reconstruction of f using the classical technique fNCz and the Hermite interpolations fNDz when N=10,20,30. As Table 1 indicates, the absolute errors decrease as N increases for both techniques. Moreover, the Hermite interpolation approximations are superior to the classical sampling representation of f. Figure 1 illustrates comparison between fz,fNCz and fNDz when z≤2,N=3.Example 4.
Consider theB1/22-function
(56)fz=e−iz2/233zcosz/3−3sinz/32π2z3.Table 1
Comparison betweenfNC· and fND· where N=10;20;30 of Example 3.
ziNfzi−fNCzifzi−fNDzi0.25101.16332×10−44.16650×10−5201.56681×10−52.90720×10−6304.75966×10−65.94953×10−70.75103.52826×10−44.28018×10−5204.71388×10−52.92797×10−6301.42974×10−55.96872×10−71.25106.01323×10−44.52465×10−5207.90176×10−52.97029×10−6302.38908×10−56.00743×10−71.75108.71706×10−44.94052×10−5201.11592×10−43.03586×10−6303.35778×10−56.06634×10−72.25101.17736×10−35.60931×10−5201.45176×10−43.12734×10−6304.33978×10−56.14651×10−7Figure 1
Illustrations associated with Example3. (a) The green continuous line is a real part of fz, while the blue and the red dashed lines are real parts of f3Cz and f3Dz, respectively. (b) The green continuous line is an imaginary part of fz, while the blue and the red dashed lines are imaginary parts of f3Cz and f3Dz, respectively.
(a)(b)Here, we takea=d=1/2,b=3/2. Table 2 and Figure 2 show the comparison between absolute error fz−fε,Nz and its bound BN,f;z+Aε,f;z where N=20 and ε=6×10−6,4×10−6,2×10−6, respectively, as well as the effect of ε on the amplitude error. We notice that the error bounds are quite realistic. Moreover, as predicted by the theory, the precision increases when ε decreases.Example 5.
In this example, we consider the function(57)fz=e−iz2/2cosπz/2−1π2z22∈Bπ/22.Table 2
Exact errorfz−fε,20z and its bound B20,f;z+Aε,f;z of Example 4.
ziε=6×10−6ε=4×10−6ε=2×10−6Absolute errorBoundAbsolute errorBoundAbsolute errorBound0.69.10196×10−65.39522×10−36.06823×10−63.89367×10−33.0345×10−62.27246×10−31.21.13933×10−56.00578×10−37.59643×10−64.50424×10−33.79955×10−62.88303×10−31.81.27884×10−56.70498×10−38.52717×10−65.20343×10−34.26597×10−63.58222×10−32.41.29357×10−57.17046×10−38.62586×10−65.66891×10−34.31603×10−64.0477×10−33.01.17621×10−57.18759×10−37.84351×10−65.68605×10−33.92497×10−64.06484×10−3Figure 2
Figures forfz−fε,20z (a), B20,f;z+Aε,f;z (b), where z∈−10,10 and ε=6×10−6,4×10−6,2×10−6, respectively.
(a)(b)Table3 and Figure 3 exhibit the absolute jitter error fz−fNδ;z and its associated bound BN,f;z+Jδ,f;z with a=b=d=1/2, N=20 and, δ=9×10−8,6×10−8,3×10−8, respectively, as well as the effect of δ on the jitter error. It can be seen that as δ decreases, accuracy improves.Table 3
Exact errorfz−f20δ;z and its bound B20,f;z+Jδ,f;z of Example 5.
ziδ=9×10−8δ=6×10−8δ=3×10−8Absolute errorBoundAbsolute errorBoundAbsolute errorBound0.62.87792×10−95.29278×10−41.97282×10−93.95763×10−41.07791×10−92.45933×10−41.21.52818×10−95.18661×10−41.03765×10−93.85145×10−45.47544×10−102.35315×10−41.87.99524×10−105.23896×10−45.79877×10−103.90381×10−43.66721×10−102.40551×10−42.44.75446×10−105.25238×10−43.6769×10−103.91723×10−42.74189×10−102.41893×10−43.08.57322×10−115.17634×10−46.85318×10−113.84118×10−45.40306×10−112.34288×10−5Figure 3
Illustrations associated with Example5. fz−f20δ,z (a) and B20,f;z+Jδ,f;z (b), where z∈−4,4 and δ=9×10−8,6×10−8,3×10−8, respectively.
(a)(b)


## 5. Conclusions

In this paper, we investigated the error analysis of the Hermite sampling theorem associated with the linear canonical transform (LCT). We provided estimates for both amplitude and jitter errors when alternative samples and nodes are implemented, respectively. The study fills a gap in the error analysis associated with Hermite sampling representation for entire functions of exponential type.



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