Abstract
There has been several Lagrange and Hermite type interpolations of entire functions whose linear canonical transforms have compact supports in . 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.
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 is defined as follows [15–17]: where are real numbers satisfying . The case , is merely a chirp multiplication, and it is excluded. Moreover, we assume that . 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).
Let be fixed. The space of -functions with compact support in the LCT domain is
It is also called the space of band-limited signals in the LCT domain. If denotes the Paley-Wiener space of -functions with compact support in the Fourier transform domain, then iff there exists such that . It is known in this case that is entire of exponential type , [18]. Moreover, both and are reproducing kernel Hilbert space, see [18].
The derivation of Lagrange-type and Hermite-type interpolations for elements of attracts the work of many researchers because of its importance in theoretical and applied problems. If , then has the Lagrange-type sampling representaion: where and see, e.g., [15, 19–23]. Series (3) converges absolutely on and uniformly on compact subsets of and on (see [24]). The Hermite sampling theorem (or derivative sampling theorem) associated with the LCT is obtained for in [15] (see also [25]) to be
, , and are arbitrary. In [26], the authors established a convergence analysis for (5). In particular it is shown that (5) converges absolutely and uniformly on and locally uniformly on . In addition, the truncation error associated with (5) is investigated in both local (pointwise) and global (uniform). For , , the truncated series of (5) is and the associated truncation error is
For and , for some , (7) is estimated in [26] via where
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 instead of the exact ones in the sampling series (5). Let , be uniformly bounded by , i.e., for a sufficiently small . The amplitude error is defined for in this case to be where the following decay conditions are presumed
For , we define .
Theorem 1. Suppose that satisfies a decay condition where and are constants, and let (11) holds. Then, for we have where and is the Euler-Mascheroni constant.
Proof. Let . From the triangle inequality and using the fact that , we obtain From (4) we have Now let be such that . Applying Hölder’s inequality and using the fact that leads to Substituting from the inequality (see [18]), in (18) yields Now, we estimate the infinite sums above. Applying Minkowski’s inequality, we obtain for also For such that , we get from (11) and (12) Similarly, Moreover, and Combining (23) and (25), as well as (24) and (26), we get for and Since satisfies Condition (12), then from (11) and Minkowski’s inequality, we obtain for : We have as follows (see [27]): Hence, Furthermore, Combining (31) and (32), we get for Substituting from (27), (28), and (33) into (20), we get When , we choose and to be Since , then and . By simple calculations, we have Combining (34), (36) and noting that , we obtain (14). If , we take 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 are perturbed from the exact nodes. Let denote the sets of perturbation values. For a sufficiently small and , , the jitter error associated with (5) is defined for by
In the following, we derive estimate for the jitter error according to the decay Condition (12).
Theorem 2. Let be such that Condition (12) holds. Then for we have where and are defined in (15).
Proof. Let such that . Appling the triangle inequality to (37) and using (4) yield From Hölder’s inequality (19) and using the fact that , we obtain From Annaby and Asharabi [27], we have and It remains to estimate For convenience, let Using the mean value theorem, leads to where , . Applying Minkowski’s inequality yields Using (12), and choosing such that leads to Hence, 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 When , we choose and to be Hence, as in (36), we get Substituting from (53) into (51) and noting , we obtain (39). When , take 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 , , and 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 , and , we set
Example 3. The function is a function where . Table 1 demonstrates the comparison between the reconstruction of using the classical technique and the Hermite interpolations when . As Table 1 indicates, the absolute errors decrease as increases for both techniques. Moreover, the Hermite interpolation approximations are superior to the classical sampling representation of . Figure 1 illustrates comparison between and when .
Example 4. Consider the -function
(a)
(b)
Here, we take . Table 2 and Figure 2 show the comparison between absolute error and its bound where and , respectively, as well as the effect of on the amplitude error. We notice that the error bounds are quite realistic. Moreover, as predicted by the theory, the precision increases when decreases.
Example 5. In this example, we consider the function
(a)
(b)
Table 3 and Figure 3 exhibit the absolute jitter error and its associated bound with , and, , respectively, as well as the effect of on the jitter error. It can be seen that as decreases, accuracy improves.
(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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author has no conflict of interest for this submission.