On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation
We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice.
In this paper, we continue investigations started in  where we considered the convergence acceleration of the classical trigonometric interpolation via polynomial corrections representing discontinuities of the function and some of its first derivatives (jumps). The resultant interpolation was called as the Krylov-Lanczos (KL-) interpolation. That approach was suggested in 1906 by Krylov  and later in 1964 by Lanczos [3, 4] (see also [1, 5–10] with references therein).
Here, we consider the convergence acceleration of the KL-interpolation by the application of rational (by ) correction functions along the ideas of the rational approximations (see [11–13] with references therein). The approach discussed here leads to the parametric (depending on parameters ) rational-trigonometric-polynomial (rtp-) interpolation . The idea of the convergence acceleration via sequential application of polynomial and rational corrections was described in [14–17]. The KL-interpolation is a special case of the rtp-interpolation corresponding to the choice of parameters , . Besides, rational corrections can be applied immediately to the classical interpolation without polynomial corrections (see interpolation ).
In this paper, we reveal the convergence properties of the rtp-interpolation, show its fast convergence compared to the KL-interpolation in the regions away from the singularities (), and discuss the problem of parameters determination in rational corrections.
2. Rational Interpolations
By , we denote the error of the classical trigonometric interpolation and write where is the th Fourier coefficient of
Rational corrections considered in this paper are based on a series of formulae of summation by parts applied to the error terms in (4). Such transformations lead to new interpolations with correction terms in the form of rational (by ) functions. Consider a vector of complex numbers . The first formula of summation by parts is easy to verify straightforwardly as If here , then the formula is valid for all . This is the main reason of including the parameters here and further. If is chosen appropriately (see (32)), then the second term in the right-hand side of (6) converges faster (however, for ) than the sum in the left-hand side. The next, slightly different formula of summation by parts is also easy to derive as Application of (7) to the second term in the right-hand side of (6) leads to the needed expansion Here, also, as we mentioned above, the sum on the right-hand side converges faster than on the left-hand side if parameter is chosen appropriately.
In a similar manner, we transform the third term in the right-hand side of (4):
For the first term in the right-hand side of (4), the formula of summation by parts is the following: where .
Substituting (8), (9), and (10) into (4), after some simplifications, provides with the following expansion of the error where . Here, we also took into account the periodicity of the coefficients For writing the expansion (11) in a short form and also for further reiterations of this transformation, we introduce the following generalized finite differences determined recurrently: for some sequence . Now, (11) can be rewritten in the form Reiteration of this transformation up to times leads to the following expansion of the error: where the first two terms in the right-hand side can be viewed as rational corrections to the error and the last two terms as the actual error. This observation leads to the following rational-trigonometric interpolation: with the error
The problem of the determination of parameters will be discussed later.
3. The KL-Interpolation
In this section, we consider the additional acceleration of the rational-trigonometric interpolation by the polynomial correction method known as the Krylov-Lanczos approach. We recap the main ideas from .
Let . By , we denote the jumps of at the end points of the interval
The polynomial correction method is based on the following representation of the interpolated function: where are 2-periodic Bernoulli polynomials with the Fourier coefficients Function is a 2-periodic and relatively smooth function on the real line with the discrete Fourier coefficients The approximation of in (19) by the classical trigonometric interpolation leads to the Krylov-Lanczos (KL-) interpolation and the approximation of by the rational-trigonometric interpolation leads to the rational-trigonometric-polynomial (rtp-) interpolation with the errors respectively.
Theorem 1 (see ). Let be even, , and . Then, the following estimate holds for , where
Theorem 2 (see ). Let be odd, , and . Then, the following estimate holds for , where
The determination of parameters is crucial for the realization of the rtp-interpolation. General method leads to the Fourier-Padé interpolation (see ).
Here, we consider a smooth function on and take where the new parameters are independent of . They can be determined differently. One approach leads to the -minimal interpolation [14, 18]. This idea was introduced and investigated in  for the Fourier-Padé approximations. The first step towards -minimal interpolation was performed in . The idea of this interpolation was in the determination of unknown parameters from the condition Paper  showed the solution of that problem for and .
Another approach for the determination of parameters was described in , where were the roots of the associated Laguerre polynomial . Below, in numerical experiments, we use this approach.
The main purpose of this paper is the derivation of the exact estimates of the when getting more accurate ones than Theorem 3 presents. Based on such estimates, we discuss the problem of parameters (, and ) determination for accurate interpolation. Theoretical estimates show the fast convergence of the rtp-interpolation compared to the KL-interpolation. Numerical experiments confirm theoretical estimates.
Throughout the paper, it is supposed that the exact values of the jumps are known and that interpolated function is smooth on .
4. Pointwise Convergence of the RTP-Interpolation
Let be defined as in (32) and by denote the coefficients of the polynomial
Theorem 4. Let be even and with for some . Let parameters be chosen as in (32). Then, the following estimate holds for , where
Proof. Expansion (19) and definition of interpolation show that
The application of transformation (14) to (17) with implies
where, by , we denoted with , and
First, we estimate the last term in the right-hand side of (40). We need to estimate for as . In view of the smoothness of , we get from expansion (19) by means of integration by parts and consequently According to Lemma A.1, and, therefore, Thus, we conclude that the last term in the right-hand side of (40) is .
Now, we estimate the third term in the right-hand side of (40). We need to estimate for as . Observing that and, applying (42), we obtain where, by , we denoted and took into account that as . Now, it follows that According to Lemma A.2, and, therefore, Hence, the third term in the right-hand side of (40) is also as .
Finally, we get and we need to estimate . Similarly, as above, In view of Lemmas A.3 and A.4, Substituting this into (52), we get the required estimate as and
We prove similar result for odd values of .
Theorem 5. Let be odd, , and with for some . Let parameters be chosen as in (32). Then, the following estimate holds for , where
Proof. As the proof of this theorem mimics the proof of the previous one, we omit some details.
The application of transformation (14) to (17) twice with implies that where, by , we denoted with , , and
We have According to Lemma A.1, and the last term in the right-hand side of (58) is as .
Similarly, and, according to Lemma A.2, Hence, the fifth term in the right-hand side of (58) is as .
Then, According to Lemma A.4, when in (64) and is odd, we get and hence the third and fourth terms in the right hand side of (58) are as . Now, from (58), we derive Taking in (64), we write Now, the application of Lemmas A.2 and A.3 completes the proof.
Note that Theorems 4 and 5 are valid also for . In that case, the rtp-interpolation coincides with the KL-interpolation as and consequently . Hence, for , Theorems 4 and 5 coincide with Theorems 1 and 2, respectively.
5. Results and Discussion
First, let us compare Theorems 4 and 5. Theorem 4 states that on the interval , the rate of convergence of is for even values of . According to Theorem 5, the rate of convergence of for odd values of is , and we have an improvement in the convergence by the factor . From the other side, Theorem 5 puts an additional smoothness requirement on the interpolated function. Moreover, while the estimate in Theorem 4 depends only on , the estimate of Theorem 5 depends on both and . All these mimic the behavior of the KL-interpolation where we have the same differences in the asymptotic estimates of Theorems 1 and 2.
Now, let us compare convergence of the KL- and the rtp-interpolations. In this section, we suppose that parameters in (32) are the roots of the associated Laguerre polynomial . Theorems concerning the rtp-interpolations put additional smoothness requirements on the interpolated function, and, in comparisons, it must be taken into account.
If an interpolated function is rather smooth (e.g., when function is infinitely differentiable) such that for given and Theorems 4 and 5 are valid, then the rtp-interpolation is more precise (however, asymptotically) than the KL-interpolation. Comparison of Theorems 1 and 2 with Theorems 4 and 5 shows that the rtp-interpolation is more precise than the KL-interpolation by factor .
Now, we consider functions with finite smoothness and suppose that all needed jumps are exactly available. Estimates proved above show (see factors and ) that the utilization of all available jumps by the KL-interpolation is not reasonable when the jumps of the interpolated function are rapidly increasing. In such cases, more accuracy can be achieved with less jumps in combination with rational corrections.
Further in this section, we suppose that parameters are the roots of the associated Laguerre polynomials .
Let , , , and let be an even number. According to Theorems 1 and 4, if the values of and satisfy the condition , then both theorems are valid, and comparison of corresponding approximations is legal. Then, asymptotic estimates of these theorems will show which values of parameters and provide with better accuracy. We show this process for a specific example. Let where is some parameter. We use the values , and in Table 1 to calculate the values of for and for different values of and with condition . Recall that corresponds to the KL-interpolation.
Table 1 shows that for and utilization of all available jumps is reasonable and the KL-interpolation has the best accuracy. For , and , the rtp-interpolation is the best, and as larger is the value of parameter as more precise is the rtp-interpolation compared to the KL-interpolation.
It is important to note that results in Table 1 are obtained based on asymptotic estimates and we must compare these results to actual errors to see how the asymptotic errors coincide with actual errors. In Table 2, the values of the actual errors for , and for different values of and with condition are calculated. Comparison with Table 1 shows that for theoretical estimates of Theorems 1 and 4 coincide with actual errors rather precisely.
We have the same situation for odd values of . Let , , , and let be an odd number. We use the same values of and in Table 3 to calculate the values of and in Table 4 to calculate the values of for and for different values of and with condition when both Theorems 2 and 5 are valid. Again, for and , the best accuracy has the KL-interpolation. For , and , the best is rtp-interpolation .
Overall conclusion based on these specific examples and on comparison of the asymptotic estimates is the following: not always the utilization of all available jumps, by the KL-interpolation, leads to the best interpolation (if we mean pointwise convergence in the regions away from the singularities where the rtp-interpolation has faster convergence rate.) This is due to the factors and in the estimates. When the values of jumps are rapidly increasing, then better accuracy can be achieved by the utilization of smaller number of jumps (consequently, with smaller and ) and appropriately chosen corrections based on the smoothness of the interpolated function. Which choice of and is the best that can be concluded from the comparison of the corresponding estimates as we did it above.
It must be also mentioned that when the jumps are rapidly increasing then getting their approximations is problematic, so in such cases, the utilization of the rational corrections is unavoidable for better accuracy.
In this section, we prove some lemmas concerning generalized finite differences . By , we denote the sequence that corresponds to , . By , we denote the differences defined by the relations Now, it is easy to verify that
Lemma A.1. The following estimate holds for as and , where are chosen as in (32)
Proof. In view of definitions of , and their connection with , we write
Taking into account that
Taking into account the explicit form of the Fourier coefficients of the Bernoulli polynomials we derive where
Thus, which completes the proof as
The proof of next lemmas can be obtained in a similar manner.
Lemma A.2. The following estimate holds for as and , where are chosen as in (32)
Lemma A.3. Let be even. Then, the following estimate holds for as , where are chosen as in (32)
Proof. Similar to (A.9), we get