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.

Here, we consider the convergence acceleration of the KLinterpolation by the application of rational (by   ) correction functions along the ideas of the rational approximations (see [11][12][13] with references therein).The approach discussed here leads to the parametric (depending on parameters  1 , . . .,   ) rational-trigonometric-polynomial (rtp-) interpolation   , (, ).The idea of the convergence acceleration via sequential application of polynomial and rational corrections was described in [14][15][16][17].The KL-interpolation is a special case of the rtp-interpolation corresponding to the choice of parameters   = 0,  = 1, . . ., .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 ( = ±1), and discuss the problem of parameters determination in rational corrections.

Rational Interpolations
In this section, we introduce a rational interpolation as a method of the convergence acceleration of the classical trigonometric interpolation.Here, we recap some details from [15,16].
By   (, ), we denote the error of the classical trigonometric interpolation Advances in Numerical Analysis 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  = { 1 , . . .,   }.The first formula of summation by parts is easy to verify straightforwardly as If here | 1 | < 1, then the formula is valid for all || ≤ 1.This is the main reason of including the parameters   here and further.If  1 is chosen appropriately (see (32)), then the second term in the right-hand side of (6) converges faster (however, for || < 1) 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  1 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   =   − f .Substituting (8), (9), and (10) into (4), after some simplifications, provides with the following expansion of the error where   =   − f .Here, we also took into account the periodicity of the coefficients f 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.

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 [1].
Let  ∈  −1 [−1, 1].By   (), we denote the jumps of  at the end points of the interval   () =  () (1) −  () (−1) ,  = 0, . . .,  − 1. ( 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 ( ∈  −1 ()) 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.We need the next results for further comparisons.Theorems 1 and 2 show the behavior of the KL-interpolation in the regions away from the singularities ( = ±1). Denote Theorem 1 (see [1]).Let  ≥ 2 be even,  ∈  +1 [−1, 1], and where The aim of this paper is the derivation of analogs of Theorems 1 and 2 for the rtp-interpolations.
The determination of parameters   is crucial for the realization of the rtp-interpolation.General method leads to the Fourier-Padé interpolation (see [16]).
Here, we consider a smooth function  on [−1, 1] and take where the new parameters   are independent of .They can be determined differently.One approach leads to the  2minimal interpolation [14,18].This idea was introduced and investigated in [14] for the Fourier-Padé approximations.The first step towards  2 -minimal interpolation was performed in [18].The idea of this interpolation was in the determination of unknown parameters   from the condition lim Paper [18] showed the solution of that problem for  = 1 and 1 ≤  ≤ 6.
Another approach for the determination of parameters   was described in [17], where   were the roots of the associated Laguerre polynomial    ().Below, in numerical experiments, we use this approach.
The following theorem, proved in [15], shows the convergence rate of the rtp-interpolation in the regions away from the singularities  = ±1 for parameters   chosen as in (32).
Throughout the paper, it is supposed that the exact values of the jumps   () are known and that interpolated function is smooth on [−1, 1].

Pointwise Convergence of the RTP-Interpolation
Let   be defined as in (32) and by   () denote the coefficients of the polynomial where The application of transformation ( 14) to (17) with where, by    (  ), we denoted    (,   ) with   = 1,  = 1, . . .,  and First, we estimate the last term in the right-hand side of (40).We need to estimate  1  (   (,   )) 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 ( −−2−1 ).

Advances in Numerical Analysis
Now, we estimate the third term in the right-hand side of (40).We need to estimate  1  (   (,   − F  )) for || ≤  as  → ∞.Observing that and, applying (42), we obtain where, by   , we denoted and took into account that   = (1) 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 ( −−2−1 ) as  → ∞.Finally, we get and we need to estimate   ± (, F  ).Similarly, as above, In view of Lemmas A.
Substituting this into (52), we get the required estimate as   → 1 and lim We prove similar result for odd values of .(57) 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 where, by    (  ), we denoted    (,   ) with   = 1,  = 1, . . ., , and We have According to Lemma A.1, and the last term in the right-hand side of (58) is ( −−2−2 ) as  → ∞.Similarly, and, according to Lemma A.2, Hence, the fifth term in the right-hand side of (58) is ( −−2−2 ) as  → ∞.Then, According to Lemma A.4, when  = 1 in (64) and  is odd, we get and hence the third and fourth terms in the right hand side of (58) are ( −−2−2 ) as  → ∞.Now, from (58), we derive Taking  = 0 in (64), we write Now, the application of Lemmas A.2 and A.3 completes the proof.

Results and Discussion
First, let us compare Theorems 4 and 5. Theorem 4 states that on the interval || < 1, the rate of convergence of   , (, ) is ( −−2−1 ) for even values of .According to Theorem 5, the rate of convergence of   , (, ) for odd values of  is ( −−2−2 ), 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  +1 ().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 rtpinterpolations.In this section, we suppose that parameters not always the utilization of all available jumps, by the KLinterpolation, 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  +1 () 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  +1 ()) 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.