Optimal Rational Approximations by the Modified Fourier Basis

and Applied Analysis 3 with the error RN,p (f, θc, θs, x) = f (x) −MN,p (f, θc, θs, x) = Rc N,p (f, θc, x) + Rs N,p (f, θs, x) . (21) A crucial step for realization of the rational approximations is determination of parameters θc and θs. Different approaches are known for solution of this problem (see [12– 19]). In general, appropriate determination of these parameters leads to rational approximations with improved accuracy compared to the classical ones in case of smooth f. However, the rational approximations are essentially nonlinear in the sense that MN,p (f + g, θc, θs, x) ̸ = MN,p (f, θc, θs, x) + MN,p (g, θc, θs, x) (22) as for each approximation we need to determine its own θc and θs vectors. In [6], those parameters were determined from the following systems of equations: Δpn (θc, fc) = 0, n = N,N − 1, . . . , N − p + 1, Δpn (θs, fs) = 0, n = N,N − 1, . . . , N − p + 1, (23) which led to the Fourier-Pade type approximations [12] with better convergence for smooth functions (see [6]) compared to the expansions by the modified Fourier basis. It is rather complex approach as parameters θc and θs depend onN and systems (23) must be solved for eachN. In this paper, assuming that f is smooth on [−1, 1], we consider simpler alternative approach, where θs and θc are determined as follows [14, 16, 19]: θc k = 1 − τ c k N, θs k = 1 − τ s k N, τc k ̸ = 0, τs k ̸ = 0, k = 1, . . . , p, (24) with τc = {τc 1, . . . , τc p} and τs = {τs 1, . . . , τs p} independent of N. Actually, in this approach, we take into consideration only the first two terms of the asymptotic expansions of θk = θk(N) in terms of 1/N. Although parameters θc and θs in (24) depend onN, we need only to determine τc and τs which are independent ofN. Hence, this approach is less complex than the modified Fourier-Pade approximations. The main results of this paper are exact constants of the main terms of asymptotic errors and optimal parameters for improved pointwise convergence of rational approximations. First, we derive the exact estimates for the main terms of asymptotic errors without specifying parameters τc and τs. Second, we determine the optimal values of parameters τc and τs which vanish the main terms and lead to approximations with substantially better pointwise convergence rates. We found that optimal values of parameters τc k and τs k, k = 1, . . . , p, are the roots of some polynomials depending on p and q, where q indicates the number of zero derivatives in (8). Moreover, the choice of optimal parameters depends on the parity of p and also on the location of x, whether |x| < 1 or x = ±1. For example, when p is odd and |x| < 1, the roots of the generalized Laguerre polynomial L(2q+1) p (x) (see Appendix A) could be used for τc and τs. In this case, the convergence rate of the MTR-approximations is O(N−2q−p−[(p+1)/2]−2) as N → ∞. It means better convergence compared to the expansions by the modified Fourier basis with improvement by factor O(Np+[(p+1)/2]). When p is odd and x = ±1, the roots of the generalized Laguerre polynomial L(2q) p (x) could be used. In this case, the convergence rate is O(N−2q−[(p+1)/2]−1) with the improvement by factor O(N[(p+1)/2]). The problem that we encountered is impossibility to get simultaneous optimality on |x| < 1 and at x = ±1. One must decide whether to use parameters that provide the minimal error on [−1, 1], but with worse accuracy on |x| < 1, or work with optimal parameters on |x| < 1 with worse accuracy at x = ±1. We expose similar observations for even values of p. It is important that the values of parameters τc and τs depend only on p and q and are independent of f. It means that if functionsf, g, andf+g have enough smoothness and obey the same derivative conditions, the optimal approach leads to linear rational approximations in the sense that MN,p (f + g, θc, θs, x) = MN,p (f, θc, θs, x) + MN,p (g, θc, θs, x) (25) with the same parameters θc and θs for all included functions. The paper is organized as follows. Section 2 presents some preliminary lemmas. Section 3 explores the pointwise convergence of the MTR-approximations away from the endpoints when |x| < 1. Section 4 considers the pointwise convergence of the MTR-approximations when x = ±1. In these sections, we show also the results of numerical experiments which confirm and explain the theoretical findings. Section 5 presents some concluding remarks. Appendix A recalls some results concerning the Laguerre polynomials and Appendix B proves some combinatorial identities that we used in the proofs of lemmas and theorems. 2. Preliminaries Throughout the paper, we assume that parameters θk, k = 1, . . . , p are defined by (see (24)) θk = 1 − τk N, τk ̸ = 0, k = 1, . . . , p. (26) Let τ = {τ1, . . . , τp} and let coefficients γk(τ) be defined by the following identity:

Let   (, ) be the truncated modified Fourier series where Moreover, the modified Fourier basis can be derived from the other classical basis H * on [0, 1] by means of a change of variable.
The first results concerning the convergence of the expansions by the modified Fourier basis appeared in the works [2,[8][9][10].We present two theorems for further comparisons.
Overall, we see better convergence rates compared to the classical Fourier expansions [11].This can be explained by faster decay of coefficients    : compared to the classical ones when  is smooth enough but nonperiodic on [−1 , 1].Estimate (11) can be explained by a nonperiodicity of the basis functions sin ( − 1/2) on Convergence acceleration of the modified Fourier expansions by means of rational corrections was considered in [6].Here, we continue those investigations.More specifically, consider a finite sequence of real numbers  = {  }  =1 ,  ≥ 1 and, by Δ   (, f), f = {  } denote the following generalized finite differences: By Δ   ( f), we denote the classical finite differences which correspond to generalized differences Δ   (, f) with  ≡ 1.It is easy to verify that Let   (, ) =  () −   (, ) =    (, ) +    (, ) , where By means of sequential Abel transformations (see details in [6]), we derive the following expansions of errors (15): where , (, , ) These expansions lead to the following modified-trigonometric-rational (MTR-) approximations: , (,  c ,   , ) =   (, ) A crucial step for realization of the rational approximations is determination of parameters   and   .Different approaches are known for solution of this problem (see [12][13][14][15][16][17][18][19]).In general, appropriate determination of these parameters leads to rational approximations with improved accuracy compared to the classical ones in case of smooth .However, the rational approximations are essentially nonlinear in the sense that as for each approximation we need to determine its own   and   vectors.
In [6], those parameters were determined from the following systems of equations: which led to the Fourier-Pade type approximations [12] with better convergence for smooth functions (see [6]) compared to the expansions by the modified Fourier basis.It is rather complex approach as parameters   and   depend on  and systems (23) must be solved for each .
In this paper, assuming that  is smooth on [−1, 1], we consider simpler alternative approach, where   and   are determined as follows [14,16,19]: with   = {  1 , . . .,    } and   = {  1 , . . .,    } independent of .Actually, in this approach, we take into consideration only the first two terms of the asymptotic expansions of   =   () in terms of 1/.Although parameters   and   in (24) depend on , we need only to determine   and   which are independent of .Hence, this approach is less complex than the modified Fourier-Pade approximations.
The main results of this paper are exact constants of the main terms of asymptotic errors and optimal parameters for improved pointwise convergence of rational approximations.First, we derive the exact estimates for the main terms of asymptotic errors without specifying parameters   and   .Second, we determine the optimal values of parameters   and   which vanish the main terms and lead to approximations with substantially better pointwise convergence rates.We found that optimal values of parameters    and    ,  = 1, . . ., , are the roots of some polynomials depending on  and , where  indicates the number of zero derivatives in (8).Moreover, the choice of optimal parameters depends on the parity of  and also on the location of , whether || < 1 or  = ±1.For example, when  is odd and || < 1, the roots of the generalized Laguerre polynomial  (2+1)  () (see Appendix A) could be used for   and   .In this case, the convergence rate of the MTR-approximations is ( −2−−[(+1)/2]−2 ) as  → ∞.It means better convergence compared to the expansions by the modified Fourier basis with improvement by factor ( +[(+1) /2] ).When  is odd and  = ±1, the roots of the generalized Laguerre polynomial  (2)  () could be used.In this case, the convergence rate is ( −2−[(+1)/2]−1 ) with the improvement by factor ( [(+1) /2] ).The problem that we encountered is impossibility to get simultaneous optimality on || < 1 and at  = ±1.One must decide whether to use parameters that provide the minimal error on [−1, 1], but with worse accuracy on || < 1, or work with optimal parameters on || < 1 with worse accuracy at  = ±1.We expose similar observations for even values of .
It is important that the values of parameters   and   depend only on  and  and are independent of .It means that if functions , , and  +  have enough smoothness and obey the same derivative conditions, the optimal approach leads to linear rational approximations in the sense that  , ( + ,   ,   , ) =  , (,   ,   , ) +  , (,   ,   , ) (25) with the same parameters   and   for all included functions.The paper is organized as follows.Section 2 presents some preliminary lemmas.Section 3 explores the pointwise convergence of the MTR-approximations away from the endpoints when || < 1. Section 4 considers the pointwise convergence of the MTR-approximations when  = ±1.In these sections, we show also the results of numerical experiments which confirm and explain the theoretical findings.Section 5 presents some concluding remarks.Appendix A recalls some results concerning the Laguerre polynomials and Appendix B proves some combinatorial identities that we used in the proofs of lemmas and theorems.

Pointwise Convergence Away from the Endpoints
In this section, we explore the pointwise convergence of the MTR-approximations away from the endpoints.Next theorem reveals the asymptotic behavior of the MTRapproximations for || < 1 without specifying the selection of parameters   and   .
Taking into account the fact that   → 1 as  → ∞, we estimate only the sums on the right-hand side of (18).By the Abel transformation, we get Lemma 5 estimates sequences Δ   (, f ), Δ 1  ( Δ (, f )), and Δ 2  ( Δ (, f )) as  → ∞ and  ≥  + 1.It shows that, for  = 1 and  = 2, we have and the third term in the right-hand side of (47) is ( −−2−2 ).Then, with  = 1 and  = 1, we have and the second term is ( −−2−3 ).Finally, using the exact estimate for Δ   (, f ), we derive which completes the proof as Note that Theorem 6 is valid also for  = 0 which corresponds to the modified Fourier expansions (compare with Theorems 1 and 2).In that case, the exact constants of the main terms in ( 44) and ( 45) coincide with similar estimate in [3] (Theorem 2.22, page 29).
Can we improve the accuracy of the rational approximations by appropriate selection of parameters   and   ?Further, in this section, we give positive answer to this question and show how the optimal values can be chosen.
Estimates of Theorem 6 show that improvement can be achieved if parameters are chosen such that   =   =  and By looking into the definition of ℎ ,2+1 (), we observe that condition (55) can be achieved, for example, if where with unknown coefficients   ,  = 1, . . ., .Then, condition (55) follows from the well-known identity Further, we determine the values of   ,  = 1, . . ., , for improved convergence of the rational approximations.
Next result is an immediate consequence of those observations and estimates of Theorem 6.
Assume the polynomial has only real-valued and nonzero roots  =   ,  = 1, . . ., , and let Then, the following estimate holds for || < 1: Theorem 7 is valid only if, for given  and , polynomial (60) has only real-valued and nonzero roots.Further, we clarify this statement by showing those cases when it is true.
By imposing extra smoothness on the underlying functions, we derive more precise estimate of (62).
Further, in Theorems 9 and 10, we show that the pointwise convergence rate of the rational approximations depends on the asymptotic Δ 0  ( Δ (, f )) and Δ 0  ( Δ (, f )).From the other side, Lemma 8 reveals that the convergence rates of those sequences depend on the value of (as  = 0) When  is odd, for the highest power of 1/, parameter  can be only  = 0.It means that   () ≡ 1.When  is even, parameter  can be  ≤ 1 which means that   () = 1 +  1 .
We determine parameter  1 later.
Next theorem unveils the convergence rate of the MTRapproximations for odd values of , when   () =  0 () ≡ 1.Note that, in this case, the roots of polynomial (60) coincide with the roots of generalized Laguerre polynomial Let   ,  = 1, . . ., , be defined by (26), where   are the roots of the generalized Laguerre polynomial  (2+1)  ().Then, the following estimates hold for || < 1: where  and σ are defined in Lemma 8.
Taking into account the fact that   → 1 as  → ∞, we estimate only the sums on the right-hand side of ( 18).
Let us return to MTR-approximations of (53). Figure 4 shows the results of approximation of (53) by the MTRapproximations with even .Parameters    =    ,  = 1, . . .,  are selected as the roots of  (2+1)  ().Compared with Figure 2, we see better accuracy on || < 1.For  = 2, the improvement is almost 27 times, and for  = 4, the improvement is almost 200 times.
Figure 5 shows similar results with parameters    =    ,  = 1, . . ., , as the roots of  (2)  ().In the next section, we will prove that those parameters provide improved accuracy also at  = ±1 for some  and .We see that both choices of parameters provide similar results on || < 1.
Note that this theorem is valid also for  = 0 which corresponds to the expansions by the modified Fourier basis (compare with Theorems 1 and 2).
Exact constants of the main terms in (90), for  = 0, can be found also in [10] (Theorem 3.2).We see that, in general, rational corrections do not increase the convergence rates of modified Fourier expansions at the endpoints  = ±1 without specifying appropriately parameters   and   .Both approaches have the same convergence rates ( −2−1 ).
Moreover, as Figure 6 shows, without reasonable selection of the parameters, modified Fourier expansions have better accuracy compared to "nonoptimal" rational approximations at  = ±1.
Is it possible to improve the accuracy by appropriate selection of parameters   and   ?The answer is positive and the solution is in the estimates of Theorem 11.Like the previous section, we put where Now, the property ℎ ,2 () = 0 follows from the identity Next theorem is the result of these observations and Theorem 11.
Assume the polynomial has only real-valued and nonzero roots  =   ,  = 1, . . ., , and let Then, Note that Theorem 12 is valid only for those parameters  and  when polynomial (97) has only real-valued and nonzero roots.Further, we clarify this property with more details.
Our next goal is derivation of the exact convergence rate of (99).
By repeating the observations of previous section, it is possible to deduce that, for getting the maximal convergence rate for odd values of , the polynomial   () can be at most degree 0 polynomial,   () =  0 () ≡ 1.For even values of ,   () =  1 () = 1 +  1 .In the first case, parameters   are the roots of  (2)  ().In the second case, if  1 = 0, we get the roots of  (2)  () and if  1 = −1/(2 + ), we get the roots of  (2−1)  ().The next two theorems immediately follow from Lemma 13 and identity (B.12) and, we omit the proofs.
Figure 7 shows the result of approximation of (53) by the rational approximations with optimal values of parameters   ,  = 1, . . ., , as in Theorem 14.We see that, by increasing  ( is odd), we increase the accuracy of approximations at the points  = ±1.Note that  = 0 corresponds to the classical expansion by the modified basis and we see that, in contrary to Figure 6, the optimal choice of parameters do have big positive impact on the accuracy.
Comparison of Theorems 9 and 14 reveals the problem of the optimal rational approximations which is in the difference of optimal values of parameters   for || < 1 and  = ±1.On || < 1 and  = ±1, the optimal values are the roots of  Next theorem explores even values of . and Assume the polynomial has only real-valued and nonzero roots  =   ,  = 1, . . ., , and let   be defined by (26) with   =   .Then, the following asymptotic expansions hold: with  and δ defined in Lemma 13.
Estimates (113) and ( 114) are valid if polynomial (112) has only real-valued and nonzero roots.As we mentioned above, in two particular cases when  1 = 0 and  1 = −1/(2 + ), the roots of the polynomial coincide with the roots of   () is reasonable as it will provide optimal rational approximation both on || < 1 (see Theorem 10) and at  = ±1 for some  and .
Figure 8 shows the result of application of the rational approximations to function (53) with optimal values of parameters   ,  = 1, . . ., , as the roots of   1 .Assume that, for given  and , it is possible to vanish Φ , ( 1 ) by appropriate selection of  1 in (113).Then, we derive or in case of smoother functions.
The problem is that we cannot vanish both Φ , ( 1 ) and Φ, ( 1 ) simultaneously by the same  1 .Hence, we decompose a function into even and odd parts and perform separate optimizations in terms of parameter  1 .In order to choose parameter  1 appropriately, we need to have and different  while approximating (53).In "nonoptimal" case, parameters    and    ,  = 1, . . ., , are the roots of  (2)  ().In "optimal" case, the parameters are chosen from Tables 1 and 2 for  = 4 and  = 1.
Figures 9 and 10 show the errors at  = ±1 while approximating (53) with rational approximations, where parameters    and    ,  = 1, . . ., , are selected according to Tables 1 and 2 for even and odd parts of the function, respectively.We called this approach "optimal" in the figures.For comparison, we showed also the result of approximations with parameters    and    ,  = 1, . . ., , as the roots of  (2)  () (see Figure 8 and Theorem 15).We called the latest "nonoptimal" in the figures.We see the impact of optimizations on the accuracy of the rational approximations at  = ±1.
Throughout the paper, we systematically required that approximated function  obeys first derivative conditions (8).Without those conditions, the convergence rate will remain slow.This is due to function jumps in certain derivatives at the endpoints  = ±1.If these jumps are known, the convergence acceleration can be achieved by well-known polynomial subtraction approach.For the classical Fourier series this approach has a very long history (see [3,[22][23][24][25][26][27][28]).For modified expansions this approach is explored in [3,5,7].More specifically, we write  (see [3]) in the terms of its Lanczos representation: where functions (polynomials)   are chosen as such to satisfy the conditions: Since  −   obeys the first  derivative conditions, the new approximation will converge with the same rate as if  obeyed those conditions.This is the polynomial subtraction technique known also as Krylov-Lanczos approach.If the jumps of  are unknown, their values can be approximated by solution of the corresponding system of linear equations (see [22]).
The same approach can be applied also for the MTRapproximations and in that case all theorems of this paper will remain valid without the requirements of the derivatives at the endpoints  = ±1.

Conclusion
In this paper, we investigated the convergence of the MTRapproximations  , with parameters   and   defined by (24).The main goal of the paper was to show that by appropriate selection of parameters   and   it was possible to improve substantially the pointwise convergence of the approximations.We accomplished the main goal by calculating the exact constants of the main terms of and different  while approximating (53).In "nonoptimal" case, parameters    and    ,  = 1, . . ., , are the roots of  (2)  ().In "optimal" case, the parameters are chosen from Tables 1 and 2 for  = 6 and  = 1.asymptotic errors and by eliminating those constants through appropriate selection of the approximation parameters.We showed that the optimization depends whether || < 1 or  = ±1 and also whether parameter  is odd or even.
Theorem 6 provides general estimate on || < 1 proving that, without optimal selection of parameters    and    ,  = 1, . . ., , the rational approximations have convergence rate ( −2−−2 ) as  → ∞ if an approximated function has enough smoothness and obeys the first  derivative conditions (see (8)).Compared with the modified Fourier expansions (see Theorems 1 and 2 or the same theorem with  = 0), the improvement is by factor (  ) as  → ∞.
In case of even values of  and || < 1 (Theorem 10), possible selection set of optimal parameters is wider.If for given  and  the polynomial has only nonzero and real-valued roots  =   ,  = 1, . . ., , then, selection    =    =   provides better convergence rate ( −2−−[/2]−2 ) compared to the estimate of Theorem 6 and improvement is by factor ( [/2] ).Improvement is by factor ( [/2]+ ) compared to the expansions by the modified Fourier basis.The problem is to find the values of  1 in (126) for which it will have only real-valued and nonzero roots.In two cases it is obvious.When  1 = 0, the roots of (126) coincide with the roots of  (2+1)  ().When  1 = −1/(2 +  + 1), the roots coincide with the ones of  (2)  ().In both cases all roots are positive.
Theorem 11 imparts the convergence rate ( −2−1 ) of the rational approximations at  = ±1 without optimal selection of parameters.Comparison with Theorems 1 and 2 shows no improvement.Moreover, as our experiments show (see Figure 6) rational approximations without reasonable selection of parameters   can perform worse at  = ±1 compared to the expansions by the modified Fourier basis.
Theorem 14 found the optimal values of parameters for odd  for better convergence rate at  = ±1.It proved that the best accuracy could be achieved when parameters    =    were the roots of the generalized Laguerre polynomial  (2)  ().For that choice, the convergence rate was ( −2−[(+1)/2]−1 ) and improvement was by factor ( [(+1)/2] ) compared to the modified Fourier expansions.
We see that when  is odd, the optimal choices for || < 1 and  = ±1 are different.The choice of polynomial  (2)  () will provide the minimal uniform error on [−1, 1], but, for || < 1, the convergence rate will be worse by factor () compared to the optimal choice  has only real-valued and nonzero roots  =   for some  1 , then, selection    =    =   will provide convergence rate ( −2−[/2]−1 ) with improvement by factor ( [/2] ) compared to the modified Fourier expansions.The problem is the same as that for polynomial (126).Polynomial (127) must have only real-valued and nonzero roots for the selected  1 .Fortunately, two such selections are known.When  1 = 0 or  1 = −1/(2 + ), the roots of (127) coincide with the ones of   () is better as it will provide optimal approximations for both || < 1 and  = ±1.However, estimates of Theorem 15 allow determining parameter  1 for even more better convergence rate.Tables 1 and 2 show some values of  1 and parameters   that will provide convergence rate ( −2−[/2]−2 ) with improvement by factor ().The problem is that the latest choice is not optimal for || < 1.It will give worse accuracy compared to the optimal selection for || < 1. Convergence rate will degrade by factor ().As in case of odd , a user of the algorithms must decide which choice will be more appropriate to select, the best approximation on overall [−1, 1] with worse accuracy on || < 1, or the best accuracy on the latest one.
In case of even , we obtained similar results.Theorem 15 outlines the set of optimal parameters.If for given  and  the polynomial