On a Pointwise Convergence of Quasi-Periodic-Rational Trigonometric Interpolation

We introduce a procedure for convergence acceleration of the quasi-periodic trigonometric interpolation by application of rational corrections which leads to quasi-periodic-rational trigonometric interpolation. Rational corrections contain unknown parameters whose determination is important for realization of interpolation. We investigate the pointwise convergence of the resultant interpolation for special choice of the unknown parameters and derive the exact constants of the main terms of asymptotic errors.

The idea of the QP interpolation is introduced in [1,2] where it is investigated based on the results of numerical experiments.Explicit representation of the interpolation is derived in [3][4][5].There, the convergence of the interpolation is considered in the framework of the  2 (−1, 1)-norm and at the endpoints  = ±1 in terms of the limit function.Pointwise convergence in the interval (−1, 1) is explored in [6].The main results there, which we need for further comparison, are the following theorems.
Then, the following estimate holds for || < 1 as  → ∞: ,0 (, ) =  0 () (−1)  2 +1  +1 sin () cos (/2) In the current paper, we consider convergence acceleration of the QP interpolation by rational corrections in terms of   which leads to quasi-periodic-rational (QPR) interpolation.We investigate the pointwise convergence of the QPR interpolation in the interval (−1, 1) and derive the exact constants of the main terms of asymptotic errors.Comparison with Theorems 1 and 2 shows the accelerated convergence for smooth functions.Some results of this research are reported also in [7].
More specifically, the QP interpolation can be realized by the following formula: where Here, V −1 ℓ, are the elements of the inverse of the Vandermonde matrix (V ,ℓ ) as and have the following explicit form [8]: where   are the coefficients of the following polynomial: Taking into account that   −   = (1/), from (13), we get ,ℓ =  ( −1 ) ,  → ∞. (16)

Quasi-Periodic-Rational Interpolation
In this section, we consider convergence acceleration of the QP interpolation by rational trigonometric corrections which leads to the QPR interpolation.Consider a vector  = {  }  =1 .By Δ   (,   ), we denote generalized finite differences defined by the following recurrent relations: for some sequence   .When  ≡ 1, we put It is easy to verify that International Journal of Analysis 3 In general, we can prove by the mathematical induction that where   () are the coefficients of the following polynomial: .By    (,   ), we denote modified finite differences defined by the following recurrent relations: for some sequence   .When  ≡ 1, we put Similar to (20), we can show that where It is easy to verify that We assume that  ∈   [−1, 1] for some  ≥ 1 and we denote where According to definition of  * , we can write Hence, Therefore, The following transformation is easy to verify (see details in [9] for similar transformation): Reiteration of it up to  times leads to the following expansion of the error: International Journal of Analysis where the first two terms can be assumed as corrections of the error.This observation leads to the following QPR interpolation: with the error The QPR interpolation is undefined until parameters   are unknown.Hence, determination of these parameters is a crucial problem for realization of the QPR interpolation.First, we assume that where   are some new parameters independent of .In the next section, we investigate convergence of the QPR interpolation independent of the choice of parameters   .Then, we discuss some choices of these parameters.We also consider an approach connected with the idea of the Fourier-Pade interpolation which leads to quasi-periodic Fourier-Pade interpolation.

Convergence Analysis
Let   be chosen as in (38) and let   () be the coefficients of the following polynomial: where  = { 1 , . . .,   }.
Let us modify (20) in view of (38).For  = 1, we write For  = 2, we have In general, we can prove by the mathematical induction the following expansion [10]: Now, let us modify (24) in view of (38).According to (20) and (42), we get (note that  + =  − ) Similar to (42), we can show that Then, from (43), we have This leads to the following needed expansion: Then, Taking into account that we find that We will frequently use the latest formula.We denote by   the th Fourier coefficient of  as Let First, we prove some lemmas.

𝑐 (𝑥) 𝑅
which concludes the proof in view of Lemma 6.
Let us compare the results of Theorems 1 and 7. Theorem 1 investigates the pointwise convergence of the QP interpolation on (−1, 1) and states that for  (+2) ∈ [−1,1] the convergence rate is ( −−−1 ) for  ≥ 1. Theorem 7 explores the pointwise convergence of the QPR interpolation and shows that convergence rate is ( −−2−1 ) for  (+2+) ∈ [−1,1] and  ≥ 1.We see that for  = 2 both theorems are provided with the same rates of convergence by putting the same smoothness requirements on , although the exact constants of the asymptotic errors are different.Then, we see that for  > /2 the QPR interpolation has improved accuracy compared to the QP interpolation and improvement is by factor ( 2− ).In this case, Theorem 7 puts additional smoothness requirement on  and comparison is valid if only the interpolated function has enough smoothness (for example, if it is infinitely differentiable).It is worth recalling that parameter  indicates the size of the Vandermonde matrix (12) that must be inverted for realization of the QP and QPR interpolations.It is wellknown that the Vandermonde matrices are ill-conditioned and standard numerical methods fail to accurately compute the entries of the inverses when the sizes of the matrices are big.Hence, from practical point of view, it is more reasonable to take  small ( ≤ 6) and additional accuracy obtain by increasing .Note that for  = 1 the second term in the brackets of estimate (79) vanishes and also  + ,1,0 (1, ) = 0. Hence, Similarly, the case  = 0 can be analyzed.We present the corresponding theorem without the proof which can be performed as the above one.
Comparison with Theorem 2 shows improvement by factor ( 2 ) for any  ≥ 1 if  has enough smoothness.

Parameter Determination in Rational Corrections
Till now, we did not discuss the problem of parameters   determination as Theorems 7 and 8 are valid for all choices.Now, let us consider some choices with the corresponding numerical results.Let One choice is   =  which shows satisfactory numerical results (see Figure 1).
The third choice is not connected with (38) and allows determining parameters   immediately along the ideas of the Fourier-Pade interpolation ( [11]).This approach is more complex as   must be recalculated for each function  and for each  and as a consequence leads to nonlinear interpolation but, however, is much more precise when || < 1.