Extremal Polynomials and Entire Functions of Exponential Type

In this paper, we discuss asymptotic relations for the approximation of xα,α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| x\right| ^{\alpha },\alpha >0$$\end{document} in L∞-1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\infty }\left[ -\,1,1\right] $$\end{document} by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind. The limiting process reveals an entire function of exponential type for which we can present an explicit formula.


The Bernstein Constant
Let α > 0 be not an even integer.Starting in year 1913 for the case α = 1, and later in 1938 for the general case α > 0, S.N. Bernstein [1], [3] established the limit denotes the error in best L p approximation of a function f on the interval [a, b] by polynomials of degree less or equal n.The proofs in [1], [3] are highly difficult and long, missing many non-trivial technical details.In his 1938 paper, Bernstein made essential use of the homogeneity property of |x| α , namely that for c > 0 one has |cx| α = c α |x| α .Using this property, one gets for a, b > 0 and all 1 ≤ p ≤ ∞ the relation (see [10], Lemma 8. Bernstein also established a formulation of the limit as the error in approximation on the real line by entire functions of exponential type, namely, Recall that an entire function f is of exponential type A ≥ 0 means that for each ε > 0 there is z 0 = z 0 (ε), such that |f (z)| ≤ exp (|z| (A + ε)) , ∀z ∈ C : |z| ≥ |z 0 | . (1.2) Moreover, A is taken to be the infimum over all possible numbers for which (1.2) holds.The elegant formulation which introduces now functions of exponential type extends to spaces other than L ∞ .Ganzburg [5] and Lubinsky [10] have shown that for all 1 ≤ p ≤ ∞ positive constants ∆ p,α exists, where ∆ p,α is defined by From now on ∆ p,α are called the Bernstein constants.
Only for p = 1, 2 are the values ∆ p,α known.In 1947, Nikolskii [12] proved that and in 1969, Raitsin [16] established In contrast to the case of the L ∞ norm, no single value of ∆ ∞,α is known.
Bernstein speculated that Over the years the speculation became known as the Bernstein conjecture in approximation theory.Some 70 years later Varga and Carpenter [20], using sophisticated high precision scientific computational methods, calculated the quantity numerically to ∆ ∞,1 = 0.28016 94990 23869 . . .
Further extensive numerical explorations for the computation of ∆ ∞,α have been provided later by Varga and Carpenter [21].Their numerical work gave an enormous impact into the analytical investigation of approximation problems, not only restricted to the Bernstein constants.We would also like to mention the numerical work of Pachón and Trefethen ([13], Figure 4.4) from 2008, when they recomputed {nE n (|x| , L ∞ [−1, 1]) : n = 1, . . ., 10 4 } again and provided an graphical illustration indicating a monotonic growth behavior.As the story continued, the approximation of entire functions of exponential type became a much studied topic in function approximation, see [4], [19], but also in connection to problems in number theory, see for instance [22].As an further application in number theory, we would like to mention a recent paper of Ganzburg [7], where he discusses new asymptotic relations between Zeta-, Dirichlet-and Catalan functions in connection with the asymptotics of Lagrange-Hermite interpolation for |x| α .

Denote by P *
n the best approximating polynomial of order n to |x| α in the L p norm.Then, for all 1 ≤ p ≤ ∞, α > − 1 p not an even integer, one has Moreover, uniformly on compact subsets of C, and there is exactly one entire function H of exponential type ≤ 1 which minimizes (1.4).While various versions of this equality and relations (1.4) have been discussed by Bernstein, Raitsin and Ganzburg, the uniqueness of H * α proved in [10] is a highly nontrivial result.From the Chebyshev alternation theorem it follows that for each integer n the best approximating polynomial P * n of order n to |x| α in the in L ∞ norm can be represented as an interpolating polynomial with unknown consecutive nodes in [−1 , 1].Thus, if one can find something about the nature of those best approximating interpolation nodes in [−1, 1], then we would successfully find an approach for a constructive analytical approximation towards some representations for the Bernstein constants ∆ ∞,α .Since |x| α is an even function a standard argument allows us to restrict ourselves to interpolation polynomials of even order n = 2m.It is not surprising that Bernstein [2] himself, in 1937, studied the interpolation process to |x| α by using the modified Chebyshev system where the x (2n) j are the zeros of the Chebyshev polynomial T 2n of first kind, defined by T n (x) = cos (n arccos x).However, x (2n) 0 is an additional choice, but not a zero of T 2n , in order to obtain the corresponding interpolation polynomial P (1) 2n of order 2n for |x| α .The final answer for its limit relation was given not before 2002 by Ganzburg ([5], Formula 2.7).For α > 0 one has dt. (1.5) Let us give some remarks on equation (1.5).Firstly, we mention that in [2] Bernstein himself established a slightly weaker solution compared to formula (1.5).Secondly, an extension of limit relation (1.5) to complex values for α was obtained recently in [6].
It is remarkable that, since the beginning with Bernstein, no one has studied in detail the interpolation process by using the node system consisting of the 2n + 1 zeros of T 2n+1 , since this node system automatically includes x = 0 as a node and apparently it seems to be the more natural choice.To go into detail, let to be the zeros of T 2n+1 and let us denote by P 2n the corresponding interpolation polynomial of order 2n for |x| α .There is one paper [23], dealing with this node system and presenting the result that the approximation order |x| α − P = O (1) /n α when α ∈ (0, 1).In other words, the interpolation process attains the Jackson order.We also would like to mention a recent monograph by Ganzburg introducing an integral of similar nature to that in formula (1.5).In this paper we continue the investigation into the precise limiting quantity of (2n for all α > 0.
The paper is organized as follows.
In section 2 we collect some definitions for several constants and functions together with some standard results for later use.
In section 3 we establish the precise limit relation (Theorem 3.1) and we show that the scaled polynomials n α P (2) n • n uniformly converge on compact subsets of the real line to an entire function H α of exponential type 1 (Theorems 3.2 and 3.3).We may also present an explicit expansion for H α as an interpolating series for |x| α (Theorem 3.3).As it can be seen later from the representation for the explicit limiting error term, i.e. from the exact determination of the quantity on the right-hand side in (1.7) for individual values for α appears to be a rather difficult challenge.
In section 5, by using an higher order asymptotics and investigating into an (itself) interesting integral inequality, see Theorem 5.1, we finally arrive in Theorem 5.3 at an asymptotic connection between In Section 6, to emphasize the importance of the interpolation formulas based on the P (1) n and P (2) n polynomials, we present a compilation of numerical results involving some non-trivial linear combinations of the just mentioned polynomials together with their corresponding Chebyshev polynomials T n , in order to present explicit formulas for near best approximation polynomials in the L ∞ norm, see formula (6.3), together with their corresponding entire functions of exponential type, see formula (6.4).Possibly and hopefully these formulas could indicate a feasible direction towards some explicit asymptotic representations of best approximation polynomials for |x| α in the L ∞ norm and thus for the Bernstein constants ∆ ∞,α themselves.

Notation
In this section we record the following constants and functions, together with properties which are used later in the paper.We denote by Γ (•) the usual Gamma function.The Chebyshev polynomials of first kind are denoted by T n , where T n (x) = cos (n arccos x).For x ∈ R, let [x] to be the floor function, namely [x] = max {m ∈ Z : m ≤ x}.Obviously, then x − 1 < [x] ≤ x.We define the following constants.
x sin (x) Note that H (α, •) should not be mixed up with the subsequent following definition of H α .We proceed further with: We collect the following easy to establish properties. (2.1) Note that (2.1f) is not an easy consequence of (2.1e).We also remark, that for α ≥ 1 equation (2.1a) remains also valid for x = 0, by interpreting both sides as their lim x→0 + .The same holds true for (2.1b) and (2.1d) for α > 0. We then have (2.2) Then, using (2.2), we define (2.4) Finally, we apologize for the repulsive notation f (x) instead of f that we occasionally use in this paper.
The objective now is to find its limiting error term in the L ∞ norm.Since the error term is symmetric in [−1, 1] we prove the following Theorem 3.1.Let α > 0. Then we have Theorem 3.3.Let α > 0 be not an even integer.Then H α (interpreted as its extension into the complex domain) is an entire function of exponential type 1, interpolating |x| α at the interpolation points {kπ : k = 0, 1, 2, . ..} and H α admits a representation as an interpolating series of the following form.
Denote by N = [α/2] .Then, for all x ∈ R, we have For the special case 0 < α < 2 the expansion is then represented by We start with the proof for Theorem 3.1 by splitting it in several Lemmas.First, we present without a proof the following two Lemmas.x Then, using Lemma 3.1 and Lemma 3.2, we estimate Then by a standard argument we arrive at Proof.This follows directly by applying the triangle inequality combined together with Lemma 3.4 and Lemma 3.5.
Lemma 3.7.Let C > 0 be fixed, ε > 0 and n > max C, C ε , 1 2ε .Then, for α > 0, we have Proof.First, we remark that for α > 0 the left-hand side in Lemma 3.7 is well defined by applying (2.2) together with Lemma 3.3.Using again the triangle inequality together with Lemma 3.6 and formula (2.1c), we arrive at Our first substantial result is now the following Lemma 3.8.Let α > 0. Then .
Using (2.1c), the latter part can be estimated to Combined together with the previous estimate and Lemma 3.7, we finally get By taking the lim the result follows.Now, we are turning to the lim case.
Lemma 3.9.Let α > 0 and C > 0 be fixed.Then Then, by applying again the triangle inequality and combining together with Lemma 3.6 and (2.1c), we estimate Now, by taking lim we establish the result.
Our second substantial result is the following Lemma 3.10.Let α > 0. Then Proof.Let ε > 0 and C > C(α) ε .Then, starting with the right-hand side in Lemma 3.10, we estimate Using again (2.1c), the latter part can be estimated to Combined together with Lemma 3.9 and the previous estimate, we arrive at .
Since the last expression holds for every ε > 0 we establish the result.
Combining now Lemma 3.8 and Lemma 3.10 together with (3.1), gives the result and we are finished.
Proof of Theorem 3.2.Let α > 0. From (3.1) it follows that for every ε > 0 we can find some n 0 = n 0 (ε), such that for all n > n 0 We proceed further by use of (2.1c), Lemma 3.3 and Lemma 3.6.
Combining together with (3.4), we obtain for every ε > 0 and n sufficiently large, Since any compact set K in [0, ∞) can be included in some interval [0, C] the result is established.

The Envelope function
In this section we consider the envelope error function H 1 (α, •) with respect to |H (α, •)|.Our next objective is to establish an asymptotics for Then, we have  x is monotonically increasing in x ∈ (0, ∞) and f (x) ≤ e.

Asymptotics of the error function
In this section we establish an asymptotic bound for the norm of the limiting error function, i.e. for H (α, •) L ∞[0,∞) .This section is the most technical part in this paper.Here, we use the generalized Watson Lemma (Laplace method for integrals with large parameter) for deriving an asymptotic expansion used to be later in the context.As it turns out, we need an higher order asymptotics up to order 5 involving the computation of certain rather complicated defined constants.However, the main idea for deriving a lower estimate is quite easy to see.Let us start, once again, with a diagram (Figure 3) involving the functions |H (α, •)| and H 1 (α, •) .•) appears to be descending.For growing values of α, the size of these maxima appear to be of the same magnitude compared to the size H 1 (α, α).We use both observations for the asymptotic analysis.First, we show that H 1 (α, •) is descending at least for values x ≥ α.Then, we derive the asymptotics for the local maximum in |H (α, β)|.It turns out that the following integral inequality plays an essential role.
Theorem 5.1.There exists a fixed constant α 0 > 0 such that for α ≥ α 0 , (5.1) We remark that (5.1) is not true for all α 0 > 0. This can be seen out from Figure 4. Also, for growing values of α, the positive magnitude becomes rather small.Numerical experiments suggest that the minimal value for α 0 such that (5.1) becomes true, is somewhere in the interval (2.54288, 2.54289).
However, since we are interested in an asymptotic expansion, the determination of the exact size of the minimal value α 0 is not important.From Theorem 5.1 we may derive our first desired property.
From Theorem 5.2 we obtain the final asymptotics.
Theorem 5.3.We have We first establish Theorem 5.2 by assuming that Theorem 5.1 holds true.Then, we present the proof for Theorem 5.1 which is completely independent of the forthcoming Lemmas related to Theorem 5.2.Finally, we present the proof for Theorem 5.3.Without proof, we first present the following Lemma 5.1.Let α > 0 be fixed and x > 0. Then S (α, x) has the representation Lemma 5.2.Let α > 0 be fixed and x > 0. Then Proof.Using (2.1a) and by differentiating under the integral, we get Lemma 5.3.Let α > 0 be fixed and x > 0. Then is an increasing function in x.
Proof.Using Lemma (5.1) and by differentiating under the integral again, we get The rescaling of R (α, •) in Lemma 5.3 is now extremely useful in proving Theorem 5.2.Considering formula (5.2) contributes to my colleague, Dr. Maximilian Thaler, for which I thank him.
Proof.We prove the relations with the generalized Watson Lemma.Let α > 0, k = 0, 1 and c ≥ 0. Then ) and g (t) = t − log t.Before applying the Watson Lemma, we have to split the integral in two parts .For the second integral 1 0 we have to apply a suitable transformation before expanding it.It is worth mentioning, that in the classical textbooks on asymptotic analysis (compare [14], p. 86) there is no general formula for the coefficients a n available.Only the first one or two coefficients are derived and as it can be easily checked, they are of rather complicated nature.Surprisingly, in the newer literature ([15], Formula 2.3.18)one can find a remarkable easy representation for these coefficients in terms of some residues as well as a reference for its derivation, namely (in our context) We used a symbolic computation software for the computation of the residues in (5.5), but we do not present the general outcome of these formulas.This would fill several pages.However, since the calculations are of crucial importance in the proof for Theorem 5.1, we present all relevant outputs.For k = 0, 1 and c = 0 we calculate = −e −c 1+3c 6 .With λ = 1 and µ = 2 we obtain for α → ∞, Proceeding in the same way for the second integral 1 0 , we compute For c ≥ 0, we compute a (0,c) 0 and µ = 2 we obtain for α → ∞, Collecting the results we finally arrive at the expansions in Lemma 5.6.
Lemma 5.7.There exists some α 1 > 0, such that Proof.From Lemma 5.6, we calculate Now, combining the last expression together with Lemma 5.6, we obtain for α → ∞ the asymptotics The assertion now follows.
We turn now to the proof for Theorem 5.3, again by establishing some Lemmas.Without proof, we first present the following Lemma 5.8.Let α > 0 and Lemma 5.9.Let α > 0 and c ≥ 0. Then Proof.From Lemma 5.6, we simply derive Proof.Using (2.1a), we obtain . (5.6) Next, using Lemma 5.4 together with a standard estimate for the zeta function, we establish and Now, combining (5.6), (5.7), (5.8) together with Lemma 5.9, we establish the result.

Approximation polynomials in L ∞
This section is devoted to an explicit construction for near best approximation polynomials to |x| α , α > 0 in the L ∞ norm.The construction involves the polynomials P n and P n together with the Chebyshev polynomials T n .The construction method is based on numerical results.The resulting formulas could indicate a general possible approach and structure for the Bernstein constants ∆ α,∞ .
Let α > 0 be not an even integer.
First, let us collect some details on the interpolating polynomials P where is an entire function of exponential type 1 that interpolates |x| α at the nodes k + 1 2 π : k ∈ Z ∪ {0}.There also exists ( [5], Formula 4.15) a representation for G α as an interpolating series, similar to formula (3.2) in Theorem 3.3.
By an analogue method as that was used in the proof for Theorem 3.2 one can show that uniformly on compact subsets of [0, ∞) we have the scaled limit lim n→∞ (2n) α P (1) 2n x 2n = G α (x) .(6.2) Now, based on numerical computations, we made the following observations.For all α > 0 not an even integer we find that, beginning with the second positive note, all interpolation points of the best approximation polynomials P * 2n are located somewhere between two consecutive interpolation points for the P (1) 2n and P (2) 2n polynomials.See Figure 5.It is well known that 1, x, . . ., x n ; x α/2 is an hypernormal Haar space of dimension n + 2 on the interval [0, 1], see ( [21], p. 199).Consequently it follows that we have always an alternation point at x = 0. Thus we cannot expect to perform in the quality of best approximation solely by using the polynomials P (1) 2n and P (2) 2n , since both of them interpolate at x = 0. Thus we consider the following polynomials where c 1,α and c 2,α are numerical constants, depending only on α.As we see later, for good choices of c 1,α and c 2,α the linear combination of P 2n results in a polynomial with almost all the same interpolation points as its best approximation P * 2n , while at the same time the last term in (6.3) establishes the alternation property at x = 0 and leaves the new interpolation points largely unchanged.
Since we are interested into the asymptotic behavior of the polynomials P we directly pass to the resulting scaled limit.From Theorem 3.2, formulas (2.3), (6.1), (6.2) and Lemma 3.6, it follows that uniformly on compact subsets of [0, ∞) we have x L∞[0,∞) .
For the moment, we cannot present an explicit formula for the constants c 1,α and c 2,α , but based on numerical calculations, we present the following where α/2 is the least integer exceeding α/2.For the alternation points it is also known that y * j ∈ [(j − 1) π, jπ] , ∀j ≥ 1.We use formula (6.4) as an approximation for H * α .In Figure 8 we present some illustrations from (6.4) for α = 0.5 and α = 1.0.In Figure 9 we illustrate the near equioscillating behavior of the error term in (6.4), again for α = 0.5 and α = 1.0, and we compare the maximal error magnitude with the corresponding numerical values for the Bernstein constants ∆ ∞,0.5 = 0.348648 . . ., ∆ ∞,1 = 0.280169 . . .The values for the Bernstein constants are taken from ([21], Table 1.1).
In the following table we present the approximations for the best interpolation points x * j for j = 1, . . ., 10 from (6.4), respectively from Figure 8.The last table suggests that, for small positive values α, all interpolation points are slightly shifted to the left.Apparently this effect becomes greater for those interpolation points which are located closer to the origin.On the other hand, the values suggest that Finally, we remark that the overall quality of the P (3) n polynomials appears to be very encouraging in search for some representations of the Bernstein constants.Their approximation properties with respect to the corresponding best approximation polynomials P * n are of high quality, even for small values of n.Thus, formula 6.4 though it is at the present time not in its full explicit form, appears to be an important step towards a possible representation for the Bernstein constants ∆ ∞,α .

Figure 2 Lemma 4 . 1 .
Figure2shows the functions |H (α, •)| and H 1 (α, •) as well as their point evaluations for values α = 1.8 and α = 6.4.The figure suggests that a useful lower estimate for H 1 (α, •) should be derivable when determining its point evaluation, i.e.H 1 (α, α), at least for large values for α.We start proving Theorem 4.1 by splitting it in several Lemmas.First, we present the following five Lemmas without proof.They can be derived by some standard analysis arguments.Lemma 4.1.The function f (x) = 1 + 1 x

Figure 3
Figure3shows the functions |H (α, •)| and its envelope H 1 (α, •) together with the point evaluations H 1 (α, α) and |H (α, β)| = H 1 (α, β), where β = β (α) = π α π + 3 2 π and α = 3.9 and α = 8.4.Geometrically, the point β is the position of the first or the second relative maximum of |H (α, •)| on the right-hand side of α, where H 1 (α, •) appears to be descending.For growing values of α, the size of these maxima appear to be of the same magnitude compared to the size H 1 (α, α).We use both observations for the asymptotic analysis.First, we show that H 1 (α, •) is descending at least for values x ≥ α.Then, we derive the asymptotics for the local maximum in |H (α, β)|.It turns out that the following integral inequality plays an essential role.

0α
, because g has exactly one single minimum at a = 1.After verifying the conditions for the Watson Lemma ([14], Theorem 8.1) it allows us to expand the integral ∞ 1 into an asymptotic series of the form ∞ a f k,c (t) e −αg(t) dt e −αg(a) (n+λ)/µ , α → ∞, with certain coefficients λ, µ and a (k,c) n

Figure 5 :
Figure 5: Interpolation points for the best approximation to |x| α .

Figure 8 :
Figure 8: Approximations for best entire functions H * α of exponential type 1.