Abstract
In this paper we compute the logarithmic energy of points in the unit interval [-1,1] chosen from different Gegenbauer Determinantal Point Processes. We check that all the different families of Gegenbauer polynomials yield the same asymptotic result to third order, we compute exactly the value for Chebyshev polynomials and we give a closed expression for the minimal possible logarithmic energy. The comparison suggests that DPPs cannot match the value of the minimum beyond the third asymptotic term.
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We want to thank the anonymous referee for helpful comments.
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CB was partially supported by grants PID2020-113887GB-I00, MTM2017-83816-P and MTM2017-90682-REDT funded by MCIN/ AEI /10.13039/501100011033, and by Banco de Santander and Universidad de Cantabria grant 21.SI01.64658. AD and LF were partially supported by FEDER/Junta de Andalucía A-FQM-246-UGR20; grant PGC2018-094932-B-I00 and IMAG-María de Maeztu grant CEX2020-001105-M both funded by MCIN/ AEI /10.13039/501100011033 and FEDER funds. JSL was partially supported by FEDER/Junta de Andalucía A-FQM-246-UGR20; grant MTM2015-71352-P funded by MINECO; and IMAG-María de Maeztu grant CEX2020-001105-M funded by MCIN/ AEI /10.13039/501100011033 and FEDER funds.
Appendices
Appendix A: Bounds and Integrals Involving Gegenabuer Polynomials
In this appendix we state some technical lemmas that have been used in the proof of the main results. Recall from [22, Eq. (7.33.6)] the following: for n ≥ 1 and \(\lambda \in (-1/2,\infty )\),
for c > 0 fixed and some constant Q(λ). Observe that when λ ≥ 0, both inequalities, though less sharp, hold in [0,π]. The following lemma follows easily:
Lemma 1
Let λ > − 1/2 and n ≥ 1. Then, for x ∈ [− 1, 1] such that \(0\leq \arccos x\leq {2} n^{-1}\) (this holds in particular if 1 − n− 2 ≤ x ≤ 1), we have
Proof
Recall that in the range of \(x=\cos \limits \theta \), for k ≥ 1 we have
where we have used some standard estimates on the Gamma function. Then,
For all \(\lambda \in (-1/2,\infty )\), the sum is bounded above by Q(λ)n2λ+ 1, and we are done. □
The following is almost immediate:
Lemma 2
Let n > 1. There exists a constant c > 0 such that for any y ∈ [− 1, 1]:
If additionally we have λ ∈ (− 1/2, 0), then
and
Proof
Recall that
the last from Eq. 4. Since for any x,y ∈ [− 1, 1] we have \(\log \frac {|x-y|}{2}\leq 0\), we conclude
The first claim of the lemma follows. For the second one, let J be the integral we want to estimate. From Holder’s inequality,
The first of these two integrals is bounded above by some universal constant. The second one is at most
For the third integral in the lemma (that we denote I(λ,n)), again Holder’s inequality yields the upper bound (independent of y):
where q > 1 is any positive number such that (1 − 2λ)q < 2 and Q(q,λ) plays the same role as Q(λ) but in this case it may depend on q. We choose q = (3 − 2λ)/(2 − 4λ) and the third inequality follows. The last one is even more elementary:
□
Lemma 20
For λ > − 1/2, n ≥ 1 and 𝜃 ∈ [0,π]
where Q(λ) is some constant depending only on λ and
One can change \(\min \limits (1,T)\) to T/(T + 1) if desired.
Proof
The classical asymptotic results for Gegenbauer polynomials [22, Eq. (8.21.18)] yield:
valid for 𝜃 ∈ [n− 1,π − n− 1], where \( S_{n}^{\lambda }=\frac {\Gamma (n+2\lambda ){\Gamma }(\lambda +1/2)}{\Gamma (2\lambda ){\Gamma }(\lambda +n+1/2)}\) satisfies \(|S_{n}^{\lambda }|\leq Q(\lambda ) n^{\lambda -1/2}. \) Then, for the orthonormal polynomials we have
Using
and multiplying by \(\sqrt {w^{\lambda }}\) we get
so Eq. 19 is proved for 𝜃 ∈ [n− 1,π − n− 1].
Now if 𝜃 ∈ [0,n− 1] ∪ [π − n− 1,π], using Eq. 18 we obtain
which is bounded above by \(n\sin \limits \theta \) if λ ≥ 0 and by \((n\sin \limits \theta )^{\lambda +1}\) otherwise, so Eq. 19 is valid for 𝜃 ∈ [0,π]. □
Lemma 21
Let λ > − 1/2 and let n ≥ 2, then
where Q(λ) is some constant depending only on λ.
Proof
For \(\theta =\arccos (x)\in [0,\pi ]\) and k ≥ 1, by Lemma 20
where we have used Lemma 20 and that the normalization coefficient \(\gamma _{k}^{\lambda }\) behaves like k1−λ. Observe that this inequality is also valid for k = 0. Then, from Lemma 29 we conclude
If λ ≥ 0 we can bound the last sums by
If λ < 0 the corresponding bounds are
(the last by dividing the integration interval with midpoint \(1/\sin \limits \theta \) if n is greater than this quantity), and
as we wanted. The reciprocal inequality
is proved the same way (now, the error bounds have a minus sign). □
Lemma 22
Let λ = 0. Then,
Proof
With the change of variables \(x=\cos \limits \theta \), the integral of the lemma becomes
the last from lemmas 25 and 27. □
Lemma 23
The following equality holds for all \(\lambda \in (-1/2,0)\cup (0,\infty )\) and integer k ≥ 0:
In other words (from the definition of \(\widehat C_{k}^{\lambda }\) and wλ),
Proof
A change of variables x → 1 − x shows that the integral in the lemma equals
that has been computed in [23, Theorem 3]. The expression in [23, Theorem 3] and ours are equivalent (use [1, 6.3.8]). □
Appendix B: Some Integrals and Sums
We have used some technical results that we include here for the reader’s convenience.
Lemma 24
Proof
Let
where \(G(n)=(n-2)!(n-3)!{\dots } 1!\) is Barnes G-function, also called the double gamma function. The asymptotics of G(z) for \(z\to +\infty \) is known (see [13, Theorem 1] or [11, 5.17.5]):
We thus have proved that
□
Lemma 25
We have
Proof
First, translate the integration interval to [0, 2π] and after that combine the change of variables α = 2πx with [14, 4.384 (3)]. □
Lemma 26
Let \(f:[-1,1]^{2}\to \mathbb {R}\) be a continuous function. Then,
In particular, from Lemma 25, if \({\int \limits }_{-\pi }^{\pi }f\left (\cos \limits \theta ,\cos \limits (\theta +\alpha ))\right ) d\theta \) is constant (i.e. if its value does not depend on α) then the integral of the lemma is 0.
Proof
Denote by I the integral in the lemma. The change of variables \(x=\cos \limits \theta \), \(y=\cos \limits \phi \) yields
where we have used the following classical fact:
The change of variables ϕ →−ϕ shows that the last two integrals above are equal and hence we have
We write this last expression as an integral in the product [−π,π] × S1, where S1 is the unit circle, getting
We have applied the isometry S1 → S1 given by z → w = e−i𝜃z. Parametrizing again the unit circle by w = eiα we get to
proving the lemma. □
Lemma 27
For any integer k ≥ 1 we have
Moreover, for integers k > ℓ ≥ 1 we have
Proof
For the first integral, we proceed as in Lemma 25
For the second integral we consider
and we note that
and adding these two equalities gives the desired result. For the last claim we similarly consider
which readily gives
and again adding these equalities gives the last integral of the lemma. □
Lemma 28
For n ≥ 2,
Proof
This is an easy exercise of induction. □
Lemma 29
The following equality holds:
Proof
From the double angle formulas, the sum in the lemma is equal to A − B + C where
These three sums are known, see [14, Sec. 1.34, 1.35], yielding the following value for the sum in the lemma:
We are done. □
Lemma 30
The following identities hold:
Proof
First identity is direct consequence of [14, 1.341.3] and the identities for the product of a pair of trigonometric functions. Second identity is consequence of the identities for the product of two trigonometric functions, and first identity. □
Lemma 31
The following identity holds:
Proof
Using the change of variable \(x=\cos \limits (v)\) and taking into account that we can write \(\sqrt {1-x^{2}}=|x-1|^{1/2} \cdot |x+1|^{1/2}\) we get
where V (x) is the equilibrium measure potential, which constantly equals \(\log 2\). So the integral vanishes. □
Lemma 32
Let Si(t) the Integral Sine function. The following asymptotic expansion holds:
being γ the Euler constant.
Proof
We will obtain this identity by means of the imaginary part of a complex line integral. Let us take x > 0 and consider \(C_{x} = \{z\in \mathbb {C} : |z|=x; \textrm {Im } z>0\}\) parametrized counterclockwise. Then,
since we are integrating an entire function along a closed curve. We now split the integral over the semicircle within three other ones,
Let us work with each one of these integrals.
For the second integral, we parametrize Cx as t = xei𝜃 and after the change of variable z = xeiσ, standard computations lead to
which can be bounded as
Now we use the Jordan’s inequality: \(\sin \limits \sigma \geq 2\sigma /\pi \) for σ ∈ (0,π/2), getting
The last integral can be computed directly using the parametrization t = xei𝜃,
Then, Eq. 20 reads as
from where, taking imaginary part we get
Finally, in the second term we use the asymptotics of the integral cosine function ([1, 5.2.2, 5.2.9, 5.2.34 and 5.2.35]),
and then the announced result is proved. □
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Beltrán, C., Delgado, A., Fernández, L. et al. On Gegenbauer Point Processes on the Unit Interval. Potential Anal 60, 139–172 (2024). https://doi.org/10.1007/s11118-022-10045-6
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DOI: https://doi.org/10.1007/s11118-022-10045-6