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Japanese Journal of Statistics and Data Science

, Volume 2, Issue 2, pp 615–639 | Cite as

On almost tight Euclidean designs for rotationally symmetric integrals

  • Masatake Hirao
  • Masanori SawaEmail author
Original Paper Information Theory and Statistics

Abstract

We present a characterization theorem of almost tight Euclidean \((2e+1)\)-designs supported by \(\lfloor e/2 \rfloor + 2\) concentric circles in terms of roots of quasi-orthogonal polynomials. We also prove that for any \(e \ge 5\) there exist no almost tight \((2e + 1)\)-designs supported by \(\lfloor e/2 \rfloor + 2\) concentric circles for Gaussian integration. Our characterization theorem is an analogue of some previous works as such by Verlinden and Cools (Numer Math 61:395–407, 1992), Cools and Schmid (Numerical integration, IV, Oberwolfach, 1992, Birkhäuser, Basel, pp 57–66, 1993) and the present authors (2010), and the nonexistence theorem provides an answer to the inverse problem for Euclidean designs, posed by Bannai et al. (Eur J Combin 31:419–422, 2010). Furthermore, the present paper includes a short review of a relationship between Euclidean designs for rotationally symmetric integrals and kernel approximation in machine learning, together with some new observations.

Keywords

Almost tightness Cubature formula Euclidean design Gaussian design Inverse Problem 

1 Introduction

For real numbers ab with \(0 \le a < b\), let \(\varOmega = \{ \varvec{x}\in \mathbb {R}^d \mid a \le \Vert \varvec{x}\Vert < b \}\), where \(\Vert \varvec{x}\Vert = (\sum _{i = 1}^{d} x_i^2)^{1/2}\) for \(\varvec{x}= (x_1, \ldots , x_d) \in \mathbb {R}^d\). Let W be a rotationally (or spherically) symmetric probability density function on \(\varOmega\), namely W depends on norm \(\Vert \varvec{x}\Vert\). Under the assumption that polynomials up to a sufficiently large degree are integrable with respect to W, we consider an integral of the form
$$\begin{aligned} \mathcal {I}[f]&= \int _{\varOmega } f (\varvec{x}) W(\Vert \varvec{x}\Vert ) \; \mathrm{d} \varvec{x}\nonumber \\&= \frac{|S^{d-1}|}{2} \int _{a^{2}}^{b^{2}} \left( \int _{S^{d-1}} f (\sqrt{u} \varvec{x}) \; \mathrm{d} \rho (\varvec{x}) \right) u^{d/2- 1} W(\sqrt{u}) \; \mathrm{d} u \end{aligned}$$
(1)
where \(\rho\) is the normalized surface measure on the \((d-1)\)-dimensional unit sphere about the origin, say Sd − 1 and \(|S^{d-1}|\) is the surface area of \(S^{d-1}\).
Let X be a finite subset of \(\varOmega\) and \(\lambda\) be a positive weight function on X. For any integrand f, we denote by \(\mathcal {Q}[f]\) a weighted summation rule
$$\begin{aligned} \mathcal {Q}[f] = \sum _{\varvec{x}\in X} \lambda (\varvec{x}) f(\varvec{x}). \end{aligned}$$
A cubature formula of degreetfor\(\mathcal {I}\) is defined by pair \((X, \lambda )\) such that
$$\begin{aligned} \mathcal {I}[f] = \mathcal {Q}[f] \quad \text { for all} \quad {f \in \mathcal {P}_{t} (\mathbb {R}^d)}. \end{aligned}$$
Here \(\mathcal {P}_t (\mathbb {R}^d)\) is the space of real polynomials of degree at most t in d variables.

While the cubature formula is originally designed as a numerical integration formula for approximating values of integrals (cf. Stroud 1971), its role is not limited to that purpose. For example, a certain class of cubature formulas for rotationally symmetric integrals has been traditionally studied in the theory of optimal experimental designs (cf. Kiefer 1960; Neumaier and Seidel 1992; Hirao et al. 2014). Also, Dao et al. (2017) discuss applications of cubature formulas for Gaussian or sub-Gaussian integration to the theory of kernel approximations in machine learning; for more details, see Sect. 4 where some new related results will be established.

The Euclidean design is a generalization of the notion of cubature formula for rotationally symmetric integrals, which was first introduced by Neumaier and Seidel (1988). Given a pair \((X, \lambda )\), let \(\{ r_{1}, \ldots , r_{p} \} = \{ \Vert \varvec{x}\Vert \mid \varvec{x}\in X \}\); for convenience we let \(r_{1}> \cdots > r_{p} \ge 0\) and \(R_{i} = r_{i}^{2}\). Let \(S_{i}\) be the \((d-1)\)-dimensional sphere with radius \(r_{i}\) centered at the origin, and let \(\varLambda _{i} = \sum _{\varvec{x}\in X_{i}} \lambda (\varvec{x})\), where \(X_{i} = X \cap S_{i}\). Further let \(\rho _i\) be the normalized surface measure on \(S_{i}\); in particular, when \(r_p = 0\), we define \(\int _{S_{p}} f(\varvec{x}) \; \mathrm{d} \rho _p (\varvec{x}) = f ({\mathbf {0}})\). Pair \((X, \lambda )\) is called a Euclideant-design supported bypconcentric spheres\(S: = S_{1} \cup \dots \cup S_{p}\) if
$$\begin{aligned} \sum _{i = 1}^p \varLambda _{i} \int _{S_{i}} f (\varvec{x}) \; \mathrm{d} \rho _i (\varvec{x}) = \mathcal {Q}[f] \quad \text { for all}\quad {f \in \mathcal {P}_{t} (\mathbb {R}^d)}. \end{aligned}$$
In particular, when \(p=1\) and \(r_1 =1\), we use the term weighted sphericalt-design of \(S^{d-1}\). Note that Euclidean design is also known as “rotatable design” in experimental design theory (Neumaier and Seidel 1988, 1992).
It is well known (see, e.g., Bannai et al. 2010; Möller 1979) that the number of points in a Euclidean \((2e+1)\)-design X supported by p concentric spheres S is bounded below as follows:
$$\begin{aligned} |X| \ge \left\{ \begin{array}{ll} 2 \dim \mathcal {P}_{e}^{*} (S) - 1 &{} \quad \text {if}~e~\text {is even and}\, {{\mathbf {0}} \in X,} \\ 2 \dim \mathcal {P}_{e}^{*} (S) &{} \quad \text {otherwise}. \end{array} \right. \end{aligned}$$
(2)
Here \(\mathcal {P}_{e}^{*} (S)\) denotes the subspace of \(\mathcal {P}_{e} (S) := \{ f|_{S} \mid f \in \mathcal {P}_{e} (\mathbb {R}^d) \}\) consisting of all even polynomials or odd polynomials of degree at most e according to the parity of e. X is called a tight Euclidean\((2e+1)\)-design of\(\mathbb {R}^{d}\) if \(\dim \mathcal {P}_{e}^{*} (\mathbb {R}^{d}) = \dim \mathcal {P}_{e}^{*} (S)\) and the equality holds in (2); see Bannai (2006) for the computation of \(\dim \mathcal {P}_{e}^{*} (S)\) or \(\dim \mathcal {P}_{e}(S)\).

It is remarked (cf. Neumaier and Seidel 1988, Theorem 2.6; Hirao and Sawa 2009, Lemma 3.1) that if a pair \((X, \lambda )\) forms a cubature formula of degree t for a rotationally symmetric integral \(\mathcal {I}\), then it is a Euclidean t-design. For this reason, X is called a Euclidean design for\(\mathcal {I}\), or shortly a design for\(\mathcal {I}\). A \((2e+1)\)-design for \(\mathcal {I}\) is tight if it is a tight \((2e+1)\)-design of \(\mathbb {R}^{d}\). A number of papers have been devoted to the existence and characterization of tight designs for rotationally symmetric integrals (Bannai and Bannai 2005; Bannai et al. 2010, 2012; Cools and Schmid 1993; Haegemans 1975; Hirao and Sawa 2012; Verlinden and Cools 1992; Xu 1998).

Bannai et al. (2010) introduce the notion of almost tightness for Euclidean design. A Euclidean \((2e+1)\)-design for a rotationally symmetric integral, say \((X, \lambda )\), is said to be almost tight if
  1. (i)

    \({\mathbf {0}} \in X\), and

     
  2. (ii)

    \(X{\setminus} \{ {\mathbf {0}} \}\) with \(\lambda\) restricted to \(X{\setminus} \{ {\mathbf {0}}\}\) is a tight Euclidean \((2e+1)\)-design of \(\mathbb {R}^d\).

     
In the present paper, we mainly focus on two-dimensional almost tight \((2e + 1)\)-designs supported by at least “\(\lfloor e/2 \rfloor + 2\) concentric circles”. Value \(\lfloor e/2 \rfloor + 2\) is the minimum number for which almost tight \((2e + 1)\)-designs supported by p concentric circles can possibly exist (Bannai et al. 2010). Remarkably, there exist almost tight 7-designs (Bannai et al. 2010) and 9-designs (Haegemans 1975), but no examples of tight 7- and 9-designs, e.g., for Gaussian integration (Hirao and Sawa 2012; Verlinden and Cools 1992); nonexistence theorems of tight 2e-designs can be also found in Bannai et al. (2012). Note that Euclidean designs for Gaussian integration often appear in the context of statistical analysis and other related fields. For example, Lyons and Victoir (2004) employ Euclidean designs for Gaussian integration to numerically approximating expectations of functionals of solutions to stochastic differential equations, which are of great importance in mathematical finance applications. Also, Euclidean designs for Gaussian integration have been studied in combinatorics (cf. Bannai and Bannai 2005; Bannai et al. 2010), where degree \(2e+1\) is understood as a measure of the symmetry of a given point configuration in the Euclidean space.

We develop a theory of almost tight \((2e+1)\)-designs supported by \(\lfloor e/2 \rfloor +2\) concentric circles for rotationally symmetric integrals, including two main theorems: (i) a characterization theorem of almost tight \((2e+1)\)-designs supported by \(\lfloor e/2 \rfloor + 2\) concentric circles in terms of roots of quasi-orthogonal polynomials, and (ii) the nonexistence theorem of such \((2e + 1)\)-designs for Gaussian integration for \(e \ge 5\).

To explain statement (i), we need some additional notations. For a nonnegative integer \(\alpha\), let \(\mathcal {J}_{\alpha }\) be the integral defined by
$$\begin{aligned} \mathcal {J}_{\alpha } [f] = \pi \int _{a^{2}}^{b^{2}} f (u) u^{\alpha } W(\sqrt{u}) \; \mathrm{d} u. \end{aligned}$$
Here \(W(\sqrt{u})\) is a positive weight function on \([a^2,b^2)\). Let \(g_{k}^{\alpha } (u)\) be an orthogonal polynomial of degree k with respect to \(\mathcal {J}_{\alpha }\), and \(\kappa _{k}^{\alpha }\) be the leading coefficient of \(g_{k}^{\alpha }(u)\).

Theorem 1

(I) Letkbe a positive integer andXbe an almost tight Euclidean\((4k + 1)\)-design supported by\(k + 2\)concentric circles\(S_{1} \cup \cdots \cup S_{k +2}\) for \(\mathcal {I}\). Then the following hold:
  1. (i)

    \(\{ R_{1}, \ldots , R_{k+1} \}\)is the set of zeros of\(g_{k + 1}^{1} (u) + \gamma _{0} g_{k}^{1} (u) + \gamma _{1} g_{k - 1}^{1} (u)\)for some real numbers\(\gamma _{0}, \gamma _{1}\).

     
  2. (ii)
    $$\begin{aligned}&\; g_{k + 1}^{1} (u) + \gamma _{0} g_{k}^{1} (u) + \gamma _{1} g_{k - 1}^{1} (u)\\&\quad = \,\left\{ \begin{array}{ll} \displaystyle \frac{\kappa _{2 \tau + 1}^{1}}{\kappa _{\tau }^{2 \tau + 1} \kappa _{\tau + 1}^{2 \tau + 1}} g_{\tau }^{2 \tau + 1} (u) \big ( g_{\tau + 1}^{2 \tau + 1} (u) + \gamma _2 g_{\tau }^{2 \tau + 1} (u) + \gamma _3 g_{\tau - 1}^{2 \tau + 1} (u) \big ) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text {if}\; {k = 2 \tau ,} \\ \displaystyle \frac{\kappa _{2 \tau + 2}^{1}}{(\kappa _{\tau + 1}^{2 \tau + 2})^{2}} \big ( g_{\tau + 1}^{2 \tau + 2} (u) + \gamma _2 g_{\tau }^{2 \tau + 2} (u) \big ) \big ( g_{\tau + 1}^{2 \tau + 2} (u) + \gamma _3 g_{\tau }^{2 \tau + 2} (u) \big ) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \text {if} \;{k = 2 \tau + 1} \end{array} \right. \end{aligned}$$
    where\(\gamma _0, \gamma _1, \gamma _2, \gamma _{3}\)are some real numbers given in each case.
     
(II) Letkbe a positive integer andXbe an almost tight Euclidean\((4k + 3)\)-design supported by\(k + 2\)concentric circles\(S_{1} \cup \cdots \cup S_{k + 2}\) for \(\mathcal {I}\). Then the following hold:
  1. (i)

    \(\{ R_{1}, \ldots , R_{k + 1} \}\)is the set of zeros of\(g_{k +1}^{1} (u) + \gamma _{0} g_{k}^{1} (u)\)for some real number\(\gamma _{0}\).

     
  2. (ii)
    $$\begin{aligned}&\; g_{k + 1}^{1}(u) + \gamma _{0} g_{k}^{1} (u)\\&\quad = \,\left\{ \begin{array}{ll} \displaystyle \frac{\kappa _{2 \tau + 1}^{1}}{\kappa _{\tau }^{2 \tau + 2} \kappa _{\tau + 1}^{2 \tau + 2}} g_{\tau }^{2 \tau + 2} (u) \big ( g_{\tau + 1}^{2 \tau + 2} (u) + \gamma _1 g_{\tau }^{2 \tau + 2} (u) + \gamma _2 g_{\tau - 1}^{2 \tau + 2} (u) \big ) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \text {if}\; {k = 2 \tau ,} \\ \displaystyle \frac{\kappa _{2 \tau + 2}^{1}}{(\kappa _{\tau + 1}^{2 \tau + 3})^{2}} \big ( g_{\tau + 1}^{2 \tau + 3} (u) + \gamma _1 g_{\tau }^{2 \tau + 3} (u) \big ) \big ( g_{\tau + 1}^{2 \tau + 3} (u) + \gamma _2 g_{\tau }^{2 \tau + 3} (u) \big ) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text {if} \;{k = 2 \tau + 1} \end{array} \right. \end{aligned}$$
    where\(\gamma _{0}, \gamma _1, \gamma _2\)are some real numbers given in each case.
     

Theorem 1 characterizes the radius set of almost tight designs in terms of quasi-orthogonal polynomials; see Xu (1994) for the precise definition of quasi-orthogonal polynomials. This is an analogue of some previous results concerning tight designs as such by Verlinden and Cools (1992), Cools and Schmid (1993) and Hirao and Sawa (2012).

As briefly mentioned above, the following is another main theorem in the present paper.

Theorem 2

(i) Let\(k \ge 3\)be an integer. Then there exists no almost tight Euclidean\((4k + 1)\)-design supported by\(k + 2\)concentric circles for Gaussian integration. (ii) Let\(k \ge 2\)be an integer. Then there exists no almost tight Euclidean\((4k + 3)\)-design supported by\(k + 2\)concentric circles for Gaussian integration.

Bannai et al. (2010) pose the inverse problem for Euclidean designs: Given a Euclidean t-design \((X, \lambda )\) and a rotationally symmetric integral \(\mathcal {I}\), does \((X, \lambda )\) form a cubature formula of degree t for \(\mathcal {I}\)? Theorem 2 gives a negative answer to this problem for Gaussian integration.

In Sects. 2 and 3 we prove Theorems 1 and 2, respectively. Section 4 is closed with a short review of a relationship between Euclidean designs for rotationally symmetric integral and kernel approximation in machine learning, together with some new observations.

2 Proof of Theorem 1

The following lemma determines the point configuration of almost tight Euclidean designs for two-dimensional rotationally symmetric integral \(\mathcal {I}\).

Lemma 1

An almost tight Euclidean\((2e + 1)\)-design supported by\(\lfloor e/2 \rfloor + 2\)concentric circles\(S_1 \cup \cdots \cup S_{\lfloor e/2 \rfloor + 2}\)for\(\mathcal {I}\)is uniquely determined, up to orthogonal transformation, as follows:
$$\begin{aligned} \mathcal {Q}[f]&= \sum _{j = 1}^{\lfloor e/2 \rfloor + 1} \frac{\varLambda _j}{h} \sum _{l = 0}^{h - 1} f \Big ( \sqrt{R_{j}} \cos \Big ( \frac{j + 2 l}{h} \pi \Big ), \sqrt{R_{j}} \sin \Big ( \frac{j + 2 l}{h} \pi \Big ) \Big ) \\&\quad + \,\varLambda _{\lfloor e/2 \rfloor + 2} f (0, 0) \end{aligned}$$
with\(h = 2 (e - \lfloor e/2 \rfloor + 1)\)and for each\(j = 2, \ldots , \lfloor e/2 \rfloor + 1\),
$$\begin{aligned} \varLambda _{j}&= \frac{R_1^{e - \lfloor e/2 \rfloor + 1}}{R_j^{e - \lfloor e/2 \rfloor + 1}} \frac{\prod _{i=2}^{j-1}(R_1 - R_i) \prod _{i=j+1}^{\lfloor e/2 \rfloor + 1}(R_1 - R_i)}{\prod _{i=2}^{j-1}(R_i - R_j) \prod _{i=j+1}^{\lfloor e/2 \rfloor + 1} (R_j - R_i) } \varLambda _{1}, \\ 1&= \varLambda _{1} + \cdots \varLambda _{\lfloor e/2 \rfloor + 2}. \end{aligned}$$

Proof

The result immediately follows by Theorems 3.1.2 and 3.1.3 in Bannai et al. (2010) (see also Bajnok 2006) and the definition of almost tightness. \(\square\)

The next lemma (Lemma 2) makes a relationship between two-dimensional Euclidean design for \(\mathcal {I}\) and one-dimensional cubature called quadrature formula. For reals ab with \(0 \le a < b\), let Y be a finite subset of \([a^{2}, b^{2})\) and \(\omega\) be a positive weight function on Y. A quadrature formula of degreetfor\(\mathcal {J}_{\alpha }\) is defined by pair \((Y, \omega )\) such that
$$\begin{aligned} \mathcal {J}_{\alpha } [f] = \sum _{\upsilon \in Y} \omega (\upsilon ) f (\upsilon ) \quad \text {for all}\quad {f \in \mathcal {P}_t (\mathbb {R})}. \end{aligned}$$
When the elements of Y and the values of \(\omega\) are indexed by \(Y = \{ \upsilon _{1}, \ldots , \upsilon _{n} \}\) and \(\omega (\upsilon _{i}) = \omega _{i}\), we may use the notation \((\{ \upsilon _{1}, \ldots , \upsilon _{n} \}\), \(\{ \omega _{1}, \ldots , \omega _{n} \})\) for \((Y, \omega )\).

Lemma 2

Assume that\((X, \lambda )\)is a Euclidean\((2e + 1)\)-design supported bypconcentric circles\(S_{1} \cup \cdots \cup S_{p}\)for\(\mathcal {I}\), where\(S_{p} = \{ (0, 0) \}\). Letsbe an integer with\(0 \le s < e\). Then\((\{ R_{1}, \ldots , R_{p - 1} \}\), \(\{ \varLambda _{1} R_{1}^{s + 1}, \ldots , \varLambda _{p-1} R_{p-1}^{s + 1} \})\)forms a quadrature formula of degree\(e - s - 1\) for \(\mathcal {J}_{s + 1}\).

Proof

Note that for any \(l = 0, \ldots , e - s - 1\),
$$\begin{aligned} \mathcal {J}_{s+1} [u^{l}]&= \pi \int _{a^{2}}^{b^{2}} u^{l} \cdot u^{s+1} W (\sqrt{u}) \; \mathrm{d} u \\&= \mathcal {I}[(x^{2} + y^{2})^{l + s +1}] = \sum _{(\xi , \eta ) \in X} \lambda (\xi , \eta ) (\xi ^{2} + \eta ^{2})^{l + s + 1}. \end{aligned}$$
Since \(R_p=0\), the last term is written by \(\sum _{i = 1}^{p- 1} ( \varLambda _{i} R_{i}^{s + 1} ) R_{i}^{l}\). \(\square\)

The following result can be found in Hirao and Sawa (2012), which originally goes back to p. 23 of Riesz (1923) for \(q-s = 2p-1\) or \(2p-1\) and Theorem 1 of Shohat (1937) for the remaining cases [see also Theorem 2.8 of Sawa and Uchida (2019)]. Polynomials of type (3) are called quasi-orthogonal polynomials (of order\(2p-2+s-1\)) (Xu 1994).

Lemma 3

Letpqbe positive integers andsbe a nonnegative integer such that\(p - 1 \le q - s \le 2 p - 1\). Assume that\((\{{\tilde{R}}_{1},\ldots ,{\tilde{R}}_{p}\},\{{\tilde{\varLambda }}_{1} {\tilde{R}}_{1}^{s},\ldots ,{\tilde{\varLambda }}_{p}{\tilde{R}}_{p}^{s}\})\) forms a quadrature formula of degree\(q - s\)for\(\mathcal {J}_{s}\). Then\(\{ {\tilde{R}}_{1}, \ldots , {\tilde{R}}_{p} \}\)is the set of zeros of
$$\begin{aligned} c_{0} g_{p}^{s} (u) + \cdots + c_{2p - 1 + s - q} g^{s}_{q - s + 1 - p} (u) \end{aligned}$$
(3)
for some real numbers\(c_{0}, \ldots , c_{2p - 1 + s - q}\).

Remark 1

It is easily seen that \(c_{0}\) in Lemma 3 is not zero. Hereafter we assume that \(c_{0} = 1/\kappa _{p}^{s}\) when using Lemma 3.

The following lemma (Lemma 4) can be found in Lemma 2 of Verlinden and Cools (1992).

Lemma 4

(cf. Verlinden and Cools 1992) Let\(\mathcal {I}\)be a two-dimensional rotationally symmetric integral defined in (1). For distinct nonnegative integers\(\alpha , \beta\), it holds that
$$\begin{aligned} \mathcal {I}[\left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta }] = 0. \end{aligned}$$

Proof

Using the Euler formula, it follows that
$$\begin{aligned} \left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta } = r^{\alpha } e^{\sqrt{-1} \alpha \theta } r^{\beta } e^{- \sqrt{-1} \beta \theta } = r^{\alpha + \beta } e^{\sqrt{-1} (\alpha - \beta ) \theta } \end{aligned}$$
where \(r = \sqrt{x^2 + y^2}\) and \(\theta\) is argument of \(x + \sqrt{-1}y\). Then it holds that
$$\begin{aligned}&\mathcal {I}[\left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta }] \\&\quad = \pi \int _{a^2}^{b^2} u^{(\alpha + \beta )/2} W(\sqrt{u}) \; \mathrm{d }u \int _{0}^{2 \pi } e^{\sqrt{-1} (\alpha - \beta ) \theta } \; \mathrm{d} \theta = 0. \end{aligned}$$
\(\square\)

We are now ready to prove Theorem 1.

Proof of Theorem 1

((I)-(i)) By Lemma 2, \((\{ R_{1}, \ldots , R_{k + 1} \}, \{ \varLambda _{1} R_{1}\), \(\ldots , \varLambda _{k + 1} R_{k + 1} \})\) forms a quadrature formula of degree \(2k - 1\) for \(\mathcal {J}_{1}\). Apply Lemma 3 when \(p = k + 1\), \(q = 2k\) and \(s = 1\). Then by Remark 1, \(R_{1}, \ldots , R_{k + 1}\) are the zeros of \(g_{k + 1}^{1}(u) + \gamma _{0} g_{k}^{1}(u) + \gamma _{1} g_{k - 1}^{1} (u)\) for some \(\gamma _{0}, \gamma _{1}\).

((I)-(ii)) By Lemma 1, an almost tight Euclidean \((4k + 1)\)-design supported by \(k + 2\) concentric circles \(S_{1} \cup \dots \cup S_{k+2}\) for \(\mathcal {I}\) has the form
$$\begin{aligned} \mathcal {Q}[f]&= \sum _{j = 1}^{k + 1} \frac{\varLambda _j}{2k + 2} \sum _{l = 0}^{2 k + 1} f \Big ( \sqrt{R_{j}} \cos \Big ( \frac{j + 2 l}{2 k + 2} \pi \Big ), \sqrt{R_{j}} \sin \Big ( \frac{j + 2 l}{2 k + 2} \pi \Big ) \Big ) \\&\quad +\, \varLambda _{k + 2} f (0, 0). \end{aligned}$$
For distinct nonnegative integers \(\alpha , \beta\) with \(0 < \alpha + \beta \le 4k + 1\), let \(f(x, y) = \left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta }\). Then we have
$$\begin{aligned} \mathcal {I}[\left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta }] = Q[\left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta }]. \end{aligned}$$
Hence, by Lemma 4, it follows that
$$\begin{aligned} 0&= \mathcal {I}[\left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta }] \nonumber \\&= Q[\left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta }] \nonumber \\&= \sum _{j = 1}^{k + 1} \frac{\varLambda _j}{2 k + 2} \sum _{l = 0}^{2 k + 1} R_j^{(\alpha + \beta )/2} \nonumber \\&\quad \times \,\Big ( \cos \Big ( \frac{(j + 2 l)(\alpha - \beta )}{2 k + 2} \pi \Big ) + \sqrt{-1} \sin \Big ( \frac{(j + 2 l)(\alpha - \beta )}{2 k + 2} \pi \Big ) \Big ) \nonumber \\&= \sum _{j = 1}^{k + 1} \frac{\varLambda _j}{2 k + 2} R_j^{(\alpha + \beta )/2} e^{\sqrt{-1} j (\alpha - \beta ) \pi /(2 k + 2)} \sum _{l = 0}^{2 k + 1} e^{\sqrt{-1} l (\alpha - \beta ) \pi / (k + 1)}. \end{aligned}$$
(4)
If \(2 k + 2\) does not divide \(\alpha - \beta\), then this equation is automatically satisfied. So we consider the case that \(\alpha = 2 k + 2 + m, \beta = m\), where \(m = 0, \ldots , k - 1\). Since
$$\begin{aligned} 0 = \sum _{j = 1}^{k + 1} \varLambda _{j} R_{j}^{k + 1 + m} e^{\sqrt{-1} j \pi } = \sum _{j = 1}^{k + 1} (-1)^{j} (\varLambda _{j} R_{j}^{k + 1} ) R_{j}^{m}, \end{aligned}$$
(5)
we have
$$\sum\limits_{\begin{subarray}{l} 1 \le j \le k + 1 \\ j:{\text{even}} \end{subarray} } {(\Lambda _{j} R_{j}^{{k + 1}} )R_{j}^{m} } = \sum\limits_{\begin{subarray}{l} 1 \le j \le k + 1 \\ j:{\text{odd}} \end{subarray} } {(\Lambda _{j} R_{j}^{{k + 1}} )R_{j}^{m} }$$
(6)
On the other hand, by Lemma 2, we have
$$\begin{aligned} \sum _{j = 1}^{k + 1} ( \varLambda _j R_j^{k + 1} ) R_j^{m} = \mathcal {J}_{k + 1} [u^{m}], \quad m = 0, \ldots , k - 1. \end{aligned}$$
(7)
By combining (6) and (7), we find that
$$\begin{aligned} \mathcal {J}_{k + 1} [u^{m}] = \sum _{\begin{subarray}{c} 1 \le j \le k + 1 \\ j: \text {even} \end{subarray}} (2 \varLambda _j R_j^{k + 1}) R_j^m = \sum _{\begin{subarray}{c} 1 \le j \le k + 1 \\ j: \text {odd} \end{subarray}} (2 \varLambda _j R_j^{k + 1}) R_j^m. \end{aligned}$$
(8)
For \(k \ge 1\) this implies that the \(R_i\), i being even, are the points of a quadrature formula (8) of degree \(k - 1\) for \(\mathcal {J}_{k + 1}\). Similarly, the \(R_i\), i being odd, are the points of a quadrature formula (8) of degree \(k - 1\) for \(\mathcal {J}_{k + 1}\). Thus, by applying Lemma 3 to (8) and using the assertion (I)-(i) above, the assertion (ii) is obtained.

((II)-(i)) By Lemma 2, \((\{ R_{1}, \ldots , R_{k + 1} \}, \{ \varLambda _{1} R_{1}, \ldots , \varLambda _{k + 1} R_{k + 1} \})\) forms a quadrature formula of degree 2k for \(\mathcal {J}_{1}\). Apply Lemma 3 to the case when \(p = k + 1\), \(q = 2k+1\) and \(s = 1\). Then by Remark 1, \(R_{1}, \ldots , R_{k + 1}\) are the zeros of \(g_{k + 1}^{1}(u) + \gamma _{0} g_{k}^{1} (u)\).

((II)-(ii)) By Lemma 1, an almost tight Euclidean \((4k + 3)\)-design supported by \(k + 2\) concentric circles \(S_{1} \cup \dots \cup S_{k + 2}\) for \(\mathcal {I}\) has the form
$$\begin{aligned} \mathcal {Q}[f]&= \sum _{j = 1}^{k + 1} \frac{\varLambda _j}{2k + 4} \sum _{l = 0}^{2 k + 3} f \Big ( \sqrt{R_{j}} \cos \Big ( \frac{j + 2 l}{2 k + 4} \pi \Big ), \sqrt{R_{j}} \sin \Big ( \frac{j + 2 l}{2 k + 4} \pi \Big ) \Big ) \\&\quad + \,\varLambda _{k + 2} f (0, 0). \end{aligned}$$
For distinct nonnegative integers \(\alpha , \beta\) with \(0 < \alpha + \beta \le 4k + 3\), let \(f(x, y) = \left(x + \sqrt{-1} y\right)^{\alpha } \left(x - \sqrt{-1} y\right)^{\beta }\). By the same arguments as in (4), it holds that
$$\begin{aligned} \sum _{j = 1}^{k + 1} \frac{\varLambda _j}{2 k + 4} R_j^{(\alpha + \beta )/2} e^{\sqrt{-1} j (\alpha - \beta ) \pi / (2 k + 4)} \sum _{l = 0}^{2 k + 3} e^{\sqrt{-1} l (\alpha - \beta ) \pi / (k + 2)} = 0. \end{aligned}$$
If \(2 k + 4\) does not divide \(\alpha - \beta\), then this equation is automatically satisfied. So we consider the case that \(\alpha = 2 k + 4 + m, \beta = m\), where \(m = 0, \ldots , k - 1\). Since
$$\begin{aligned} 0 = \sum _{j = 1}^{k + 1} \varLambda _{j} R_{j}^{k + 2 + m} e^{\sqrt{-1} j \pi } = \sum _{j = 1}^{k + 1} (-1)^{j} (\varLambda _{j} R_{j}^{k + 2} ) R_{j}^{m}, \end{aligned}$$
we have
$$\begin{aligned} \sum _{\begin{subarray}{c} 1 \le j \le k + 1 \\ j: \text {even} \end{subarray}} ( \varLambda _j R_j^{k + 2} ) R_j^m = \sum _{\begin{subarray}{c} 1 \le j \le k + 1 \\ j: \text {odd} \end{subarray}} ( \varLambda _j R_j^{k + 2} ) R_j^m. \end{aligned}$$
(9)
On the other hand, by Lemma 2, we have
$$\begin{aligned} \sum _{j = 1}^{k + 1} ( \varLambda _j R_j^{k + 2} ) R_j^{m} = \mathcal {J}_{k + 2} [u^{m}], \quad m = 0, \ldots , k-1. \end{aligned}$$
(10)
By combining (9) and (10), we see that
$$\begin{aligned} \mathcal {J}_{k + 2} [u^{m}] = \sum _{\begin{subarray}{c} 1 \le j \le k + 1\\ j: \text {even} \end{subarray}} (2 \varLambda _j R_j^{k + 2}) R_j^m = \sum _{\begin{subarray}{c} 1 \le j \le k + 1\\ j: \text {odd} \end{subarray}} (2 \varLambda _j R_j^{k + 2}) R_j^m. \end{aligned}$$
(11)
For \(k \ge 1\) this implies that the \(R_i\), i being even, are the points of a quadrature formula (11) of degree \(k - 1\) for \(\mathcal {J}_{k + 2}\). Similarly the \(R_i\), i being odd, are the points of a quadrature formula (11) of degree \(k - 1\) for \(\mathcal {J}_{k + 2}\). Thus, by applying Lemma 3 to (11) and using the assertion (II)-(i) above, the assertion (ii) is obtained. \(\square\)

Remark 2

Theorem 1 can be thought of as an analogue of some previous results concerning tight designs as such by Cools and Schmid (1993) and the present authors Hirao and Sawa (2012).

3 Proof of Theorem 2

Let us consider Gaussian integration of the form
$$\begin{aligned} \mathcal {I}[f] = \int _{\mathbb {R}^{2}} f(x,y) \pi ^{-1} e^{- (x^{2} + y^{2})} \; \mathrm{d} x \mathrm{d} y. \end{aligned}$$
Before proceeding to the proof of Theorem 2, we prepare some lemmas. Note that \(\mathcal {J}_{\alpha }\) is expressed as
$$\begin{aligned} \mathcal {J}_{\alpha } [f] = \int _{0}^{\infty } f(u) u^{\alpha } e^{- u} \; \mathrm{d} u \end{aligned}$$
since the probability density function of \(\mathcal {I}\) is \(\pi ^{-1} \exp (- (x^{2} + y^{2}))\). The orthogonal polynomials with respect to \(\mathcal {J}_{\alpha }\) are known as Laguerre polynomials with parameter \(\alpha\). The Laguerre polynomial of degreekwith parameter\(\alpha\), denoted by \(L_{k}^{\alpha } (u)\), is explicitly given by
$$\begin{aligned} L_{k}^{\alpha } (u) = \sum _{j = 0}^{k} (-1)^{j} \left( {\begin{array}{c}\alpha + k\\ k - j\end{array}}\right) \frac{u^{j}}{j!} \end{aligned}$$
and satisfies the orthogonality relation
$$\begin{aligned} \mathcal {J}_{\alpha } [L_{k}^{\alpha } L_{l}^{\alpha }] = \left\{ \begin{array}{ll} 0 &{} \quad \text {if} \;{k \ne l, } \\ (\alpha + k)!/k! &{} \quad \text {if} \;{k = l}. \\ \end{array} \right. \end{aligned}$$
(12)
The following identity is well known:
$$\begin{aligned} L_k^{\alpha } (u) =&\ \sum _{j = 0}^{k} \left( {\begin{array}{c}\alpha - \beta + j - 1\\ j\end{array}}\right) L_{k - j}^{\beta } (u) \end{aligned}$$
(13)
where \(\alpha > \beta\); for more information about Laguerre polynomials, see (Szegő 1975, Chapter 5.1).

Lemma 5

(I) Letkbe a positive integer andXbe an almost tight Euclidean\((4k + 1)\)-design supported by\(k + 2\)concentric circles\(S_{1} \cup \cdots \cup S_{k + 2}\)for Gaussian integration\(\mathcal {I}\). Then the following hold:
  1. (i)

    \(\{ R_{1}, \ldots , R_{k+1} \}\)is the set of zeros of\(L_{k + 1}^{1} (u) + \gamma _{0} L_{k}^{1} (u) + \gamma _{1} L_{k -1}^{1} (u)\)for some real numbers\(\gamma _{0}, \gamma _{1}\).

     
  2. (ii)
    $$\begin{aligned}&\; L_{k + 1}^{1}(u) + \gamma _{0} L_{k}^{1} (u) + \gamma _{1} L_{k -1}^{1} (u)\\&\quad = \, \left\{ \begin{array}{ll} \frac{\tau ! (\tau + 1)!}{(2 \tau + 1)!} L_{\tau }^{2 \tau + 1} (u) \big ( L_{\tau + 1}^{2 \tau + 1} (u) + \gamma _2 L_{\tau }^{2 \tau + 1} (u) + \gamma _3 L_{\tau - 1}^{2 \tau + 1} (u) \Big ) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text {if}\; {k = 2 \tau ,} \\ \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \Big ( L_{\tau + 1}^{2 \tau + 2} (u) + \gamma _2 L_{\tau }^{2 \tau + 2} (u) \Big ) \Big ( L_{\tau + 1}^{2 \tau + 2} (u) + \gamma _3 L_{\tau }^{2 \tau + 2} (u) \Big ) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \text {if}\; {k = 2 \tau + 1} \end{array} \right. \end{aligned}$$
    where\(\gamma _0, \gamma _1, \gamma _2, \gamma _{3}\)are some real numbers given in each case.
     
(II) Letkbe a positive integer andXbe an almost tight Euclidean\((4k + 3)\)-design supported by\(k + 2\)concentric circles\(S_{1} \cup \dots \cup S_{k+2}\)for Gaussian integration\(\mathcal {I}\). Then the following hold:
  1. (i)

    \(\{ R_{1}, \ldots , R_{k+1} \}\)is the set of zeros of\(L_{k + 1}^{1} (u) + \gamma _{0} L_{k}^{1} (u)\)for some real number\(\gamma _{0}\).

     
  2. (ii)
    $$\begin{aligned}&\; L_{k + 1}^{1}(u) + \gamma _{0} L_{k}^{1} (u)\\&\quad = \,\left\{ \begin{array}{ll} \frac{\tau ! (\tau + 1)!}{(2 \tau + 1)!} L_{\tau }^{2 \tau + 2} (u) \Big ( L_{\tau + 1}^{2 \tau + 2} (u) + \gamma _1 L_{\tau }^{2 \tau + 2} (u) + \gamma _2 L_{\tau - 1}^{2 \tau + 2} (u) \Big ) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text {if}\;{k = 2 \tau ,} \\ \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \Big ( L_{\tau + 1}^{2 \tau + 3} (u) + \gamma _1 L_{\tau }^{2 \tau + 3} (u) \Big ) \Big ( L_{\tau + 1}^{2 \tau + 3} (u) + \gamma _2 L_{\tau }^{2 \tau + 3} (u) \Big ) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \text {if}\; {k = 2 \tau + 1} \end{array} \right. \end{aligned}$$
    where\(\gamma _{0}, \gamma _1, \gamma _2\)are some real numbers given in each case.
     

Proof

The result immediately follows from Theorem 1. \(\square\)

Lemma 6

Let\(\alpha , \beta\)be integers such that\(0 \le \alpha \le \beta\). Then the following identities hold:
  1. (i)

    \(\displaystyle \sum _{j=0}^\alpha \frac{(\beta -j)!}{(\alpha - j)!} = \frac{(\beta +1)!}{\alpha ! (\beta - \alpha + 1)}\),

     
  2. (ii)

    \(\displaystyle \sum _{j=0}^\alpha j \frac{(\beta - j)!}{(\alpha - j)!} = \frac{(\beta +1)!\alpha }{\alpha !(\beta - \alpha + 1)(\beta - \alpha + 2)}\),

     
  3. (iii)

    \(\displaystyle \sum _{j=0}^{\alpha } j^2 \frac{(\beta - j)!}{(\alpha - j)!} = \frac{(\beta + 1)!(\beta + \alpha + 1)\alpha }{\alpha !(\beta - \alpha + 1)(\beta - \alpha + 2)(\beta - \alpha + 3)}\),

     
  4. (iv)

    \(\displaystyle \sum _{j = 0}^{\alpha } j^3 \frac{(\beta - j)!}{(\alpha - j)!} = \frac{(\beta + 1)! (5 \alpha + \alpha ^{2} + \beta + 4 \alpha \beta + \beta ^{2}) \alpha }{\alpha !(\beta - \alpha + 1)(\beta - \alpha + 2)(\beta - \alpha + 3)(\beta - \alpha + 4)}\),

     
  5. (v)

    \(\displaystyle \sum _{j = 0}^{\alpha } j^4 \frac{(\beta - j)!}{(\alpha - j)!} = \frac{(\beta + 1)! (\beta + \alpha + 1)(-4 + 15 \alpha + \alpha ^{2} - 3 \beta + 10 \alpha \beta + \beta ^{2})\alpha }{\alpha !(\beta - \alpha + 1)(\beta - \alpha + 2)(\beta - \alpha + 3)(\beta - \alpha + 4)(\beta -\alpha + 5)}\)

     
where for our convenience we let\(0! = 1\).

Remark 3

For example, Lemma 6 (i) is a special case of the Gauss sum. Similarly (ii)–(v) may be well known somewhere else, but the authors could not find appropriate literature. Nevertheless, Lemma 6 can be readily proved by elementary calculations of combinatorial identities.

Let us now prove Theorem 2.

Proof of Theorem 2(i)

Assume that there exists an almost tight Euclidean \((4k+1)\)-design supported by \(k + 2\) concentric circles \(S_{1} \cup \cdots \cup S_{k + 2}\) for Gaussian integration \(\mathcal {I}\). The proof is divided into two cases.

The first case to consider is that \(k = 2 \tau + 1, \tau \ge 1\). It follows from (12), (13) and Lemma 5 (I)-(ii) that
$$\begin{aligned} 0&= \int _{0}^{\infty } \Big ( L_{k +1}^{1} (u) + \gamma _{0} L_{k}^{1} (u) + \gamma _{1} L_{k -1}^{1} (u) \Big ) u^{2 \tau - 1} \cdot u e^{- u} \; \mathrm{d} u \\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!}\int _{0}^{\infty } \Big (L_{\tau + 1}^{2 \tau + 2} (u) + \gamma _{2} L_{\tau }^{2 \tau + 2} (u) \Big ) \\&\quad \times \, \Big (L_{\tau + 1}^{2 \tau + 2} (u) + \gamma _{3} L_{\tau }^{2 \tau + 2} (u) \Big ) u^{2 \tau } e^{- u} \; \mathrm{d} u \\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \Bigg \{ \sum _{j = 0}^{\tau + 1} (j + 1)^{2} \int _{0}^{\infty } L_{\tau + 1 - j}^{2 \tau }(u)^{2} \cdot u^{2 \tau } e^{- u} \; \mathrm{d} u \\&\quad +\, (\gamma _{2} + \gamma _{3}) \sum _{j = 0}^{\tau } (j + 1) (j + 2) \int _{0}^{\infty } L_{\tau - j}^{2\tau } (u)^{2} \cdot u^{2 \tau } e^{ - u} \; \mathrm{d} u \\&\quad +\, \gamma _{2} \gamma _{3} \sum _{j = 0}^{\tau } (j + 1)^{2} \int _{0}^{\infty } L_{\tau - j}^{2 \tau } (u)^{2} \cdot u^{2 \tau } e^{- u} \; \mathrm{d} u\Bigg \} \\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \Bigg \{ \sum _{j = 0}^{\tau + 1} (j + 1)^{2} \frac{(3 \tau + 1 - j)!}{(\tau + 1 - j)!} \\&\quad +\, (\gamma _{2} + \gamma _{3}) \sum _{j = 0}^{\tau } (j + 1) (j + 2) \frac{(3 \tau - j)!}{(\tau - j)!} + \gamma _{2} \gamma _{3} \sum _{j = 0}^{\tau } (j + 1)^{2} \frac{(3 \tau - j)!}{(\tau - j)!} \Bigg \}. \end{aligned}$$
By Lemma 6, this can be reduced to the equation
$$\begin{aligned} 0 = 3 (5 + 4 \tau ) + 6 (1 + \tau ) (\gamma _{2} + \gamma _{3}) + (3 + 4 \tau ) \gamma _{2} \gamma _{3}. \end{aligned}$$
(14)
Moreover, it follows from (12), (13) and Lemma 5 (I)-(ii) that
$$\begin{aligned} 0&= \int _{0}^{\infty } \Big ( L_{k +1}^{1} (u) + \gamma _{0} L_{k}^{1} (u) + \gamma _{1} L_{k -1}^{1} (u) \Big ) u^{2 \tau - 2} \cdot u e^{- u} \; \mathrm{d} u \\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \int _{0}^{\infty } (L_{\tau + 1}^{2 \tau + 2} (u) + \gamma _{2} L_{\tau }^{2 \tau + 2} (u)) \\&\quad \times \, (L_{\tau + 1}^{2 \tau + 2} (u) + \gamma _{3} L_{\tau }^{2 \tau + 2} (u)) u^{2 \tau - 1} e^{- u} \; \mathrm{d} u \\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \Bigg \{ \sum _{j = 0}^{\tau + 1} \frac{(j + 1)^{2} (j + 2)^{2}}{4} \int _{0}^{\infty } L_{\tau + 1 - j}^{2 \tau - 1}(u)^{2} \cdot u^{2 \tau -1} e^{- u} \mathrm{d} u \\&\quad +\, (\gamma _{2} + \gamma _{3}) \sum _{j = 0}^{\tau } \frac{(j + 1) (j + 2)^{2} (j + 3)}{4} \int _{0}^{\infty } L_{\tau - j}^{2\tau -1} (u)^{2} \cdot u^{2 \tau - 1} e^{ - u} \; \mathrm{d} u \\&\quad +\, \gamma _{2} \gamma _{3} \sum _{j = 0}^{\tau } \frac{(j + 1)^{2} (j + 2)^{2}}{4} \int _{0}^{\infty } L_{\tau - j}^{2 \tau - 1} (u)^{2} \cdot u^{2 \tau - 1} e^{- u} \; \mathrm{d} u\Bigg \}\\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \Bigg \{ \sum _{j = 0}^{\tau + 1} \frac{(j + 1)^{2} (j + 2)^{2}}{4} \frac{(3 \tau - j)!}{(\tau + 1 - j)!}\\&\quad + \, (\gamma _{2} + \gamma _{3}) \sum _{j = 0}^{\tau } \frac{(j + 1) (j + 2)^{2} (j + 3)}{4} \frac{(3 \tau - 1 - j)!}{(\tau - j)!} \\&\quad +\, \gamma _{2} \gamma _{3} \sum _{j = 0}^{\tau } \frac{(j + 1)^{2} (j + 2)^{2}}{4} \frac{(3 \tau - 1 - j)!}{(\tau - j)!} \Bigg \}. \end{aligned}$$
By Lemma 6, this can be reduced to the equation
$$\begin{aligned} 0 = 3 (18 + 28 \tau + 11 \tau ^{2}) + 18 (1 + \tau )^{2} (\gamma _{2} + \gamma _{3}) + (6 + 16 \tau + 11 \tau ^{2}) \gamma _{2} \gamma _{3}. \end{aligned}$$
(15)
By solving (14) and (15), we have
$$\begin{aligned} \begin{array}{l} \displaystyle \gamma _{2} = - \frac{12 + 26 \tau + 13 \tau ^{2} \pm h_{1} (\tau )}{2 (3 + 8 \tau + 6 \tau ^{2} + \tau ^{3})}, \\ \displaystyle \gamma _{3} = - \frac{12 + 26 \tau + 13 \tau ^{2} \mp h_{1} (\tau )}{2 (3 + 8 \tau + 6 \tau ^{2} + \tau ^{3})} \end{array} \end{aligned}$$
(16)
with \(h_{1}(\tau ) = \sqrt{36 + 192 \tau + 388 \tau ^{2} + 388 \tau ^{3} + 217 \tau ^{4} + 72 \tau ^{5} + 12 \tau ^{6}}\). On the other hand, by contrasting the coefficients of \(u^{k}\) (or \(u^{k - 1}, u^{k - 2}\), resp.) on both sides of the equation in Lemma 5 (I)-(ii), we find that
$$\begin{aligned} 0 =&\, 2 \gamma _{0} - \gamma _{2} - \gamma _{3} - 2 \tau , \end{aligned}$$
(17)
$$\begin{aligned} 0 =&\, 4 (1 + \tau ) (1 + 2 \tau ) \gamma _{0} + 2 (1 + 2 \tau ) \gamma _{1} - (1 + \tau ) \gamma _{2} \gamma _{3} \nonumber \\&-\, (3 + 8 \tau + 6 \tau ^{2}) (\gamma _{2} + \gamma _{3}) - (1 + \tau ) (3 + 8 \tau + 10 \tau ^{2}), \end{aligned}$$
(18)
$$\begin{aligned} 0 =&\, 6 (1 + \tau ) (1 + 2 \tau )^{2} \gamma _{0} + 6 (1 + 2 \tau )^{2} \gamma _{1} - 3 (2 + 3 \tau ) (2 + 4 \tau + 3 \tau ^{2}) (\gamma _{2} + \gamma _{3}) \nonumber \\&- \, 3 (1 + \tau ) (2 + 3 \tau ) \gamma _{2} \gamma _{3} - 2 (1 + \tau ) (9 + 28 \tau + 34 \tau ^{2} + 19 \tau ^{3}). \end{aligned}$$
(19)
Substituting \(\gamma _{2}, \gamma _{3}\) of (16) into (17) leads to the following equation:
$$\begin{aligned} \gamma _{0} = \frac{- 12 -20 \tau + 3 \tau ^{2} + 12 \tau ^{3} + 2 \tau ^{4}}{2(1 + \tau ) (3 + 5 \tau + \tau ^{2})}. \end{aligned}$$
(20)
By substituting \(\gamma _{0}\) of (20) and \(\gamma _{2}, \gamma _{3}\) of (16) into (18), we see that
$$\begin{aligned} \gamma _{1} = \frac{6 + 16 \tau + 15 \tau ^{2} + 18 \tau ^{3} + 30 \tau ^{4} + 18 \tau ^{5} + 2 \tau ^{6}}{2 (1 + \tau ) (1 + 2 \tau ) (3 + 5 \tau + \tau ^{2})}. \end{aligned}$$
(21)
Similarly, by substituting \(\gamma _{0}\) of (20), \(\gamma _{1}\) of (21) and \(\gamma _{2}, \gamma _{3}\) of (16) into (19), we obtain
$$\begin{aligned} 0 = (-1 + \tau ) (2 + \tau ) (9 + 30 \tau + 22 \tau ^{2} + 2 \tau ^{3}). \end{aligned}$$
Obviously if \(\tau \ge 2\), then the left-hand side of this equation is greater than 0, and so \(\tau = 1\). Then by Lemma 5 (I)-(i), we can explicitly compute the roots \(R_{1}, R_{2}, R_{3}, R_{4}\) of \(L_{4}^{1} (u) + \gamma _{0} L_{3}^{1} (u) + \gamma _{1} L_{2}^{1} (u)\):
$$\begin{aligned} R_{1}&= \frac{55 + \sqrt{145} + \sqrt{10 (89 - \sqrt{145})}}{12}, \;\quad R_{2} = \frac{55 - \sqrt{145} + \sqrt{10 (89 + \sqrt{145})}}{12}, \\ R_{3}&= \frac{55 + \sqrt{145} - \sqrt{10 (89 - \sqrt{145})}}{12}, \;\quad R_{4} = \frac{55 - \sqrt{145} - \sqrt{10 (89 + \sqrt{145})}}{12}. \end{aligned}$$
By Lemma 2, \((\{R_{1}, \ldots , R_{4} \}, \{ \varLambda _{1} R_{1}, \ldots , \varLambda _{4} R_{4} \})\) forms a quadrature formula of degree 5 for \(\mathcal {J}_{1}\). Since \(R_{1}, \ldots , R_{4}, \varLambda _{1}, \ldots , \varLambda _{4}\) satisfy that
$$\begin{aligned}&\varLambda _{1} R_{1} + \cdots + \varLambda _{4} R_{4}=1, \qquad \qquad (\varLambda _{1} R_{1}) R_{1} + \cdots + (\varLambda _{4} R_{4}) R_{4} = 2, \\&(\varLambda _{1} R_{1}) R_{1}^{2} + \cdots + (\varLambda _{4} R_{4}) R_{4}^{2} = 6, \quad (\varLambda _{1} R_{1}) R_{1}^{3} + \cdots + (\varLambda _{4} R_{4}) R_{4}^{3} = 24, \end{aligned}$$
it holds that
$$\begin{aligned} \varLambda _{1}&= \frac{54 (7 \sqrt{5} - 5 \sqrt{29})- 56 \sqrt{3 (89 + \sqrt{145})} + 8 \sqrt{435 (89 + \sqrt{145})}}{25 \Big ( 116 \sqrt{5} + 76 \sqrt{29} + 29 \sqrt{2(89 - \sqrt{145})} + 5 \sqrt{290 (89 - \sqrt{145})}\Big )}, \\ \varLambda _{2}&= \frac{-54 (7 \sqrt{5} + 5 \sqrt{29}) + 56 \sqrt{3 (89 - \sqrt{145})} + 8 \sqrt{435 (89 - \sqrt{145})}}{25 \Big ( -116 \sqrt{5} + 76 \sqrt{29} - 29 \sqrt{2 (89 + \sqrt{145})} + 5 \sqrt{290 (89 + \sqrt{145})}\Big )}, \\ \varLambda _{3}&= \frac{54 (- 7 \sqrt{5} + 5 \sqrt{29}) - 56 \sqrt{3 (89 + \sqrt{145})} + 8 \sqrt{435 (89 + \sqrt{145})}}{25 \Big ( -116 \sqrt{5} - 76 \sqrt{29} + 29 \sqrt{2 (89 - \sqrt{145})} + 5 \sqrt{290 (89 - \sqrt{145})}\Big )}, \\ \varLambda _{4}&= \frac{54 (7 \sqrt{5} + 5 \sqrt{29}) + 56 \sqrt{3 (89 - \sqrt{145})} + 8 \sqrt{435 (89 - \sqrt{145})}}{25 \Big ( 116 \sqrt{5} - 76 \sqrt{29} - 29 \sqrt{2 (89 + \sqrt{145})} + 5 \sqrt{290 (89 + \sqrt{145})}\Big )}. \end{aligned}$$
Therefore, \(\sum _{j = 1}^{4} (-1)^{j} \varLambda _{j} R_{j}^{4} = -48 \sqrt{5/29} \ne 0\), which contradicts (5).
The next case to consider is when \(k = 2 \tau , \tau \ge 2\). By contrasting the coefficients of \(u^{k}\) (or \(u^{k -1}, u^{k - 2}, u^{k -3}\), resp.) on both sides of the equation in Lemma 5 (II)-(ii), we find that
$$\begin{aligned} \begin{array}{ll} 0 &{}= (1 + 2 \tau ) \gamma _{0} - (1 + \tau ) \gamma _{2} - 2 \tau ^{2}, \\ 0 &{}= 2 (1 + 2 \tau )^{2} \gamma _{0} + 2 (1 + 2 \tau ) \gamma _{1} - 2 (1 + \tau )(1 + 3 \tau ) \gamma _{2}\\ &{}\quad - (1 + \tau ) \gamma _{3} - (1 + 5 \tau + 8 \tau ^{2} + 10 \tau ^{3}), \\ 0 &{}= 3 (-1 + 2 \tau ) (1 + 2 \tau )^{2} \gamma _{0} + 6 (-1 + 2 \tau ) (1 + 2 \tau ) \gamma _{1} \\ &{}\quad - 3 (1 + \tau ) (-1 + 3 \tau ) (1 + 3 \tau ) \gamma _{2} - 3 (1 + \tau ) (-1 + 3 \tau ) \gamma _{3} \\ &{}\quad -2 (-2 - 3 \tau + 13 \tau ^{2} + 15 \tau ^{3} + 19 \tau ^{4}), \\ 0 &{}= 4 (-1 + 2 \tau )^{2} (1 + 2 \tau )^{2} \gamma _{0} + 12 (-1 + 2 \tau )^{2} (1 + 2 \tau ) \gamma _{1} \\ &{}\quad - 6 (1 + \tau ) (1 + 3 \tau ) (1 - 2 \tau + 6 \tau ^{2}) \gamma _{2}- 9 (1 + \tau ) (1 - 2 \tau + 6 \tau ^{2}) \gamma _{3} \\ &{}\quad -10 (1 + \tau - 2 \tau ^{2} + 16 \tau ^{3} + 13 \tau ^{4} + 13 \tau ^{5}). \end{array} \end{aligned}$$
(22)
By solving (22), we have
$$\begin{aligned} \begin{array}{l} \displaystyle \gamma _{0} = \frac{1 + 3 \tau - 6 \tau ^{2} - 18 \tau ^{3} - 6 \tau ^{5} + 2 \tau ^{6}}{\tau (1 + 2 \tau ) (4 + 13 \tau + 2 \tau ^{2} + 2 \tau ^{3})}, \\ \displaystyle \gamma _{1} = \frac{-2 - 2 \tau + 51 \tau ^{2} + 153 \tau ^{3} + 96 \tau ^{4} + 36 \tau ^{5} - 10 \tau ^{6} + 2 \tau ^{7}}{6 \tau (1 + 2 \tau ) (4 + 13 \tau + 2 \tau ^{2} + 2 \tau ^{3})}, \\ \displaystyle \gamma _{2} = - \frac{-1 -2 \tau + 8 \tau ^{2} + 18 \tau ^{3} + 8 \tau ^{4} + 2 \tau ^{5}}{\tau (4 + 13 \tau + 2 \tau ^{2} + 2 \tau ^{3})}, \\ \displaystyle \gamma _{3} = \frac{2 (- 1 -9 \tau - 24 \tau ^{2} - 6 \tau ^{3} + 24 \tau ^{4} + 6 \tau ^{5} + \tau ^{6})}{3 \tau (4 + 13 \tau + 2 \tau ^{2} + 2 \tau ^{3})}. \end{array} \end{aligned}$$
(23)
On the other hand, it follows from (12), (13) and Lemma 5 (I)-(ii) that
$$\begin{aligned} 0&= \int _0^{\infty } \Big (L_{k + 1}^1 (u) + \gamma _{0} L_k^1 (u) + \gamma _{1} L_{k - 1}^1 (u) \Big ) u^{2 \tau - 2} \cdot u e^{- u} \; \mathrm{d} u \\&= \frac{\tau ! (\tau + 1)!}{(2\tau + 2)!} \int _0^{\infty } L_{\tau }^{2 \tau + 1} (u) \\&\quad \times \, \Big ( L_{\tau + 1}^{2 \tau + 1} (u) + \gamma _{2} L_{\tau }^{2 \tau + 1} (u) + \gamma _{3} L_{\tau - 1}^{2 \tau + 1} (u )\Big ) u^{2 \tau -1} e^{- u} \; \mathrm{d} u \\&= \,\frac{\tau ! (\tau + 1)!}{(2\tau + 2)!} \Big \{ \sum _{j = 0}^{\tau } (j + 1) (j + 2) \int _{0}^{\infty } L_{\tau - j}^{2 \tau -1}(u)^{2} \cdot u^{2 \tau - 1} e^{- \tau } \; \mathrm{d} u \\&\quad +\, \gamma _{2} \sum _{j = 0}^{\tau } (j + 1)^{2} \int _{0}^{\infty } L_{\tau - j}^{2 \tau -1}(u)^{2} \cdot u^{2 \tau - 1} e^{- \tau } \; \mathrm{d} u \\&\quad +\, \gamma _{3} \sum _{j = 0}^{\tau - 1} (j + 1) (j + 2) \int _{0}^{\infty } L_{\tau - j -1}^{2 \tau -1}(u)^{2} \cdot u^{2 \tau - 1} e^{- \tau } \; \mathrm{d} u \Big \} \\&= \,\frac{\tau ! (\tau + 1)!}{(2 \tau + 2)!} \Big \{ \sum _{j = 0}^{\tau } (j + 1) (j + 2) \frac{(3 \tau -1 - j)!}{(\tau - j)!} \\&\quad + \gamma _{2} \sum _{j = 0}^{\tau } (j + 1)^{2} \frac{(3 \tau - 1 - j)!}{(\tau - j)!} + \gamma _{3} \sum _{j = 0}^{\tau - 1} (j + 1)(j + 2) \frac{(3 \tau - 2 - j)!}{(\tau - 1 - j)!} \Big \}. \end{aligned}$$
Thus, by Lemma 6, we have
$$\begin{aligned} 0 = 2 + 3 \tau + (1 + 2 \tau ) \gamma _{2} + \tau \gamma _{3}. \end{aligned}$$
(24)
Substituting \(\gamma _{0}, \gamma _{1}, \gamma _{2}, \gamma _{3}\) of (23) into (24) leads to the following equation:
$$\begin{aligned} 0 = (-1 + \tau ) (- 1 - 3 \tau + \tau ^{2}). \end{aligned}$$
It is easily seen that this equation has no integer solutions as \(\tau \ge 2\). \(\square\)

Proof of Theorem 2 (ii)

Assume that there exists an almost tight Euclidean \((4k + 3)\)-design supported by \(k + 2\) concentric circles \(S_{1} \cup \cdots \cup S_{k + 2}\) for Gaussian integration \(\mathcal {I}\). The proof is divided into two parts.

The first case is that \(k = 2 \tau + 1, \tau \ge 1\). By contrasting the coefficients of \(u^{k}\) (or \(u^{k -1}, u^{k - 2}\), respectively) on both sides of equation in Lemma 5 (II)-(ii), we see that
$$\begin{aligned} 0 =&\, 2 \gamma _{0} - \gamma _{1} - \gamma _{2} - 2 (\tau + 1), \end{aligned}$$
(25)
$$\begin{aligned} 0 =&\, 4 (1 + 2 \tau ) \gamma _{0} - 2 (2 + 3 \tau ) (\gamma _{1} + \gamma _{2}) - \gamma _{1} \gamma _{2} - 10 (1 + \tau )^{2} , \end{aligned}$$
(26)
$$\begin{aligned} 0 =&\, 12 (1 + 2 \tau )^{2} \gamma _{0} - 9 (5 + 10 \tau + 6 \tau ^{2}) (\gamma _{1} + \gamma _{2}) - 18 (1 + \tau ) \gamma _{1} \gamma _{2} \nonumber \\&-\, 2 (54 + 137 \tau + 122 \tau ^{2} + 38 \tau ^{3}). \end{aligned}$$
(27)
Moreover, it follows from (12), (13) and Lemma 5 (II)-(ii) that
$$\begin{aligned} 0&= \int _{0}^{\infty } \Big ( L_{k +1}^{1} (u) + \gamma _{0} L_{k}^{1} (u) \Big ) u^{2 \tau } \cdot u e^{- u} \; \mathrm{d} u \\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \int _{0}^{\infty } \Big (L_{\tau + 1}^{2 \tau + 3} (u) + \gamma _{1} L_{\tau }^{2 \tau + 3} (u) \Big ) \\&\quad \times \Big (L_{\tau + 1}^{2 \tau + 3} (u) + \gamma _{2} L_{\tau }^{2 \tau + 3} (u) \Big ) u^{2 \tau + 1} e^{- u} \mathrm{d} u \\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \Bigg \{ \sum _{j = 0}^{\tau + 1} (j + 1)^{2} \int _{0}^{\infty } L_{\tau + 1 - j}^{2 \tau + 1}(u)^{2} \cdot u^{2 \tau + 1} e^{- u} \; \mathrm{d} u \\&\quad + (\gamma _{1} + \gamma _{2}) \sum _{j = 0}^{\tau } (j + 1) (j + 2) \int _{0}^{\infty } L_{\tau - j}^{2\tau + 1} (u)^{2} \cdot u^{2 \tau + 1} e^{ - u} \; \mathrm{d} u \\&\quad + \gamma _{1} \gamma _{2} \sum _{j = 0}^{\tau } (j + 1)^{2} \int _{0}^{\infty } L_{\tau - j}^{2 \tau + 1} (u)^{2} \cdot u^{2 \tau + 1} e^{- u} \; \mathrm{d} u\Bigg \} \\&= \frac{((\tau + 1)!)^{2}}{(2 \tau + 2)!} \Bigg \{ \sum _{j = 0}^{\tau + 1} (j + 1)^{2} \frac{(3 \tau + 2 - j)!}{(\tau + 1 - j)!} \\&\quad + (\gamma _{1} + \gamma _{2}) \sum _{j = 0}^{\tau } (j + 1) (j + 2) \frac{(3 \tau + 1 - j)!}{(\tau - j)!} + \gamma _{1} \gamma _{2} \sum _{j = 0}^{\tau } (j + 1)^{2} \frac{(3 \tau + 1 - j)!}{(\tau - j)!} \Bigg \}. \end{aligned}$$
By Lemma 6, this can be reduced to the equation
$$\begin{aligned} 0 = (3 + 2 \tau ) (4 + 3 \tau ) + (1 + \tau ) (4 + 3 \tau ) (\gamma _{1} + \gamma _{2}) + 2 (1 + \tau )^{2} \gamma _{1} \gamma _{2}. \end{aligned}$$
(28)
By solving (25), (26) and (28), we have
$$\begin{aligned} \begin{array}{l} \displaystyle \gamma _{0} = \frac{- 13 - 14 \tau + 2 \tau ^{2} + 4 \tau ^{3}}{2 (1 + \tau ) (5 + 4 \tau )}, \\ \displaystyle \gamma _{1} = - \frac{23 + 42 \tau + 24 \tau ^{2} + 4 \tau ^{3} \pm h_{2}(\tau )}{2 (1 + \tau ) (5 + 4 \tau )}, \\ \displaystyle \gamma _{2} = - \frac{23 + 42 \tau + 24 \tau ^{2} + 4 \tau ^{3} \mp h_{2}(\tau )}{2 (1 + \tau ) (5 + 4 \tau )} \end{array} \end{aligned}$$
(29)
with \(h_{2}(\tau ) = \sqrt{209 + 636 \tau + 796 \tau ^{2} + 568 \tau ^{3} + 280 \tau ^{4} + 96 \tau ^{5} + 16 \tau ^{6}}\). Thus, substituting \(\gamma _{0}, \gamma _{1}, \gamma _{2}\) of (29) into (27) gives the following equation:
$$\begin{aligned} 129 + 286 \tau + 232 \tau ^{2} + 96 \tau ^{3} + 32 \tau ^{4} + 8 \tau ^{5} = 0, \end{aligned}$$
which is obviously a contradiction since \(\tau \ge 1\).
The next case to be considered is when \(k = 2 \tau , \tau \ge 1\). By contrasting the coefficients of \(u^{k}\) (or \(u^{k - 1}, u^{k - 2}\), resp. ) on both sides of the equation in Lemma 5 (II)-(ii), we find that
$$\begin{aligned} 0&= (1 + 2 \tau ) \gamma _{0} - (1 + \tau ) \gamma _{1} - (1 + 2 \tau + 2 \tau ^{2}), \\ 0&= 2 (1 + 2 \tau )^{2} \gamma _{0} - 2 (1 + \tau ) (2 + 3 \tau ) \gamma _{1} - (1 + \tau ) \gamma _{2} - 2 (3 + 9 \tau + 10 \tau ^{2} + 5 \tau ^{3}), \\ 0&= 6 \tau (-1 + 2 \tau ) (1 + 2 \tau )^{2} \gamma _{0} - 3 (1 + \tau ) (2 + 3 \tau ) (- 1 + 6 \tau ^{2}) \gamma _{1} \\&\quad - 3 (1 + \tau ) (-1 + 6 \tau ^{2}) \gamma _{2} -2 (-6 - 19 \tau + 12 \tau ^{2} + 80 \tau ^{3} + 84 \tau ^{4} + 38 \tau ^{5}). \end{aligned}$$
Solving these equations, we have
$$\begin{aligned} \gamma _{0} =&\, \frac{-5 - 18 \tau - 26 \tau ^{2} - 12 \tau ^{3} + 4 \tau ^{4}}{3 (1 + 2 \tau ) (-1 + 2 \tau ^{2})}, \nonumber \\ \gamma _{1} =&\, - \frac{2 (1 + \tau ) (1 + 2 \tau )^{2}}{3 (-1 + 2 \tau ^{2})}, \nonumber \\ \gamma _{2} =&\, \frac{2 (1 + \tau ) (8 + 13 \tau + 4 \tau ^{2} + 2 \tau ^{3})}{3 (-1 + 2 \tau ^{2})}. \end{aligned}$$
(30)
On the other hand, it follows from (12), (13) and Lemma 5 (II)-(ii) that
$$\begin{aligned} 0&= \,\int _{0}^{\infty } \Big ( L_{k +1}^{1} (u) + \gamma _{0} L_{k}^{1} (u) \Big ) u^{2 \tau - 1} \cdot u e^{- u} \; \mathrm{d} u \\&= \frac{\tau !(\tau + 1)!}{(2 \tau + 1)!}\int _{0}^{\infty } L_{\tau }^{2 \tau + 2} (u) \\&\quad \times \, \Big (L_{\tau + 1}^{2 \tau + 2} (u) + \gamma _{1} L_{\tau }^{2 \tau + 2} (u) + \gamma _{2} L_{\tau - 1}^{2 \tau + 2} \Big ) u^{2 \tau } e^{- u} \;\mathrm{d} u \\&= \,\frac{\tau !(\tau + 1)!}{(2 \tau + 1)!} \Bigg \{ \sum _{j = 0}^{\tau } (j + 1) (j + 2) \int _{0}^{\infty } L_{\tau - j}^{2 \tau }(u)^{2} \cdot u^{2 \tau } e^{- u} \; \mathrm{d} u \\&\quad +\, \gamma _{1} \sum _{j = 0}^{\tau } (j + 1)^{2} \int _{0}^{\infty } L_{\tau - j}^{2\tau } (u)^{2} \cdot u^{2 \tau } e^{ - u} \; \mathrm{d} u \\&\quad +\, \gamma _{2} \sum _{j = 0}^{\tau -1} (j + 1) (j + 2) \int _{0}^{\infty } L_{\tau -1 - j}^{2 \tau } (u)^{2} \cdot u^{2 \tau } e^{- u} \; \mathrm{d} u\Bigg \} \\&= \,\frac{\tau ! (\tau + 1)!}{(2 \tau + 1)!} \Bigg \{ \sum _{j = 0}^{\tau } (j + 1) (j + 2) \frac{(3 \tau - j)!}{(\tau - j)!} \\&\quad +\, \gamma _{1}\sum _{j = 0}^{\tau } (j + 1)^{2} \frac{(3 \tau - j)!}{(\tau - j)!} + \gamma _{2} \sum _{j = 0}^{\tau -1} (j + 1)(j + 2) \frac{(3 \tau -1 - j)!}{(\tau -1 - j)!} \Bigg \}. \end{aligned}$$
By Lemma 6, this can be reduced to the equation
$$\begin{aligned} 0 = 6 (1 + \tau ) + (3 + 4 \tau ) \gamma _{1} + 2 \tau \gamma _{2}. \end{aligned}$$
Substituting \(\gamma _{1}, \gamma _{2}\) of (30) into this equation leads to the following equation:
$$\begin{aligned} 0 =(-1 + \tau ) (3 + 3 \tau - \tau ^{2} + \tau ^{3}). \end{aligned}$$
It is easily checked that, if \(\tau \ge 2\), the left-hand side of this equation is nonzero and therefore \(\tau = 1\). Then by Lemma 5 (II)-(i), the roots \(R_{1}, R_{2}, R_{3}\) of \(L_{3}^{1} (u) + \gamma _{0} L_{2}^{1} (u)\) are explicitly computed as \(R_{1} = 5\), \(R_{2} = 3 (-2 + \sqrt{6})\), \(R_{3} = 3 (-2 - \sqrt{6})\). This is a contradiction since \(R_{3} < 0\). \(\square\)

Remark 4

While some sporadic examples of almost tight 5-, 7- and 9-designs supported by 3, 4 and 5 concentric circles can be found for Gaussian integration (cf. Bannai et al. 2010; Haegemans 1975), there do not exist almost tight \((2e+1)\)-designs supported by \(\lfloor e/2 \rfloor + 2\) concentric circles for \(e\ge 2\) in general. On the other hand, the following is a new example of almost tight 7-designs supported by 4 (\(= \lfloor e/2 \rfloor + 3\)) concentric circles for Gaussian integration:
$$\begin{aligned} \mathcal {Q}[f]&= \sum _{l=0}^{3} \frac{3}{4r^2 (2r^4 - 6r^2 + 9)} f \Big ( r \cos \Big ( \frac{2l + 1}{4} \pi \Big ), r \sin \Big ( \frac{2l + 1}{4} \pi \Big ) \Big ) \\&\quad + \sum _{l=0}^{3} \frac{1}{36} f \Big ( \sqrt{3} \cos \Big ( \frac{2 l + 2}{4} \pi \Big ), \sqrt{3} \sin \Big ( \frac{2l + 2}{4} \pi \Big ) \Big ) \\&\quad + \sum _{l=0}^{3} \frac{(2 r^2 -3)^3}{36 (r^2 - 3) (2r^4 - 6r^2 + 9)} \\&\quad \times f \Big ( \sqrt{\frac{3r^2 - 9}{2r^2 - 3}} \cos \Big ( \frac{2 l + 3}{4} \pi \Big ), \sqrt{\frac{3r^2 - 9}{2r^2 - 3}} \sin \Big ( \frac{2l + 3}{4} \pi \Big ) \Big ) \\&\quad + \frac{4r^4-18r^2+9}{9r^2 (r^2 - 3)} f(0,0). \end{aligned}$$
This example might, in general, imply that more and more almost tight \((2e+1)\)-designs supported by at least \(\lfloor e/2 \rfloor + 3\) concentric circles can be found for rotationally symmetric integrals.

4 A short review: relationships between Euclidean designs and machine learning theory

While two-dimensional Euclidean designs are of profound theoretical interest from the viewpoints of combinatorics and numerical analysis, higher-dimensional designs serve to find applications in statistics. For example, the construction theory for Euclidean designs in high dimensions has gradually drawn attention in the study of kernel approximation in machine learning theory (see Dao et al. 2017; Munkhoeva et al. 2018). The final section is devoted to a short review of a relationship between kernel approximation and some constructions of high-dimensional Euclidean designs.

Let \(K: \varOmega \times \varOmega \rightarrow \mathbb {R}\) be a kernel (function) on \(\varOmega\), namely K satisfies
  1. (i)

    \(\sum _{i,j=1}^n c_ic_j K(\varvec{x}_i,\varvec{x}_j) \ge 0\) for all \(n \ge 1\), \(c_i \in \mathbb {R}\) and \(\varvec{x}_i \in \varOmega\);

     
  2. (ii)

    \(K(\varvec{x}, \varvec{x}') = K(\varvec{x}', \varvec{x})\) for all \(\varvec{x}, \varvec{x}' \in \varOmega\).

     
Kernel K is shift-invariant if there exists a univariate function \(\psi\) such that
$$\begin{aligned} K(\varvec{x},\varvec{x}') = \psi (\varvec{x}- \varvec{x}') \quad \text { for all}\quad {\varvec{x},\varvec{x}' \in \varOmega }. \end{aligned}$$
Note that Gaussian kernel \(K(\varvec{x},\varvec{x}') = \exp (-\Vert \varvec{x}- \varvec{x}' \Vert ^2/2 \sigma ^2)\) is a typical example of shift-invariant kernels.

The following plays a role in the study of kernel approximations.

Theorem 3

(Bochner’s theorem (cf. Rudin 1990)) A continuous shift-invariant kernel\(K(\varvec{x}, \varvec{x}') = \psi (\varvec{x}- \varvec{x}')\)on\(\mathbb {R}^d\)is positive definite, i.e., \(\sum _{i,j=1}^n c_ic_j \psi (\varvec{x}_i-\varvec{x}_j) \ge 0\)for all\(c_1,\ldots ,c_n \in \mathbb {R}\)and\(\varvec{x}_1, \ldots , \varvec{x}_n \in \mathbb {R}^d\), if and only if, there exists a probability measure\(\mu\)on\(\mathbb {R}^d\)such that
$$\begin{aligned} K(\varvec{x},\varvec{x}') = \int _{\mathbb {R}^d} \exp (\sqrt{-1} \langle \varvec{x}- \varvec{x}', \varvec{y}\rangle ) \; \mathrm{d}\mu (\varvec{y}). \end{aligned}$$
(31)
Recently, Dao et al. (2017) has proposed a deterministic construction of kernel approximation through Euclidean designs for Gaussian or sub-Gaussian integrations. Assume that a probability measure \(\mu\) in (31) has rotational symmetry (e.g., Gaussian kernel corresponds to Gaussian measure \(\mu\)). Given a positive integer t, let \((X, \lambda )\) be a Euclidean t-design for \(\int _{\mathbb {R}^d} \cdot \; \mathrm{d} \mu\) with \(|X| = n\), i.e.,
$$\begin{aligned} \sum _{i = 1}^{n} \lambda _i f(\varvec{x}_i) = \int _{\mathbb {R}^d} f(\varvec{x}) \; \mathrm{d} \mu (\varvec{x}) \quad \text {for all }\quad {f \in \mathcal {P}_t(\mathbb {R}^d)} \end{aligned}$$
where \(\lambda _i := \lambda (\varvec{x}_i)\) for all i. The key idea of Dao et al. (2017) is the use of approximate kernels \({\tilde{K}}\) defined as
$$\begin{aligned} {\tilde{K}} (\varvec{x}, \varvec{x}') = \sum _{i = 1}^{n} F(\varvec{x}) \overline{F(\varvec{x}')}^T = \sum _{i = 1}^{n} \lambda _i \exp (\sqrt{-1} \langle \varvec{x}- \varvec{x}', {\varvec{x}_i} \rangle ) \end{aligned}$$
where kernel features are given as
$$\begin{aligned} F(\varvec{x}) = [\sqrt{\lambda _1} \exp (\sqrt{-1} \langle \varvec{x}, {\varvec{x}_1} \rangle ), \ldots , \sqrt{\lambda _n} \exp (\sqrt{-1} \langle \varvec{x}, {\varvec{x}_n} \rangle )]. \end{aligned}$$
Dao et al. (2017) deals with constructions of cubature formulas such as Smolyak methods, product rules and some other methods. Afterward, Munkhoeva et al. (2018) discusses a different approach that combines Gaussian quadrature with a certain random point distribution so-called “stochastic spherical rule” on spheres. Stochastic spherical rule is an approximation for spherical integration by using random orthogonal matrix Q, i.e., given \(\varvec{x}_1, \ldots , \varvec{x}_n \in S^{d-1}\), by considering a weighted summation rule as
$$\begin{aligned} \sum _{i = 1}^{n} w_i f(Q \varvec{x}_i) \approx \int _{S^{d-1}} f(\varvec{x}) \; \mathrm{d} \rho (\varvec{x}) \end{aligned}$$
where Q is a random orthogonal matrix. Note that Sect. 2.4 of Munkhoeva et al. (2018)  also explains how to generate uniformly random orthogonal matrices.

These authors point out that approximate kernels \({\tilde{K}}\) so constructed have “good” statistical features compared to random constructions as such by Rahimi and Recht (2008); for detailed information about this, see Dao et al. (2017) and Munkhoeva et al. (2018).

The idea of Munkhoeva et al. (2018) can be modified in more deterministic manner. Let
$$\begin{aligned} \mathcal {J}[f] := \frac{|S^{d-1}|}{2} \int _{a^2}^{b^2} f(u) u^{d/2 - 1} W(\sqrt{u}) \; \mathrm{d} u. \end{aligned}$$

Proposition 1

Assume that there exists a weighted sphericalt-design (Yw) and a quadrature formula\((\{ r_i^2 \}, \{ \varLambda _i \})\)of degree\(\lfloor t/2 \rfloor\)for integral\(\mathcal {J}\). Then a pair\((\{ r_i Y \}, \{ \varLambda _i w \})\)forms a Euclideant-design for a rotationally symmetric integral\(\mathcal {I}\).

Proof

It is known (e.g., Neumaier and Seidel 1992) that the polynomial space \(\mathcal {P}_{t} (\mathbb {R}^d)\) is decomposed as
$$\begin{aligned} \mathcal {P}_{t} (\mathbb {R}^d) = \bigoplus _{0 \le 2j +\ell \le t} \Vert \varvec{x}\Vert ^{2j} \text {Harm}_{\ell } (\mathbb {R}^d) \end{aligned}$$
where \(\text {Harm}_{\ell } (\mathbb {R}^d)\) denotes the space of all harmonic homogeneous polynomials of degree exactly \(\ell\).
For any \(f \in \Vert \varvec{x}\Vert ^{2j} \text {Harm}_l (\mathbb {R}^d)\) with \(1 \le \ell \le t\) and \(0 \le j \le \lfloor (t - \ell )/2 \rfloor\), it holds that
$$\begin{aligned} \sum _{i = 1}^{p} \sum _{\varvec{x}\in Y} (\varLambda _i w(\varvec{x})) f(r_i \varvec{x})&= \sum _{i = 1}^{p} r_i^{\ell + 2j} \varLambda _i \sum _{\varvec{x}\in Y} w(\varvec{x}) f(\varvec{x}) \nonumber \\&= \sum _{i = 1}^{p} r_i^{\ell + 2j} \varLambda _i \int _{S^{d-1}} f(\varvec{x}) \; \mathrm{d} \rho (\varvec{x}) = 0 = \mathcal {I}[f] \end{aligned}$$
(32)
where the last equality in (32) follows since
$$\begin{aligned} \mathcal {I}[f]&= \int _{a}^{b} \left( |S^{d-1} |\int _{S^{d-1}} f(r \varvec{x}) \; \mathrm{d} \rho (\varvec{x}) \right) r^{d-1} W(r) \; \mathrm{d} r \\&= \int _{a}^{b} r^{2j + l + d - 1} W(r) \; dr \cdot |S^{d-1}| \int _{S^{d-1}} f(\varvec{x}) \; \mathrm{d} \rho (\varvec{x}) = 0. \end{aligned}$$
Moreover, for any \(f(\varvec{x}) = \Vert \varvec{x}\Vert ^{2j}\) with \(j = 0, 1, \ldots , \lfloor t/2 \rfloor\), we find that
$$\begin{aligned} \sum _{i = 1}^{p} \varLambda _i r_i^{2j} = \mathcal {J}[u^j] = \mathcal {I}[\Vert \varvec{x}\Vert ^{2j}] \; \left( > 0 \right) . \end{aligned}$$
(33)
Thus, the desired result is obtained from (32) and (33). \(\square\)

We know, by the existence of Gaussian quadrature (cf. Stroud (1971)), that there always exists a quadrature formula of degree t for \(\mathcal {J}\) with at least \(\lfloor t/2 \rfloor\) points. Therefore, if a spherical design can be explicitly constructed, then an approximate kernel \({\tilde{K}}\) is generated from Proposition 1 in a fairly deterministic way compared to the method by Munkhoeva et al. (2018).

A famous theorem by Seymour and Zaslavsky (1984) states that there exists a spherical design for sufficiently large number of points. More surprisingly, Bondarenko et al. (2013) shows the following theorem.

Theorem 4

(Bondarenko et al. (2013)) Given\(d \ge 2\), there exists a sphericalt-design of\(S^{d-1}\)with |X| points for all\(|X| \ge c_d t^{d-1}\)where the constant\(c_d\)does only depend ond.

The following immediately follows from Proposition 1 and Theorem 4.

Corollary 1

Given\(d \ge 2\), there exists a Euclideant-design for a rotationally symmetric integral withnpoints for all\(n \ge c'_d t^d\)where the constant\(c'_d\)does only depend ond.

Note that this lower bound has the same asymptotic behavior as bound (2) for fixed d.

Then what about constructions of spherical designs? A classical approach in combinatorics and numerical analysis is the use of orbits of finite reflection groups, as exemplified by the hyperoctahedral group (cf. Bajnok 2007; Nozaki and Sawa 2012, 2013; Sali 1994; Sawa and Xu 2014). Details on this group-theoretic approach can be available in Bannai and Bannai (2009), Stroud (1971) and references therein. Although it is not easy to explicitly construct spherical t-designs with \({\mathcal {O}}(t^{d-1})\) points for sufficiently large t, numerous good examples have been found for small values of t.

Meanwhile, Brauchart et al. (2014) and Hirao (2018) consider probabilistic methods for generating QMC designs through random point processes. While a drawback is the computational cost, a great advantage of QMC designs is the flexibility of the size of design.

Notes

Acknowledgements

The first author is partially supported by Grant-in-Aid for Young Scientists (B) 16K17645 by the Japan Society for the Promotion of Science (JSPS), and the second author is partially supported by Grant-in-Aid for Scientific Research (C) 18K03414 by JSPS. Part of this paper was started during the second author’s visit at the University of Texas at Brownsville from July to September, 2010, under the sponsorship of JSPS. We would like to express appreciations to Eiichi Bannai and Etsuko Bannai for valuable comments to this work.

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Copyright information

© Japanese Federation of Statistical Science Associations 2019

Authors and Affiliations

  1. 1.Department of Information and Science TechnologyAichi Prefectural UniversityNagakuteJapan
  2. 2.Graduate School of System InformaticsKobe UniversityKobeJapan

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