Entrainment Limit of Weakly Forced Nonlinear Oscillators

Abstract

Nonlinear oscillators exhibit entrainment (injection locking) to external periodic forcings. Despite its history of entrainment, and the wide utility of injection locking to date, it has been an open problem to establish an ideally efficient injection locking in a given oscillator. In this chapter, I identify a universal mechanism governing the entrainment limit under weak forcings, which enables us to understand how and why the ideal injection locking is realized. Using this mechanism, I prove that the maximization of the entrainment range or the stability of entrainment for general forcings, including pulse trains, and a fundamental limit of general m:n entrainment are realized and systematically designed. These results support the design principle of efficient injection locking, which is easily applied to experimental systems in a real environment, and this experimental systems applicability is tested in the Hodgkin–Huxley neuron model here as an example.

Appendices

Appendix 1 Assumptions on the Phase Response Function and Outlines of the Presented Proofs

In this appendix, we consider assumptions on the phase response functions Z(θ) which are required to prove the existence of entrainment limits, i.e., the optimal forcingsf _opt,  p , and we present outlines of the corresponding proofs for P1 and P2 in the main text, for the cases (1) 1 < p < ∞ and p = ∞, and(2) p = 1,respectively.

[ Case of 1 < p < ∞ and p = ∞]

For 1 < p < ∞, we assume that Z(θ) satisfies the following assumptions (i–iv):

• (i) Z is twice differentiable (and hence, locally Lipschitz continuous);

• (ii) $g(\theta ) =\bar{ Z}(\theta )+\lambda = 0$ has (a finite number of) isolated zeros θ _∗;

• (iii) $H(\Delta \phi,\lambda )$ has a finite number of isolated optimal solutions $(\Delta \phi _{{\ast}},\lambda _{{\ast}})$; and,

• (iv) At each optimal solution of H, the bordered Hessian $\vert \mathcal{H}(H)\vert \neq 0$, and $\bar{Z}^{{\prime}}(\theta _{{\ast}})\neq 0$ for each θ _∗ in (ii).

Namely, assume that Z is smooth and generic in the above sense; then the best forcing $(\equiv \overline{f}_{\mathrm{opt},\ p} \in L^{p}(S))$ is uniquely given by f _∗ in Eq. (4.4) in the main text, if the “best” solution $(\overline{\Delta \phi }_{{\ast}},\overline{\lambda }_{{\ast}})$ exists, which is determined by the nonlinear equations (4.5) in the main text; in this case, $F(\overline{\Delta \phi }_{{\ast}},\overline{\lambda }_{{\ast}})$ is the largest among all possible optima of $F(\Delta \phi _{{\ast}},\lambda _{{\ast}})$ under the constraint $G(\Delta \phi,\lambda ) = 0$.

The outline of the proof to the above statement is as follows. Firstly, f _∗ ∈ L ^p(S) and g ∈ L ^q(S) are satisfied by assumption (i). Then, from Hölder’s inequality, we start from a candidate f _∗ of Eq. (4.4) in the main text, and the problem reduces to finite-dimensional optimization of $H(\Delta \phi,\lambda )$ as shown in the main text. This optimization of H is possible, owing to the following facts. First, from assumption (iii), H has only isolated optimal points. Second, from assumptions (i), (ii), and (iv), the derivatives $\frac{\partial ^{2}H} {\partial \Delta \phi ^{2}}$, $\frac{\partial ^{2}H} {\partial \Delta \phi \partial \lambda }$, $\frac{\partial ^{2}H} {\partial \lambda \partial \Delta \phi }$, and $\frac{\partial ^{2}H} {\partial \lambda ^{2}}$ are continuous in a neighborhood of the optimal points; this is directly verified by the (ε, δ)-definition of limit, using the following (a) and (b): (a) all required derivatives of H are explicitly given as in Eqs. (4.15) below, and (b) although some integrals in Eqs. (4.15) are singular, all of them are verified to have finite values.

Then, as the next step, we obtain the necessary and sufficient conditions for existence of the optimal solutions of H. Finally, using these conditions, all (locally) optimal forcings f _opt,  p are captured, and we verify that the best (optimal) forcing indeed maximizes the locking range in the phase Eq. (4.1) in the main text.

As for the case of p = ∞, the same assumptions (i–iv) are required and almost the same argument as above is repeated. First, for $f_{\mathrm{opt},\ \infty }(\theta ) \equiv M\mathrm{sig}[g(\theta )]$, this satisfies equality in Hölder’s inequality: $2\pi J[f_{\mathrm{opt},\ \infty }] =\Vert f_{\mathrm{opt},\ \infty }\ g\Vert _{1} =\Vert f_{\mathrm{opt},\ \infty }\Vert _{\infty }\Vert g\Vert _{1} = M\Vert g\Vert _{1}$, which implies that f _opt,  ∞ is optimal in L ^∞(S). (We note that uniqueness of f _opt,  ∞ can be proved, but the proof is omitted here.)   Next, to determine the parameters $(\Delta \phi,\lambda )$, the same argument as in the case of P1 for 1 < p < ∞ is repeated, resulting in the following equations: $\left \langle \mathrm{sig}[g(\theta )]Z^{{\prime}}(\theta +\Delta \phi )\right \rangle = 0$, $\left \langle \mathrm{sig}[g(\theta )]\right \rangle = 0$, and $\vert \mathcal{H}(H)\vert = \mathcal{H}_{13}(\mathcal{H}_{12}^{2} -\mathcal{H}_{13}\mathcal{H}_{22}) > 0$. Note, we can verify that such f _opt,  ∞ indeed maximizes (minimizes) Γ at ϕ ₊ (ϕ ₋).

[ Case of p = 1]

For p = 1, we assume that Z(θ) satisfies the following assumptions (i), (ii), and (iii):

• (i) Z is locally Lipschitz continuous,

• (ii) $g(\theta ) =\bar{ Z}(\theta )+\lambda$ achieves the maximum and the minimum, respectively, at some θ = θ _max and θ = θ _min, and

• (iii) $\mathrm{The\ maximum\ of}\ \bar{Z}(\theta ) -\mathrm{the\ minimum\ of}\ \bar{Z}(\theta )$ [$=\bar{ Z}(\theta _{\mathrm{max}}) -\bar{ Z}(\theta _{\mathrm{min}})$] is maximized for a particular value of $\Delta \phi = \Delta \phi _{\mathrm{max}}$ and this choice of $\Delta \phi _{\mathrm{max}}$ (and its associated θ _max,  θ _min) is unique for a given Z.

Namely, assume that Z is continuous and generic in the above sense; then the pair of two pulses

$$\displaystyle\begin{array}{rcl} f_{\mathrm{opt},\ 1}(\theta ) \equiv -M[\Delta (\theta +\Delta \phi _{\mathrm{max}}) - \Delta (\theta )],& &{}\end{array}$$

(4.7)

where

$$\displaystyle\begin{array}{rcl} \Delta (\theta ) = \frac{1} {2n\epsilon }\ (\mathrm{for}\ \vert \theta \vert \leq \epsilon ),\ 0\ (\mathrm{otherwise}),& &{}\end{array}$$

(4.8)

realizes the ideal largest locking range $\frac{M} {2\pi } \Vert g\Vert _{\infty }$ as ε → +0 (in $\Delta (\theta ) = \frac{1} {2n\epsilon }$), where $\Delta \phi _{\mathrm{max}}$ satisfies $\Delta \phi _{\mathrm{max}} =\theta _{\mathrm{max}} -\theta _{\mathrm{min}}$.

The outline of the proof to the above statement is as follows. Firstly, g ∈ L ^∞(S) is satisfied by assumption (i). Then, we construct a candidate of optimal f _opt,  1 ( ∈ L ¹(S)) as shown in Eq. (4.7), taking care of the constraints (4.2) in the main text, from Hölder’s inequality under assumptions (ii) and (iii). Next, we verify that this f _opt,  1 realizes the best possible locking range ($= \frac{M} {2\pi } \Vert g\Vert _{\infty }$) obtained from Hölder’s inequality as ε → +0, again using assumptions (i), (ii), and (iii).

More precisely, for f _opt,  1, we assume that $\bar{Z}(\theta ) \equiv Z(\theta +\Delta \phi ) - Z(\theta )$ has a maximum and minimum at θ = θ _max and θ = θ _min, respectively (assumption (ii)), and further, without loss of generality, we assume that the value of $(\mathrm{the\ maximum\ of\ }\bar{Z}) - (\mathrm{the\ minimum\ of\ }\bar{Z}) =\bar{ Z}(\theta _{\mathrm{max}}) -\bar{ Z}(\theta _{\mathrm{min}})$ is largest at some unique value of $\Delta \phi = \Delta \phi _{\mathrm{max}}$ (assumption (iii)). Note that this assumption is satisfied if Z is (locally Lipschitz) continuous (assumption (i)), and a simple mathematical argument shows that $\Delta \phi _{\mathrm{max}} =\theta _{\mathrm{max}} -\theta _{\mathrm{min}}$ if such $\Delta \phi _{\mathrm{max}}$ exists. Then, we represent f _opt,  1 as $f_{\mathrm{opt},\ 1}(\theta ) = -M[\Delta (\theta +\Delta \phi _{\mathrm{max}}) - \Delta (\theta )]$, as shown above. (Intuitively, this $f_{\mathrm{opt},\ 1}$ is an arbitrarily tall pair of pulses satisfying the constraints (4.2) in the main text.) For this f _opt,  1, the equality in Hölder’s inequality $2\pi J[f_{\mathrm{opt},\ 1}] =\Vert f_{\mathrm{opt},\ 1}\Vert _{1}\Vert g\Vert _{\infty } = M\Vert g\Vert _{\infty }$ is satisfied as ε → 0, which implies that f _opt,  1 becomes optimal (in L ¹(S)) as ε → 0. Also, the associated $\varGamma (\phi ) \rightarrow \frac{M} {4\pi } [Z(\theta _{\mathrm{max}}+\phi ) - Z(\theta _{\mathrm{min}}+\phi )] \equiv \varGamma _{0}(\phi )$ (uniformly, ε → 0) holds if Z is locally Lipschitz continuous. Similarly to in the above cases of 1 < p < ∞ and p = ∞, we can verify that the associated Γ ₀(ϕ) is maximized (minimized) at ϕ ₊ (ϕ ₋), and $\varGamma _{0}(\phi _{+}) -\varGamma _{0}(\phi _{-}) = \frac{M} {2\pi } \Vert g\Vert _{\infty }$, where $\phi _{+} -\phi _{-}$ is defined by $\phi _{+} -\phi _{-} = \Delta \phi _{\mathrm{max}}$, if $\Delta \phi _{\mathrm{max}}$ exists.

So far, we have assumed the existence of the best choice of $\Delta \phi = \Delta \phi _{\mathrm{max}}$, which results in a possible optimal forcing f _opt,  1. Now, we are in a position to determine $\Delta \phi _{\mathrm{max}}$ for a given Z. This $\Delta \phi _{\mathrm{max}}$ is numerically obtained as follows (if Z is continuous): we plot the graph $\varGamma _{0}(\phi ) = M[Z(\phi +\Delta \phi ) - Z(\phi )] = M\bar{Z}(\theta )$ for a given $\Delta \phi \in [-\pi,0]$ and then gradually vary this parameter, again plotting the graph of Γ ₀(ϕ) for each value. Thus, the locking range R = (the maximum of Γ ₀) − (the minimum of Γ ₀) is obtained as a function of $\Delta \phi$. Note that $R(\Delta \phi )$ is an even function, due to the symmetry of the wave form of f _opt, 1. Once $\Delta \phi _{\mathrm{max}}$ is obtained, $\Delta \phi _{\mathrm{max}}$ determines θ _max and θ _min from the graph of Γ ₀.

Appendix 2 Derivation of the Nonlinear Equations Determining Optimal Forcings

First, note that $\left (\frac{\partial G} {\partial \Delta \phi }, \frac{\partial G} {\partial \lambda } \right )\neq \mathbf{0}$ is always satisfied, since $\frac{\partial G} {\partial \lambda } > 0$ as obtained in Eq. (4.13) below. Hence, the Lagrange multiplier rule is applied, and some value of μ (which can be 0) in $H = F +\mu G$ exists and its associated optimal solution $(\Delta \phi _{{\ast}},\lambda _{{\ast}})$ to $H(\Delta \phi,\lambda )$ satisfies

$$\displaystyle\begin{array}{rcl} \left (\frac{\partial H} {\partial \Delta \phi }, \frac{\partial H} {\partial \lambda } \right ) = \mathbf{0},& &{}\end{array}$$

(4.9)

if it exists. Namely, candidates for the optimal solution to the optimization of $H(\Delta \phi,\lambda )$ are obtained by solving Eq. (4.9) for $\Delta \phi _{{\ast}},\lambda _{{\ast}},$ and μ _∗; the derivation process is as follows.

For conciseness of expressions, we start by defining $\alpha = \frac{p} {p-1}$ ( > 0) and $\beta = \frac{1} {p-1}$ ( > 0). Then, the derivatives of $F(\Delta \phi,\lambda )$ are obtained as:

$$\displaystyle\begin{array}{rcl} & & \frac{\partial F} {\partial \Delta \phi } =\alpha \left \langle \mathrm{sig}[g(\theta )]\vert g(\theta )\vert ^{\beta }Z^{{\prime}}(\theta +\Delta \phi )\right \rangle \equiv F_{ 1}(\Delta \phi,\lambda ),{}\end{array}$$

(4.10)

$$\displaystyle\begin{array}{rcl} & & \frac{\partial F} {\partial \lambda } =\alpha \left \langle \mathrm{sig}[g(\theta )]\vert g(\theta )\vert ^{\beta }\right \rangle =\alpha G \equiv F_{2}(\Delta \phi,\lambda ),{}\end{array}$$

(4.11)

where $g(\theta ) =\bar{ Z}(\theta )+\lambda$. Likewise, the derivatives of $G(\Delta \phi,\lambda )$ are obtained as:

$$\displaystyle\begin{array}{rcl} \frac{\partial G} {\partial \Delta \phi }& =& \beta \left \langle \vert g(\theta )\vert ^{\beta -1}Z^{{\prime}}(\theta +\Delta \phi )\right \rangle,{}\end{array}$$

(4.12)

$$\displaystyle\begin{array}{rcl} \frac{\partial G} {\partial \lambda } & =& \beta \left \langle \vert g(\theta )\vert ^{\beta -1}\right \rangle > 0.{}\end{array}$$

(4.13)

The derivation of Eqs. (4.10)–(4.13) is straightforward, and it is omitted here. Note that in Eq. (4.11), $F_{2}(\Delta \phi,\lambda ) =\alpha \left \langle \mathrm{sig}(g(\theta ))\vert g(\theta )\vert ^{\beta }\right \rangle = 0$ is simply the constraint $G(\Delta \phi,\lambda ) = 0$. Since in Eq. (4.9) we have $\frac{\partial H} {\partial \lambda } = \frac{\partial F} {\partial \lambda } +\mu \frac{\partial G} {\partial \lambda } = 0$, and $\frac{\partial F} {\partial \lambda } =\alpha G(\Delta \phi,\lambda ) = 0$ and $\frac{\partial G} {\partial \lambda } > 0$ follow from the above arguments, μ _∗ is uniquely determined as μ _∗ = 0. (This sounds a bit contradictory, as the μ G term vanishes in $H = F +\mu G$ if $\mu =\mu _{{\ast}} = 0$; however, for μ _∗ = 0, Eq. (4.9) reduces to $F_{1}(\Delta \phi,\lambda ) = F_{2}(\Delta \phi,\lambda ) = 0$, and the solutions to this automatically satisfy the charge–balance constraint (4.2) in the main text. Hence, the situation here does not contradict the result from the Lagrange multiplier rule.) Thus, the candidates for optimal solutions are obtained from $F_{1} = F_{2} = 0$.

Now we are in a position to distinguish optimal solutions from nonoptimal ones. For this purpose, the so-called bordered Hessian matrix of H [30] is introduced as:

$$\displaystyle\begin{array}{rcl} \mathcal{H}(H) = \left [\begin{array}{ccc} 0 &\mathcal{H}_{12} & \mathcal{H}_{13} \\ \mathcal{H}_{21} & \mathcal{H}_{22} & \mathcal{H}_{23} \\ \mathcal{H}_{31} & \mathcal{H}_{32} & \mathcal{H}_{33}\\ \end{array} \right ],& &{}\end{array}$$

(4.14)

whose elements are given by:

$$\displaystyle\begin{array}{rcl} & & \mathcal{H}_{12} = \mathcal{H}_{21} = \frac{\partial G} {\partial \Delta \phi } =\beta \left \langle \vert g(\theta )\vert ^{\beta -1}Z^{{\prime}}(\theta +\Delta \phi )\right \rangle,{}\end{array}$$

(4.15a)

$$\displaystyle\begin{array}{rcl} & & \mathcal{H}_{13} = \mathcal{H}_{31} = \frac{\partial G} {\partial \lambda } =\beta \left \langle \vert g(\theta )\vert ^{\beta -1}\right \rangle > 0,{}\end{array}$$

(4.15b)

$$\displaystyle\begin{array}{rcl} \mathcal{H}_{22} = \frac{\partial ^{2}F} {\partial \Delta \phi ^{2}}& =& \alpha \beta \left \langle \left \vert g(\theta )\right \vert ^{\beta -1}Z^{{\prime}}(\theta +\Delta \phi )^{2}\right \rangle \\ & +& \alpha \left \langle \mathrm{sig}[g(\theta )]\vert g(\theta )\vert ^{\beta }Z^{{\prime\prime}}(\theta +\Delta \phi )\right \rangle,{}\end{array}$$

(4.15c)

$$\displaystyle\begin{array}{rcl} & & \mathcal{H}_{23} = \frac{\partial ^{2}F} {\partial \Delta \phi \partial \lambda } = \alpha \beta \left \langle \vert g(\theta )\vert ^{\beta -1}Z^{{\prime}}(\theta +\Delta \phi )\right \rangle,{}\end{array}$$

(4.15d)

$$\displaystyle\begin{array}{rcl} \mathcal{H}_{32} = \frac{\partial ^{2}F} {\partial \lambda \partial \Delta \phi }& =& \alpha \frac{\partial } {\partial \Delta \phi }\left \langle \mathrm{sig}[g(\theta )]\vert g(\theta )\vert ^{\beta }\right \rangle \\ & =& \alpha \beta \left \langle \vert g(\theta )\vert ^{\beta -1}Z^{{\prime}}(\theta +\Delta \phi )\right \rangle,{}\end{array}$$

(4.15e)

$$\displaystyle\begin{array}{rcl} \mathcal{H}_{33} = \frac{\partial ^{2}F} {\partial \lambda ^{2}} & =& \alpha \frac{\partial } {\partial \lambda }\left \langle \mathrm{sig}[g(\theta )]\vert g(\theta )\vert ^{\beta }\right \rangle \\ & =& \alpha \beta \left \langle \vert g(\theta )\vert ^{\beta -1}\right \rangle =\alpha \mathcal{H}_{ 13} > 0,{}\end{array}$$

(4.15f)

where $g(\theta ) =\bar{ Z}(\theta )+\lambda$. The Hessian $\vert \mathcal{H}(H)\vert$ is then obtained as:

$$\displaystyle\begin{array}{rcl} \vert \mathcal{H}(H)\vert = \mathcal{H}_{13}(\alpha \mathcal{H}_{12}^{2} -\mathcal{H}_{ 13}\mathcal{H}_{22}),& &{}\end{array}$$

(4.16)

which turns out to be particularly useful because the solution $(\Delta \phi _{{\ast}},\lambda _{{\ast}})$ to $F_{1}(\Delta \phi,\lambda ) = F_{2}(\Delta \phi,\lambda ) = 0$ is maximal if it satisfies $\vert \mathcal{H}(H)\vert > 0$, and it is minimal if $\vert \mathcal{H}(H)\vert < 0$ [30]. Hence, from the above calculations, the optimal solution $(\Delta \phi,\lambda )$ to $F(\Delta \phi,\lambda ) = 0$ under the charge–balance constraint (4.2) in the main text is found to exist if the following conditions are satisfied:

$$\displaystyle\begin{array}{rcl} & & \left \langle \mathrm{sig}[\bar{Z}(\theta )+\lambda ]\vert \bar{Z}(\theta )+\lambda \vert ^{\beta }Z^{{\prime}}(\theta +\Delta \phi )\right \rangle = 0,{}\end{array}$$

(4.17)

$$\displaystyle\begin{array}{rcl} & & \left \langle \mathrm{sig}[\bar{Z}(\theta )+\lambda ]\vert \bar{Z}(\theta )+\lambda \vert ^{\beta }\right \rangle = 0,{}\end{array}$$

(4.18)

$$\displaystyle\begin{array}{rcl} & & \vert \mathcal{H}(H)\vert = \mathcal{H}_{13}(\alpha \mathcal{H}_{12}^{2} -\mathcal{H}_{ 13}\mathcal{H}_{22}) >\ 0,{}\end{array}$$

(4.19)

as in the nonlinear equations (4.5) in the main article.

Appendix 3 Detailed Information Regarding Optimal Forcings

Here we present detailed information regarding the optimal forcings for the example in Fig. 4.3 in the main text. These are numerically obtained by the algorithms related to P1 and P2, presented in the main text. We note these optimals are consistent with results from a brute force search for optimal forcings with a genetic algorithm, which are omitted here due to space limitations.

Fig. 4.3

All the optimal forcings for various p ( = 1, 1. 01, 2, 5, ∞) obtained for the Hodgkin–Huxley neuron phase model [17]: (a) p = 1, 1. 01, (b) p = 2, ∞, and (c) p = 5

All the solutions $(\Delta \phi _{{\ast}},\lambda _{{\ast}})$ to $H(\Delta \phi,\lambda ) = 0$, the associated locking range, and the sign of the Hessian $\vert \mathcal{H}(H)\vert$ are listed below for p = 1. 01, 2, 5, ∞.

• (a) p = 1. 01 (cf. Fig. 4.3a)

  solution 1::

  $(\Delta \phi,\lambda ) = (-\pi,0)$,

  locking range R = 2. 2585700, $\vert \mathcal{H}(H)\vert < 0$,

  solution 2::

  $(\Delta \phi,\lambda ) = (-1.3649363,0.57040837)$,

  R = 2. 7613265, $\vert \mathcal{H}(H)\vert > 0$;

• (b) p = 2 (cf. Fig. 4.3b)

  solution 1::

  $(\Delta \phi,\lambda ) = (-\pi,0)$,

  locking range R = 1. 3287487, $\vert \mathcal{H}(H)\vert < 0$,

  solution 2::

  $(\Delta \phi,\lambda ) = (-1.6150653,0.0000000)$,

  R = 1. 4926698, $\vert \mathcal{H}(H)\vert > 0$;

• (c) p = 5 (cf. Fig. 4.3c)

  solution 1::

  $(\Delta \phi,\lambda ) = (-\pi,0)$,

  locking range R = 1. 21853096, $\vert \mathcal{H}(H)\vert > 0$,

  solution 2::

  $(\Delta \phi,\lambda ) = (-2.5085617,-0.23864305)$,

  R = 1. 209864499, $\vert \mathcal{H}(H)\vert < 0$,

  solution 3::

  $(\Delta \phi,\lambda ) = (-1.948572843,-0.23837645)$,

  R = 1. 2161630, $\vert \mathcal{H}(H)\vert > 0$;

• (d) p = ∞ (cf. Fig. 4.3b)

  solution 1::

  $(\Delta \phi,\lambda ) = (-\pi,0)$,

  locking range R = 1. 1784578, $\vert \mathcal{H}(H)\vert > 0$.

  These solutions and the corresponding optimal wave forms, which have a positive $\vert \mathcal{H}(H)\vert$ in the above list, are shown in Fig. 4.3.

For the case of p = 1, $\Delta \phi _{\mathrm{max}}$ is (uniquely) obtained as $\Delta \phi _{\mathrm{max}}\ \approx -1.36094$ from the algorithm regarding P2. Note that this value of $\Delta \phi _{\mathrm{max}}$ corresponds to the optimal solution of $\Delta \phi \ \approx -1.364$ for p = 1. 01 shown in Fig. 4.3a.

Appendix 4 Derivation of Optimal Forcings in Two Limits

Here, we consider the optimal forcings in the following two cases: (1) p → ∞ and (2) p → 1.

[ Case of p → ∞]

First, we assume $0 <\Vert g\Vert _{q} < \infty$ and $0 \leq \vert g(\theta )\vert < \infty$, $\forall \theta \in S$, which is natural since g(θ) is given as $g(\theta ) =\bar{ Z}(\theta )+\lambda$. Then, it is obvious that $0 \leq \vert g(\theta )\vert /\Vert g\Vert _{q} < \infty$, and hence, as p → ∞,

$$\displaystyle\begin{array}{rcl} \left (\frac{\vert g(\theta )\vert } {\Vert g\Vert _{q}} \right )^{ \frac{1} {p-1} } \rightarrow \left \{\begin{array}{ll} 1,&\mathrm{for}\ \vert g(\theta )\vert > 0\\ 0, &\mathrm{for }\ \vert g(\theta )\vert = 0. \end{array} \right.& &{}\end{array}$$

(4.20)

Thus, taking the limit of Eq. (4.20) for f _∗ in Eq. (4.4) in the main text, the limit of f _∗(θ) → Msig[g(θ)] is obtained.

[ Case of p → 1]

The limiting value of $(\vert g(\theta )\vert /\Vert g\Vert _{q})^{ \frac{1} {p-1} }$ for f _∗ in Eq. (4.4) in the main text is obtained by the following calculations at θ = θ _∗ and at θ ≠ θ _∗. First, rescaling $\vert g(\theta )\vert /\Vert g\Vert _{q}$ by using $\bar{g}(\theta ) = g(\theta )/\vert g(\theta _{{\ast}})\vert$, we have $\vert \bar{g}(\theta _{{\ast}})\vert ^{ \frac{1} {p-1} } = 1$ (for any p), and for any θ we have:

$$\displaystyle\begin{array}{rcl} \left (\frac{\vert g(\theta )\vert } {\Vert g\Vert _{q}} \right )^{ \frac{1} {p-1} } = \left (\frac{\vert C\bar{g}(\theta )\vert } {\Vert C\bar{g}\Vert _{q}} \right )^{ \frac{1} {p-1} } ={\Bigl ( \frac{\vert \bar{g}(\theta )\vert } {\Vert \bar{g}\Vert _{q}} \Bigr )}^{ \frac{1} {p-1} },& &{}\end{array}$$

(4.21)

where C denotes $\vert g(\theta _{{\ast}})\vert \ (< \infty )$. Here we assume θ _∗ to have 0 measure. Then, from $\vert \bar{g}(\theta )\vert < 1$ and $\frac{1} {p} \rightarrow 1$ (p → 1), we have $\vert \bar{g}(\theta )\vert ^{ \frac{p} {p-1} } \rightarrow 0$ a.e. on S, resulting in $\langle \vert \bar{g}(\theta )\vert ^{ \frac{p} {p-1} }\rangle ^{ \frac{1} {p} } \rightarrow 0$ (p → 1). Thus, for θ = θ _∗ we obtain $\left (\frac{\vert g(\theta _{{\ast}})\vert } {\Vert g\Vert _{q}} \right )^{ \frac{1} {p-1} } \rightarrow +\infty$ (p → 1) from Eq. (4.21).

In contrast, for θ ≠ θ _∗, we have $\Vert \bar{g}\Vert _{q} \rightarrow \Vert \bar{ g}\Vert _{\infty } = 1\ (q \rightarrow +\infty )$, and $\bar{g}(\theta ) < 1$. This implies $\frac{\vert \bar{g}(\theta )\vert } {\Vert \bar{g}\Vert _{q}} \rightarrow \vert \bar{ g}(\theta )\vert < 1$ (q → +∞). Then, by taking the logarithm of Eq. (4.21), $\log {\Bigl [{\Bigl (\frac{\vert \bar{g}(\theta )\vert } {\Vert \bar{g}\Vert _{q}} \Bigr )}^{ \frac{1} {p-1} }\Bigr ]} = (q - 1)\log \left (\frac{\vert \bar{g}(\theta )\vert } {\Vert \bar{g}\Vert _{q}} \right ) \rightarrow -\infty \ (q \rightarrow +\infty )$, and hence ${\Bigl ( \frac{\vert g(\theta )\vert } {\Vert \bar{g}(\theta )\Vert _{q}}\Bigr )}^{ \frac{1} {p-1} } \rightarrow 0$ (p → 1). Thus, from these calculations, we obtain the limit of f _∗(θ) in Eq. (4.6b) in the main text.