1 Erratum to: Constr Approx (2014) 40:141–171 DOI 10.1007/s00365-014-9235-5

Several forms of uncertainty principles on the unit sphere are established in [1]. When stated in term of the vector

$$\begin{aligned} \tau (f):=\frac{1}{{\omega }_d}\int _{\mathbb {S}^{d-1}} x |f(x)|^2 \, d{\sigma }(x) \end{aligned}$$

of \({\mathbb R}^d\) (normalization constant \(1/{\omega }_d\) was missing in [1]), our main result is in:

Corollary 4.4

Let \(f\in W_2^1(\mathbb {S}^{d-1})\) be such that \(\int _{\mathbb {S}^{d-1}} f(y)\, d{\sigma }(y)=0\) and \(\Vert f\Vert _2=1\). If \(d \ge 2\), then

$$\begin{aligned} (1-\Vert \tau (f)\Vert ) \Vert \nabla _0 f\Vert _{2}^2\ge C_d^{-1}. \end{aligned}$$
(4.11)

Here \(C_d\) is a constant given in Theorem 4.1. We next attempted to remove the condition that \(\int _{\mathbb {S}^{d-1}} f(y)\, d{\sigma }(y)=0\) and stated:

Theorem 4.5

Assume that \(d\ge 2\), and let \(f\in W_2^1(\mathbb {S}^{d-1})\) be such that \(\Vert f\Vert _{2}=1\). Then

$$\begin{aligned} (1-\Vert \tau (f)\Vert ) \Vert \nabla _0 f\Vert _{2}^2 \ge c_d \Vert \tau (f)\Vert . \end{aligned}$$
(4.14)

This theorem, however, is incorrect. This was pointed out to us by Stefan Steinerberger, who showed that the inequality (4.14) does not hold for the function \(f(\cos {\theta }, \sin {\theta }) = 1+ \varepsilon \sin {\theta }\) for small enough \(\varepsilon \) when \(d=2\). The mistake in the proof appeared on the line 6 of page 166, which states that \(\Vert \tau (f)\Vert \le (2 |m_f| +1) \Vert g||_2^2\), but it should have been \(\Vert \tau (f)\Vert \le \Vert g\Vert _2^2 + 2 |m_f| \Vert g\Vert _2\). As a consequence, the right-hand side of (4.14) has to be replaced by \(c_d \Vert \tau (f)\Vert ^2\). Since \(\Vert \tau (f) \Vert \le \Vert f\Vert _2^2\), the resulted inequality is then equivalent to

$$\begin{aligned} (1-\Vert \tau (f)\Vert ^2) \Vert \nabla _0 f\Vert _{2}^2 \ge c_d \Vert \tau (f)\Vert ^2, \end{aligned}$$
(1)

which was already known in the literature; see the discussion in [1] and references therein.

Since (4.14) no longer holds, an immediate question is whether the uncertainty principle in (4.11) and that in (1) are equivalent, assuming \(\int _{\mathbb {S}^{d-1}} f(y)\, d{\sigma }(y)=0\). That they are not equivalent is shown in the following proposition.

Proposition 1

For \(n \ge 3\), let \(Y \in {\mathcal H}_n^d\), a spherical harmonic of degree \(n\) on \(\mathbb {S}^{d-1}\), and let \(Q\) be a polynomial of degree at most \(n-2\) such that \(\int _{\mathbb {S}^{d-1}} Q(x) d{\sigma }=0\). Assume that both \([Y(x)]^2\) and \([Q(x)]^2\) are even in every coordinate. Let

$$\begin{aligned} f = b (Y + Q), \quad \hbox {where}\quad b^{-1} := \Vert Y+Q\Vert _2 > 0. \end{aligned}$$

Then \(\tau (f) = 0\). In particular, (1) becomes the trivial inequality \(\Vert \nabla _0 f\Vert _2^2 \ge 0\), whereas (4.11) shows that \(\Vert \nabla _0 f\Vert _2^2 \ge c > 0\).

Proof

Since the degree of \(Q\) is at most \(n-2\), it follows from the orthogonality of \(Y\) and the even parity of \(Y^2\) and \(Q^2\) that

$$\begin{aligned} \int _{\mathbb {S}^{d-1}} x_i |f(x)|^2 d{\sigma }= \int _{\mathbb {S}^{d-1}} x_i \left( Y(x)^2 + 2 Y(x)Q(x) + Q(x)^2\right) d{\sigma }(x) =0 \end{aligned}$$

for \(1 \le k \le d\). Hence, \(\tau (f) =0\). By its definition, \(\Vert f\Vert _2 =1\), and, by the orthogonality of \(Y\) and the zero mean of \(Q\), we see that \(\int _{\mathbb {S}^{d-1}} f(x) d{\sigma }=0\) so that (4.11) is applicable on \(f\). \(\square \)

As a simple example of the function \(f\), we can choose \(Q(x) = x_1^k\) and \(Y(x) = C_n^{\lambda }(x_1)\) for \(x =(x_1,\ldots , x_d) \in \mathbb {S}^{d-1}\), where \({\lambda }= (d-2)/2\) and \(1 \le k \le n -2\).