##
**Trois problèmes sur les sommes trigonométriques.**
*(French)*
Zbl 0318.42002

Astérisque 1, 1-86 (1973).

This paper includes several interesting results, at various levels of generality, on trigonometric sums

\[ s(t)= a_0 + \sum a_k \cos\lambda_kt + \sum b_k \sin\lambda_kt. \tag{1} \]

Part I is the most specialized. Here the frequencies are chosen to be \(\lambda_k= \sqrt{k(k+n-2)}\), for a fixed \(n\geq 3\); these are the natural frequencies of vibration of the unit sphere \(S^{n-1}\) in \(\mathbb R^n\). The crucial result is

Proposition 1. There exists a constant \(C_n\) such that if \(a_0 =0\) and the sum in (1) is finite then \[ \sum (| a_k |+| b_k|)\leq \limsup_{t\to\infty} s(t). \]

Note that this would be false if \(n\) were 2; then (1) would be periodic. The proof depends on a number-theoretic fact: the set \(\{\lambda_k\}\) is linearly independent over \(\mathbb Z\). Applications are found in the theory of vibrations on \(S^{n-1}\); in this context, \(a_k\) and \(b_k\) belong to the space \(H_k\) of spherical harmonics of order \(k\) on \(S^{n-1}\). The reader may pause to reflect on theorems in a very classical topic of mathematical physics being proved with the aid of number-theoretic tools, including \(\ell\)-adic number fields.

Part II is devoted to the concept of harmonic density. If \(\Lambda\) is a set of real numbers, then let \(S_\Lambda\) denote the vector space of finite trigonometric sums with frequencies in \(\Lambda\). On \(S_\Lambda\) we consider the norms \(\sup_{\mathbb R} | f|\) and \(\sup_{[0,T]}| f|\) for finite \(T\). We say that \(\Lambda\) is coherent if for some finite \(T\) these norms are equivalent; then the infimum of the set of such \(T\) is called the harmonic density of \(\Lambda\) and denoted \(d_n(\Lambda)\). This definition is comparable with other notions of density. Finding \(d_n(\Lambda)\), even when \(\Lambda\) is a set of rational numbers, can be quite difficult.

Theorem 6 states that \(d_n(\Lambda)\) is the lower bound of the densities of the periodic sets which contain \(\Lambda\), provided that \(\Lambda\subset\mathbb Q\) and a certain technical condition is satisfied; this condition involves equidistribution of certain subsets of \(\Lambda\) modulo irrational numbers. The value of \(d_n(\Lambda)\) is computed for a family of sets not covered by the foregoing results. If \(\theta\) is real, let \(d\Lambda_\theta\) be the set of all finite sums of the form \(\sum \varepsilon_k\theta^k\) where \(\varepsilon_k=0\) or \(1\). In Theorem 8, \(d_n(\Lambda_\theta)\) is found for all \(\theta\); the most interesting cases are those for which \(|\theta|\) is a Pisot number and \(1\leq|\theta|\leq 2\).

Theorem 9 states that the harmonic density of a sufficiently regular model \(\Lambda\) is the lower bound of the densities of the regular models containing \(\Lambda\). This generalizes Theorem 6, and it leads to the proof of Theorem 8; the connection is that \(\Lambda_\theta\) is a sufficiently regular model when \(|\theta|\) is a Pisot number between 1 and 2.

Part III of the paper proves some refined theorems about spectral synthesis, again involving Pisot numbers. The questions raised in the theory of spectral synthesis are concerned with to generalizations of the set \(S_\Lambda\) defined above. If \(E\) is a compact subset of \(\mathbb R\), we let \(S_E\) denote the space of finite trigonometric sums with frequencies belonging to \(E\). We denote by the larger space of continuous bounded functions \(\varphi\) such that the Fourier transform of \(\varphi\) is a distribution supported by \(E\). The “problem of spectral synthesis” is this: whether or not \(S_E\) is dense in \(B_E\) for the topology \(\sigma(L^\infty,L^1)\).

\[ s(t)= a_0 + \sum a_k \cos\lambda_kt + \sum b_k \sin\lambda_kt. \tag{1} \]

Part I is the most specialized. Here the frequencies are chosen to be \(\lambda_k= \sqrt{k(k+n-2)}\), for a fixed \(n\geq 3\); these are the natural frequencies of vibration of the unit sphere \(S^{n-1}\) in \(\mathbb R^n\). The crucial result is

Proposition 1. There exists a constant \(C_n\) such that if \(a_0 =0\) and the sum in (1) is finite then \[ \sum (| a_k |+| b_k|)\leq \limsup_{t\to\infty} s(t). \]

Note that this would be false if \(n\) were 2; then (1) would be periodic. The proof depends on a number-theoretic fact: the set \(\{\lambda_k\}\) is linearly independent over \(\mathbb Z\). Applications are found in the theory of vibrations on \(S^{n-1}\); in this context, \(a_k\) and \(b_k\) belong to the space \(H_k\) of spherical harmonics of order \(k\) on \(S^{n-1}\). The reader may pause to reflect on theorems in a very classical topic of mathematical physics being proved with the aid of number-theoretic tools, including \(\ell\)-adic number fields.

Part II is devoted to the concept of harmonic density. If \(\Lambda\) is a set of real numbers, then let \(S_\Lambda\) denote the vector space of finite trigonometric sums with frequencies in \(\Lambda\). On \(S_\Lambda\) we consider the norms \(\sup_{\mathbb R} | f|\) and \(\sup_{[0,T]}| f|\) for finite \(T\). We say that \(\Lambda\) is coherent if for some finite \(T\) these norms are equivalent; then the infimum of the set of such \(T\) is called the harmonic density of \(\Lambda\) and denoted \(d_n(\Lambda)\). This definition is comparable with other notions of density. Finding \(d_n(\Lambda)\), even when \(\Lambda\) is a set of rational numbers, can be quite difficult.

Theorem 6 states that \(d_n(\Lambda)\) is the lower bound of the densities of the periodic sets which contain \(\Lambda\), provided that \(\Lambda\subset\mathbb Q\) and a certain technical condition is satisfied; this condition involves equidistribution of certain subsets of \(\Lambda\) modulo irrational numbers. The value of \(d_n(\Lambda)\) is computed for a family of sets not covered by the foregoing results. If \(\theta\) is real, let \(d\Lambda_\theta\) be the set of all finite sums of the form \(\sum \varepsilon_k\theta^k\) where \(\varepsilon_k=0\) or \(1\). In Theorem 8, \(d_n(\Lambda_\theta)\) is found for all \(\theta\); the most interesting cases are those for which \(|\theta|\) is a Pisot number and \(1\leq|\theta|\leq 2\).

Theorem 9 states that the harmonic density of a sufficiently regular model \(\Lambda\) is the lower bound of the densities of the regular models containing \(\Lambda\). This generalizes Theorem 6, and it leads to the proof of Theorem 8; the connection is that \(\Lambda_\theta\) is a sufficiently regular model when \(|\theta|\) is a Pisot number between 1 and 2.

Part III of the paper proves some refined theorems about spectral synthesis, again involving Pisot numbers. The questions raised in the theory of spectral synthesis are concerned with to generalizations of the set \(S_\Lambda\) defined above. If \(E\) is a compact subset of \(\mathbb R\), we let \(S_E\) denote the space of finite trigonometric sums with frequencies belonging to \(E\). We denote by the larger space of continuous bounded functions \(\varphi\) such that the Fourier transform of \(\varphi\) is a distribution supported by \(E\). The “problem of spectral synthesis” is this: whether or not \(S_E\) is dense in \(B_E\) for the topology \(\sigma(L^\infty,L^1)\).

Reviewer: C. J. Heinrich