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Theorem 28. For every complex measure \( \lambda \), there is a positive measure \( \mu \) and a function \( \mathbf{j} \) locally \( \mu \) -integrable such that\n\n\[ {d\lambda }\left( x\right) = \mathbf{j}\left( x\right) {d\mu }\left( x\right) \] | Let us set \( \lambda = {\lambda }_{1} + i{\lambda }_{2} \), where \( {\lambda }_{1} \) and \( {\lambda }_{2} \) are real and suppose there are positive measures such that\n\n\[ {\lambda }_{1} = {\lambda }_{1}^{\prime } - {\lambda }_{1}^{\prime \prime },\;{\lambda }_{2} = {\lambda }_{2}^{\prime } - {\lambda }_{2}^{\pri... | Yes |
For every continuous linear functional \( \nu \) on the Banach space \( {L}^{p}\left( {X;\mu }\right) \) with \( 1 < p < + \infty \) (resp. \( p = 1 \) ), there is a function \( \mathbf{j} \in {L}^{q}\left( {X;\mu }\right) \) (resp. bounded) such that\n\n\[ \nu \left( f\right) = \mu \left( {f\mathbf{j}}\right) \]\n\nfo... | This shows that for \( p > 1 \), the norm of the given linear functional \( \nu \) on \( {L}^{p} \) is\n\n\[ \parallel \mathbf{j}{\parallel }_{q} = \sup \left| {\nu \left( f\right) }\right| /\parallel f{\parallel }_{p}. \]\n\nFor \( p = 1,{L}^{q} \) needs to be replaced by the space \( {L}^{\infty }\left( {X;\mu }\righ... | Yes |
Theorem 31 (M. Fréchet, 1907). For any continuous linear functional \( f \) on \( \mathcal{H} \), there is a unique \( a \in \mathcal{H} \) such that \( f\left( x\right) = \left( {x \mid a}\right) \) for all \( x \in H \) . | One may assume that \( f \neq 0 \) . The subspace \( \mathcal{E} \) defined by \( f\left( x\right) = 0 \) is then closed and distinct from \( \mathcal{H} \) . Let us suppose that there exists \( b \neq 0 \) orthogonal to \( \mathcal{E} \) ; the relation \( f\left( x\right) = 0 \) then implies \( \left( {x \mid b}\right... | Yes |
Lemma 1. Let \( C \) be a closed convex set in \( \mathcal{H} \). For all \( a \in \mathcal{H} \), there is a unique \( {a}^{\prime } \in C \) such that\n\n(19.2)\n\n\[ \parallel x - a\parallel < \begin{Vmatrix}{{a}^{\prime } - a}\end{Vmatrix} \]\n\nfor all \( x \in C \) other than \( {a}^{\prime } \) . | Indeed, let \( m \) be the distance from \( a \) to \( C \) and let us choose elements \( {a}_{n} \in C \) such that \( \lim \begin{Vmatrix}{{a}_{n} - a}\end{Vmatrix} = m \). Applying to \( a - x \) and \( a - y \), the identity\n\n(19.3)\n\n\[ \parallel x + y{\parallel }^{2} + \parallel x - y{\parallel }^{2} = 2\left(... | Yes |
Lemma 2. Let \( \mathcal{E} \) be a closed vector subspace of \( \mathcal{H} \). Then \( \mathcal{H} \) is the direct sum of \( \mathcal{E} \) and of the orthogonal complement of \( \mathcal{E} \). | For all \( a \in \mathcal{H} \), let \( {a}^{\prime } \) be the unique point of \( \mathcal{E} \) at a minimum distance from \( a \). For all \( x \in \mathcal{E} \) and all \( \lambda \in \mathbb{C} \), \[ {\begin{Vmatrix}a - \left( {a}^{\prime } + \lambda x\right) \end{Vmatrix}}^{2} = {\begin{Vmatrix}a - {a}^{\prime ... | Yes |
Lemma 3. Let \( \mathcal{E} \) be a closed subspace of \( \mathcal{H} \) and \( \mathbf{A} \) a self-adjoint operator algebra on \( \mathcal{H} \) . The following properties are then equivalent:\n\n(i) \( \mathcal{E} \) is \( \mathbf{A} \) -invariant; \( {}^{75} \)\n\n(ii) the operator \( P \) of orthogonal projection ... | (i) \( \Rightarrow \) (ii): if \( x \) is orthogonal to \( \mathcal{E} \), then \( 0 = \left( {x \mid {Ty}}\right) = \left( {{T}^{ * }x \mid y}\right) \) for all \( y \in \mathcal{E} \) and all \( T \in \mathbf{A} \), so that the orthogonal complement of \( \mathcal{E} \) is \( \mathbf{A} \) -invariant.\n\n(ii) \( \Rig... | Yes |
Lemma 4. Let \( H \) be a Hermitian operator on a Hilbert space \( \mathcal{H} \). Then \( H - \zeta \) is invertible for all \( \zeta \notin \mathbb{R} \). | If \( \zeta = \alpha + {i\beta } \) with \( \beta \neq 0 \), then \( H - \zeta = \beta \left\lbrack {\left( {H - \alpha }\right) /\beta - i}\right\rbrack \) . As \( \left( {H - \alpha }\right) /\beta \) is again Hermitian, it suffices to prove the lemma for \( \zeta = i \) . First of all, \( H - i \) is injective since... | Yes |
Lemma 5. The spectrum of a Hermitian operator \( H \) is contained in the interval \( \left\lbrack {{m}_{H},{M}_{H}}\right\rbrack \) and contains its endpoints. | Let us suppose that for example \( \zeta < {m}_{H} \) and replace \( H \) by \( H - \zeta \), which subtracts \( \zeta \) from \( {m}_{H} \) and \( {M}_{H} \) . For the new operator, \( {m}_{H} > 0 \) and the proof reduces to deducing that \( H \) is invertible. As \( \left( {{Hx} \mid x}\right) \geq {m}_{H}\left( {x \... | Yes |
Lemma 6. Let \( \\left( {a}_{n}\\right) \) be a sequence of vectors such that \( \\lim \\left( {{a}_{n} \\mid y}\\right) \) exists for all \( y \) . Then \( \\sup \\begin{Vmatrix}{a}_{n}\\end{Vmatrix} < + \\infty \) and the sequence \( \\left( {a}_{n}\\right) \) converges weakly. | Setting \( {f}_{n}\\left( x\\right) = \\left( {{a}_{n} \\mid x}\\right) \) and \( f\\left( x\\right) = \\lim {f}_{n}\\left( x\\right) \), the proof reduces to showing that the linear functional \( f \) is continuous, and to obtain this, that the norms of \( {f}_{n} \) (i.e. of \( {a}_{n} \) ) are bounded. For all \( N ... | Yes |
Lemma 7. Let \( \\left( {A}_{n}\\right) \) be a sequence of continuous linear operators such that \( \\lim \\left( {{A}_{n}x \\mid y}\\right) \) exists for all \( x, y \) . Then \( \\sup \\begin{Vmatrix}{A}_{n}\\end{Vmatrix} < + \\infty \) and there is a continuous linear operator \( A \) such that | The corollary of theorem 31 tells us that it is sufficient to prove the first proposition. By the previous lemma, we already know that \( \\sup \\begin{Vmatrix}{{A}_{n}x}\\end{Vmatrix} < + \\infty \) for all \( x \\in \\mathcal{H} \), which is sufficient to prove the lemma \( \\left\\lbrack {{\\mathrm{n}}^{ \\circ }{15... | No |
Lemma 8. Let \( {H}_{1} \leq {H}_{2} \leq \ldots \) be an increasing sequence of Hermitian operators. Suppose that \( \sup \left( {{H}_{n}x \mid x}\right) < + \infty \) for all \( x \in \mathcal{H} \) . Then there is a continuous linear operator \( H \) such that \( \lim \begin{Vmatrix}{{Hx} - {H}_{n}x}\end{Vmatrix} = ... | As \( {H}_{pq} = {H}_{q} - {H}_{p} \geq 0 \) for \( q \geq p \), \[ {\left| \left( {H}_{pq}x \mid y\right) \right| }^{2} \leq \left( {{H}_{pq}x \mid x}\right) \left( {{H}_{pq}y \mid y}\right) . \] Hence like the sequences \( \left( {{H}_{n}x \mid x}\right) \) and \( \left( {{H}_{n}y \mid y}\right) \), \( \left( {{H}_{n... | Yes |
Lemma 9. Let \( \mathbf{A} \) be a self-adjoint operator algebra on a Hilbert space \( \mathcal{H} \) . \( \mathbf{A} \) is everywhere dense in \( {\mathbf{A}}^{\prime \prime } \) with respect to the ultrastrong topology if and only if\n\n(19.17)\n\n\[ \n{Ax} = 0\text{ for all }A \in \mathbf{A} \Rightarrow x = 0.\n\] | (17) is obviously necessary. So suppose that (17) holds.\n\nThe previous theorem tells us operators \( T + ╏ \), where \( T \in \mathbf{A} \), are ultra-strongly dense in \( {\mathbf{A}}^{\prime \prime } \) . If one shows that the operator 1 is in the ultrastrong closure of \( \mathbf{A} \), the result will follow.\n\n... | Yes |
Lemma 1. Let \( \\mathbf{A} \) be a complete normed algebra. For all \( x \\in \\mathbf{A} \), the spectrum of \( x \) is a non-empty compact set contained in the disc \( \\left| \\zeta \\right| \\leq \\parallel x\\parallel \) . | To prove this, one sets that the geometric series \( \\sum {x}^{n} \) converges for \( \\parallel x\\parallel < \) 1. Since calculations already encountered in Chap. II show that\n\n\[ \n\\left( {1 - x}\\right) \\left( {1 + x + \\ldots }\\right) = 1 \n\]\n\nit follows that \( 1 - x \) is invertible for \( \\parallel x\... | Yes |
Lemma 2. If every non-trivial element of a complete normed algebra \( \mathbf{A} \) is invertible, then \( \mathbf{A} \) has dimension 1. | Indeed, for all \( x \in \mathbf{A} \), there exists \( \zeta \in \mathbb{C} \) such that \( x - \zeta \) is not invertible, whence \( x - \zeta = 0 \) . | Yes |
Lemma 3 (Gelfand-Mazur). Let \( I \) be a maximal ideal of a commutative complete normed space \( \mathbf{A} \). Then \( \dim \left( {\mathbf{A}/I}\right) = 1 \). | Indeed, the quotient of a commutative ring by an ideal \( I \) can be considered to be a commutative ring by observing that, for \( x, y \in \mathbf{A} \), the coset of \( {xy}{\;\operatorname{mod}\;I} \) only depends on those of \( I \) : if \( a, b \in I \), then indeed\n\n\[ \left( {x + a}\right) \left( {y + b}\righ... | Yes |
Theorem 32. Let \( I \) be a maximal ideal of a commutative complete normed space \( \mathbf{A} \) . Then there is a linear functional \( \chi \) on \( \mathbf{A} \) such that \( \chi \left( 1\right) = 1 \), \[ \chi \left( {xy}\right) = \chi \left( x\right) \chi \left( y\right) \;\text{ for }x, y \in \mathbf{A} \] and ... | Since \( \mathbf{A}/I \) has dimension 1, for all \( x \in \mathbf{A} \) there is a unique scalar \( \chi \left( x\right) \) such that \[ x - \chi \left( x\right) {.1} \in I\text{.} \] The multiplicativity formula follows trivially. So does the characterization of \( x \in I \) . Conversely, any non-trivial solution of... | Yes |
Lemma 4. For all \( x \in \mathbf{A} \) ,\n\n\[ \parallel \widehat{x}\parallel = \lim {\begin{Vmatrix}{x}^{n}\end{Vmatrix}}^{1/n}. \] | For all \( n \) and character \( \chi ,\left| {\chi {\left( x\right) }^{n}}\right| = \left| {\chi \left( {x}^{n}\right) }\right| \leq \begin{Vmatrix}{x}^{n}\end{Vmatrix} \) . Thus \( \left| {\chi \left( x\right) }\right| \leq \) \( {\begin{Vmatrix}{x}^{n}\end{Vmatrix}}^{1/n} \) for all \( \chi \) and so\n\n\[ {\begin{V... | Yes |
Lemma 5. Let \( \mathcal{H} \) be a Banach space, \( {\mathcal{H}}^{\prime } \) the space of continuous linear functionals on \( \mathcal{H} \) and \( {B}^{\prime } \) the unit ball \( \parallel f\parallel \leq 1 \) of \( {\mathcal{H}}^{\prime } \) . Then \( {B}^{\prime } \) is compact with respect to the weak topology... | The proof is easy provided any (finite or infinite) Cartesian product of compact spaces is admitted to be compact, which is another application of Zorn’s theorem mentioned above in a footnote. This being admitted, let \( D \) be the disc \( \left| z\right| \leq 1 \) of \( \mathbb{C} \) and \( B \) the unit ball \( \par... | Yes |
Theorem 33. Let \( \mathbf{A} \) be a commutative GN algebra. Then the Gelfand transform \( x \mapsto \widehat{x} \), defined by\n\n\[ \widehat{x}\left( \chi \right) = \chi \left( x\right) \]\n\nis an isomorphism from \( \mathbf{A} \) onto the GN algebra of continuous functions on the spectrum \( \widehat{\mathbf{A}} \... | A corollary of this result is that, if \( f\left( \zeta \right) \) is a complex-valued function defined and continuous (but not necessarily analytic!) on the spectrum of some \( x \in \mathbf{A} \), then there is a unique \( y \in \mathbf{A} \) such that\n\n\[ \widehat{y}\left( \chi \right) = f\left\lbrack {\widehat{x}... | Yes |
Corollary 2 (Schur’s lemma II). Let \( \left( {\mathcal{H}, U}\right) \) be an irreducible unitary representation of some \( \operatorname{lcg}G \) and \( S \) a self-adjoint or positive symmetric operator defined on an everywhere dense subspace \( \mathcal{D} \) of \( \mathcal{H} \) . Suppose \( \mathcal{D} \) is \( U... | If \( S \) is positive symmetric, the canonical self-adjoint extension \( S \) commutes with all \( U\left( x\right) \), hence so do its spectral projections, which being continuous, are scalars, proving the result in this case.\n\nIf \( S \) is not positive, but self-adjoint, consider \( S + i \) . As in \( {\mathrm{n... | No |
Lemma 1. Let \( f \) be a function on \( G \) with values in a topological space. The function \( \left( {x, y}\right) \mapsto f\left( {xy}\right) \) is measurable on \( G \times G \) if and only if \( f \) is measurable on \( G \) . | Indeed, the most primitive version of LF shows that l'on a\n\n\[\n\iint F\left( {x, y}\right) {dxdy} = \iint F\left( {x,{y}^{-1}}\right) \Delta \left( y\right) {dxdy} = \int \Delta \left( y\right) {dy}\int F\left( {x,{y}^{-1}}\right) {dx}\n\]\n\n\[= \int \Delta \left( y\right) {dy}\int F\left( {{xy},{y}^{-1}}\right) \D... | Yes |
If \( \lambda \) is a bounded measure, then \( \lambda * \varphi \) is defined for all \( \varphi \in {L}^{p}\left( G\right) \) and all \( p \in \left\lbrack {1, + \infty }\right\rbrack \), and is again in \( {L}^{p} \) and \[ \parallel \lambda * \varphi {\parallel }_{p} \leq \parallel \lambda \parallel \cdot \parallel... | \( \lambda \) and \( \varphi \) may be supposed to be positive by replacing them with their absolute values. We first need to show that, for all \( f \in {L}_{ + }\left( G\right) \), the function \( f\left( {xy}\right) \) is integrable with respect to \( {d\lambda }\left( x\right) \varphi \left( y\right) {dy} \), i.e. ... | Yes |
Theorem 37. Let \( G \) be a locally compact group. For all \( \varphi \in {L}^{1}\left( G\right) \) and \( \psi \in {L}^{p}\left( G\right) ,1 \leq p \leq + \infty \), the function \( y \mapsto \varphi \left( {xy}\right) \psi \left( {y}^{-1}\right) \) is integrable for almost all \( x \in G \), the function \[ \int \va... | If \( G \) is unimodular, the product \( \varphi * \psi \) exists when \( \varphi \in {L}^{p},\psi \in {L}^{q} \) with \( 1/p + 1/q = 1 \) . To see this it suffices to verify that \( f\left( {xy}\right) \varphi \left( x\right) \psi \left( y\right) \) is integrable over \( G \times G \) for all \( f \in L\left( G\right)... | No |
Lemma 2. Let \( X \) be a locally compact space, a a point of \( X \) and \( \left( {\mu }_{n}\right) \) a sequence \( {}^{97} \) of bounded complex measures on \( X \) . Assume the following conditions hold:\n\n(D 1) \( \sup \begin{Vmatrix}{\mu }_{n}\end{Vmatrix} < + \infty \) ;\n\n(D 2) \( \lim \int d{\mu }_{n}\left(... | Dividing \( {\mu }_{n} \) by \( {\mu }_{n}\left( 1\right) \) which, by \( \left( {\mathrm{D}2}\right) \), tends to 1 reduces the proof to the case where \( {\mu }_{n}\left( 1\right) = 1 \) for all \( n \) . The function \( f \) being Borel and bounded, and hence integrable with respect to any bounded measure,\n\n\[ {\m... | Yes |
If \( G = \mathbb{R} \), the Fourier transform immediately provides us with such homomorphisms, namely\n\n\[ f \mapsto \widehat{f}\left( y\right) = \int f\left( x\right) \overline{\mathbf{e}\left( {xy}\right) }{dx} \] | where recall that \( \mathbf{e}\left( x\right) = \exp \left( {2\pi ix}\right) \) . If \( \mathbb{R} \) is replaced by a Cartesian space \( E \) with dual \( {E}^{ * } \), then likewise, setting\n\n(26.1)\n\n\[ \mathbf{e}\left( {x, y}\right) = \exp \left( {{2\pi i}\langle x, y\rangle }\right) \]\n\nwhere \( \langle x, y... | Yes |
Lemma 1. Let \( \chi \) be a homomorphism from \( {L}^{1}\left( G\right) \) to \( \mathbb{C} \) which is not identically zero. Then there is a unique continuous function \( \chi \left( x\right) \) on \( G \) such that\n\n(26.2)\n\n\[ \chi \left( {xy}\right) = \chi \left( x\right) \chi \left( y\right) \]\n\nand for whic... | The existence of a bounded and measurable function \( \chi \left( x\right) \) satisfying (3) follows from the duality of \( {L}^{p} \) (n° 18, theorem 30). By definition of \( f * g \) , relation \( \chi \left( {f * g}\right) = \chi \left( f\right) \chi \left( g\right) \) can then be written\n\n\[ \iint \overline{\chi ... | Yes |
For \( G = \mathbb{R} \), we recover the exponentials \( \mathbf{e}\left( {xy}\right) \) of Fourier transforms. | There are no other solutions given the characterization of exponential functions by their functional equation (Chap. IV). | No |
Lemma 2. The weak topology is identical to the topology of compact convergence on \( \widehat{G} \) , \( t \) | (This shows that, for \( G = \mathbb{R} \), this is the usual topology of \( \mathbb{R} \) ). First, compact convergence implies weak convergence in the unit ball of \( {L}^{\infty } \) . Indeed, for \( f \in {L}^{1} \) and \( \varphi ,{\varphi }_{0} \) in the unity ball of \( {L}^{\infty } \) ,\n\n\[ \left| {{\int }_{... | Yes |
Lemma 3. Let \( E \) be a Banach space, \( {E}^{\prime } \) its topological dual, \( {B}^{\prime } \) the unit \( {\text{ball}}^{100} \) of \( {E}^{\prime } \) and \( K \) a compact subset of \( E \) . Weak convergence in \( {B}^{\prime } \) is equivalent to uniform convergence in \( K \) . | We need to show that, for all \( {f}_{0} \in {B}^{\prime } \) and all \( r > 0 \), there exists a neighbourhood \( V \) of \( {f}_{0} \) in \( {B}^{\prime } \) equipped with the weak topology such that\n\n\[ f \in V \Rightarrow \left| {f\left( \mathbf{x}\right) - {f}_{0}\left( \mathbf{x}\right) }\right| < r\;\text{ for... | Yes |
Theorem 39. The Fourier transform \( f \mapsto \widehat{f}, f \in {L}^{1}\left( G\right) \), is injective. For any pair of distinct elements \( x, y \) of \( G \), there is a character \( \chi \) of \( G \) such that \( \chi \left( x\right) \neq \chi \left( y\right) \) . | If \( \widehat{f}\left( \chi \right) = 0 \) for all \( \chi \in \widehat{G} \), then, by (6), \( \chi \left\lbrack {U\left( f\right) }\right\rbrack = 0 \) must perforce hold for any character \( \chi \) of \( \mathbf{A}\left( G\right) \) . But in a GN algebra, the Gelfand transform is injective (theorem 33). Hence \( U... | Yes |
Lemma 2. For all \( f, g, p, q \in {L}^{12}\left( G\right) \), \[ \widehat{f}\left( \chi \right) \overline{\widehat{g}\left( \chi \right) }d{\mu }_{p, q}\left( \chi \right) = \widehat{p}\left( \chi \right) \overline{\widehat{q}\left( \chi \right) }d{\mu }_{f, g}\left( \chi \right) . \] | As (20) is an identity between two measures, we need to show that \[ \int \varphi \left( \chi \right) \widehat{f}\left( \chi \right) \overline{\widehat{g}\left( \chi \right) }d{\mu }_{p, q}\left( \chi \right) = \int \varphi \left( \chi \right) \widehat{p}\left( \chi \right) \overline{\widehat{q}\left( \chi \right) }d{\... | Yes |
Lemma 3. For \( p, q \in {L}^{12}\left( G\right) \), the measure \( d{\mu }_{p, q} \) does not depend on the function \( p * \widetilde{q} \) . | Indeed by (17), under the assumptions of the lemma, for \( f \in L\left( G\right) \)\n\n\[ \n{\mu }_{p, q}\left( \widehat{f}\right) = \left( {M\left( \widehat{f}\right) p \mid q}\right) = \left( {U\left( f\right) p \mid q}\right) = \left( {f * p \mid q}\right) .\n\]\n\nSo by (25.24'),\n\n\[ \n{\mu }_{p, q}\left( \wideh... | Yes |
Theorem 40. The invariant measure on \( \widehat{G} \) can be chosen in such a way that\n\n\[ d{\mu }_{p, q}\left( \chi \right) = \widehat{p}\left( \chi \right) \overline{\widehat{q}\left( \chi \right) }{d\chi } \] \n\nfor all \( p, q \in {L}^{12}\left( G\right) \) . | To simplify calculations, I will denote by \( \Lambda \) the set of functions of the form \( P\left( \chi \right) = \widehat{p}\left( \chi \right) \overline{\widehat{q}\left( \chi \right) } \) on \( G \), where \( p, q \in {L}^{12}\left( G\right) \) . Lemma 3 enables us to associate the measure\n\n\[ d{\mu }_{P}\left( ... | Yes |
Lemma 4. Let \( X \) be a locally compact space and \( {\left( {\Omega }_{i}\right) }_{i} \in I \) an open cover of \( X \) . Suppose that a measure \( {\mu }_{i} \) is given on each \( {\Omega }_{i} \) . There is a measure \( \mu \) on \( X \) such that \[ \mu \left( f\right) = {\mu }_{i}\left( f\right) \] for all \( ... | (25) is clearly necessary. If this condition holds and if some \( f \in L\left( X\right) \) vanishes outside a compact set \( K \), there exist \( {p}_{i} \in L\left( X\right) \) whose supports are contained in the sets \( {\Omega }_{i} \), which are trivial for almost all \( i \), and such that \( \sum {p}_{i} = 1 \) ... | Yes |
Corollary 1 (Plancherel). There is a unique isomorphism \( f \mapsto \widehat{f} \) from \( {L}^{2}\left( G\right) \) onto \( {L}^{2}\left( \widehat{G}\right) \) which reduced to the Fourier transform on \( {L}^{12}\left( G\right) \) . | The proof reduces to showing that the set of \( \widehat{f}, f \in {L}^{12} \), is everywhere dense on \( {L}^{2}\left( \widehat{G}\right) \) . Otherwise, there is a non-zero function \( F \in {L}^{2}\left( \widehat{G}\right) \) orthogonal to all \( \widehat{f} \), in particular to all \( P \in \Lambda \) used in the p... | Yes |
Corollary 2. The Fourier transform of the product \( {fg} \) of two square integrable functions is the convolution \( \widehat{f} * \widehat{g} \) of their Fourier transforms. | Let us prove this by replacing \( g \) by its conjugate. The Fourier transform of the convolution is the function\n\n\[ \chi \mapsto \int f\left( x\right) \overline{g\left( x\right) \mathbf{e}\left( {x;\chi }\right) }{dx}. \]\n\nThat of the function \( g\left( x\right) \mathbf{e}\left( {x;\chi }\right) \) is clearly th... | Yes |
Theorem 41. Let \( f \) be a function belonging either to \( {L}^{1}\left( G\right) \) or to \( {L}^{2}\left( G\right) \), and whose Fourier transform is integrable; then\n\n\[ f\left( x\right) = \int \mathbf{e}\left( {x;\chi }\right) \widehat{f}\left( \chi \right) {d\chi }\;\text{ almost everywhere } \]\n\nand the rig... | (a) If \( f \in {L}^{1}\left( G\right) \), then the operator \( U\left( f\right) : g \mapsto f * g \) is in the algebra \( \mathbf{A}\left( G\right) \) and\n\n\[ \left( {U\left( f\right) p \mid q}\right) = \int \widehat{f}\left( \chi \right) \widehat{p}\left( \chi \right) \overline{\widehat{q}\left( \chi \right) }{d\ch... | Yes |
Theorem 42. The canonical homomorphism \( j : G \rightarrow \widehat{\widehat{G}} \) is bijective. | The Fourier transform on \( G \) is an isomorphism \( f \mapsto \widehat{f} \) from \( {L}^{2}\left( G\right) \) onto \( {L}^{2}\left( \widehat{G}\right) \) . Likewise, the Fourier transform on \( \widehat{G} \) is an isomorphism onto \( {L}^{2}\left( \widehat{\widehat{G}}\right) \) mapping \( \widehat{f} \) onto \( \w... | Yes |
Let \( \left( {\mathcal{H}, U}\right) \) be a representation of \( G \) . Every \( \mathbf{a} \in \mathcal{H} \) is the limit of vectors of the form \( U\left( f\right) \mathbf{a} \) . | The first proposition was proved at the end of \( {\mathrm{n}}^{ \circ }{25} \) : given measures of the form \( {f}_{n}\left( x\right) {dx} \), with \( {f}_{n} \geq 0 \) having integral equal to 1 and vanishing outside compact on decreasingly small neighbourhoods of \( e,\lim U\left( {f}_{n}\right) \mathbf{a} = \mathbf... | Yes |
Lemma 2. Let \( \mathcal{H} \) be a Hilbert space and \( f \mapsto U\left( f\right) \) a homomorphism from the algebra \( L\left( G\right) \) to the algebra \( \mathcal{L}\left( \mathcal{H}\right) \) . Suppose that\n\n\[ U{\left( f\right) }^{ * } = U\left( \widetilde{f}\right) ,\;\parallel U\left( f\right) \parallel \l... | First, the subspace \( {\mathcal{H}}_{0} \) generated by all vectors \( U\left( f\right) \mathbf{a} \) is everywhere dense in \( \mathcal{H} \) because a vector orthogonal to \( {\mathcal{H}}_{0} \) is a zero of all \( U{\left( f\right) }^{ * } \), hence of all \( U\left( f\right) \), and so is zero.\n\nTo define \( U\... | Yes |
Lemma 2. Let \( \varphi \) be a continuous function of positive type. \( \varphi \) is elementary if and only if every continuous function of positive type \( \psi \ll \varphi \) is proportional to \( \varphi \) . Every measure \( \mu \) of positive type such that \( \mu \ll \varphi \) is then proportional to the measu... | An easy calculation shows that, for any finite number of \( {x}_{i} \in G \) and scalars \( {\xi }_{i} \in \mathbb{C} \), in notation (10), (30.14) \[ \sum \varphi \left( {{x}_{j}^{-1}{x}_{i}}\right) {\xi }_{i}{\bar{\xi }}_{j} = {\begin{Vmatrix}\sum {\xi }_{i}U\left( {x}_{i}\right) \mathbf{a}\end{Vmatrix}}^{2} \geq 0. ... | No |
Theorem 45 (S. Bochner). A continuous function \( \varphi \) on a commutative \( \log G \) is of positive type if and only if there is a bounded positive measure \( {}^{120} \) \( d\widehat{\varphi }\left( \chi \right) \) on \( \widehat{G} \) such that\n\n\[ \varphi \left( x\right) = \int \mathbf{e}\left( {x,\chi }\rig... | The measure \( \widehat{\varphi } \) is unique.\n\nIndeed, applying theorem 44 to the measure \( {d\mu }\left( x\right) = \varphi \left( x\right) {dx} \), we get a measure \( \widehat{\mu } \geq 0 \) on \( \widehat{G} \) such that\n\n\[ \int \widehat{f}\left( \chi \right) \overline{\widehat{g}\left( \chi \right) }d\wid... | Yes |
Theorem 47. Let \( \\mathbf{A} \) be a Hilbert algebra and \( \\mathcal{H} \) its completion. If \( A \) and \( B \) are continuous operators on \( \\mathcal{H} \) such that\n\n\[ \n{AR}\\left( f\\right) = R\\left( f\\right) A,\\;{BL}\\left( f\\right) = L\\left( f\\right) B \n\]\n\nfor all \( f \\in \\mathbf{A} \), the... | The first step of the proof consists in associating to each \( \\varphi \\in \\mathcal{H} \) a linear map \( R\\left( \\varphi \\right) : \\mathbf{A} \\rightarrow \\mathcal{H} \) by setting\n\n(31.15)\n\n\[ \nR\\left( \\varphi \\right) f = L\\left( f\\right) \\varphi = f.\\varphi \\;\\text{ for }f \\in \\mathbf{A}. \n\... | No |
Lemma 1. If \( \varphi \in \mathcal{H} \) is right moderate, so is \( {B\varphi } \) and\n\n\[ R\left( {B\varphi }\right) = {BR}\left( \varphi \right) . \] | Indeed, for all \( f \in \mathbf{A} \) ,\n\n\[ {BR}\left( \varphi \right) f = {BL}\left( f\right) \varphi = L\left( f\right) {B\varphi }.\]\n\nAs \( B \) and \( R\left( \varphi \right) \) are continuous, the map \( f \mapsto L\left( f\right) {B\varphi } \) is continuous, so that \( {B\varphi } \) is right moderate, qed... | Yes |
Lemma 2. If \( \varphi \) is right moderate, then so is \( \widetilde{\varphi } \) and\n\n\[ R\left( \widetilde{\varphi }\right) = R{\left( \varphi \right) }^{ * }.\] | Indeed by (15), for \( f, g \in \mathbf{A} \), \n\n\[ \left( {R\left( \varphi \right) f \mid g}\right) = \left( {L\left( f\right) \varphi \mid g}\right) = \left( {\varphi \mid L\left( \widetilde{f}\right) g}\right) = \left( {\varphi \mid \widetilde{f}.g}\right) = \left( {\widetilde{g}.f \mid \widetilde{\varphi }}\right... | Yes |
Lemma 3. Any right moderate \( \varphi \in \mathcal{H} \) is left moderate and | \[ {SR}\left( \varphi \right) {S}^{-1} = L\left( \widetilde{\varphi }\right) = L{\left( \varphi \right) }^{ * } \] For \( f \in \mathbf{A} \), \[ {SR}\left( \varphi \right) {S}^{-1}f = {SR}\left( \varphi \right) \widetilde{f} = {SL}\left( \widetilde{f}\right) \varphi = R\left( f\right) {S}^{-1}\varphi \;\text{ by (10) ... | Yes |
Lemma 4. If \( \varphi \) and \( \psi \) are moderate, then\n\n\[ L\left( \varphi \right) \psi = R\left( \psi \right) \varphi . \] | For \( f \in \mathbf{A} \) ,\n\n\[ \left( {L\left( \varphi \right) \psi \mid f}\right) = \left( {\psi \mid L\left( \widetilde{\varphi }\right) f}\right) \;\left( {\text{ lemma }2}\right) \]\n\n\[ = \left( {\psi |R\left( f\right) \widetilde{\varphi }}\right) = \left( {R\left( \widetilde{f}\right) \psi |\widetilde{\varph... | Yes |
Theorem 48. The function \( \operatorname{Tr}\left( H\right) \) has the following properties:\n\n(a) \( \operatorname{Tr}\left( {{UH}{U}^{ * }}\right) = \operatorname{Tr}\left( H\right) \) for any unitary operator \( U \in \mathcal{L} \) ; | (a) The operator \( U{H}^{1/2}{U}^{ * } \) is positive Hermitian and \( {U}^{ * }U = 1 \) since \( U \) is isometric. So \( {\left( U{H}^{1/2}{U}^{ * }\right) }^{2} = U{H}^{1/2}{U}^{ * }U{H}^{1/2}{U}^{ * } = {UH}{U}^{ * } \) and\n\n\[ U{H}^{1/2}{U}^{ * } = {\left( UH{U}^{ * }\right) }^{1/2}. \]\n\nThis being so, we fir... | Yes |
Theorem 51 (Bargmann orthogonality relations). Let \( \left( {\mathcal{H}, U}\right) \) be an irreducible unitary representation of a locally compact unimodular group \( G \) and \( Z \) the centre of \( G \) . Set \( U\left( z\right) = \alpha \left( z\right) 1 \) for all \( z \in Z \) . (i) If the representation \( \l... | Proof of (i) and (ii). Let us suppose that \( {\varphi }_{{a}^{\prime }, a} \in {L}^{2}\left( {G/Z;\alpha }\right) \) for a pair of vectors \( a,{a}^{\prime } \in \mathcal{H} \) . The set \( \mathcal{D} \) of \( u \in \mathcal{H} \) such that \( {\varphi }_{u, a} \) has the same property is invariant by (11), and so is... | Yes |
For all \( f \in \mathcal{S}\left( \mathbb{R}\right) \), the function \( {\theta }_{f}^{ * } \) is \( {C}^{\infty } \) on \( {\mathbb{R}}^{ * } \) and rapidly decreasing at infinity and so are all its derivatives. | As, for all \( N \in \mathbb{N} \), there is an upper bound \( \left| {{x}^{N}f\left( x\right) }\right| \leq {c}_{N} \) valid in all of \( \mathbb{R} \) [the left hand side is continuous on \( \mathbb{R} \) and bounded for large \( \left| x\right| \), hence on \( \mathbb{R}\rbrack \), it follows that, for \( N \geq 2 \... | Yes |
Theorem 1. The function\n\n\[ \xi \left( s\right) = {\pi }^{-s/2}\Gamma \left( {s/2}\right) \zeta \left( s\right) ,\;\operatorname{Re}\left( s\right) > 1 \]\n\nextends analytically to a meromorphic function on \( \mathbb{C} \) whose only singularities are simple poles at \( s = 1 \) and \( s = 0 \), where its residues ... | The \( \zeta \) function extends analytically to all of \( \mathbb{C} \), except for a simple pole at \( s = 1 \) where\n\n\[ \operatorname{Res}\left( {\zeta ,1}\right) = 1 \]\n\n\( \zeta \left( s\right) = 0 \) for \( s = - 2, - 4,\ldots \) .\n\nThe assertions about \( \xi \left( s\right) \) follow from properties of \... | Yes |
Theorem 4 (Jacobi,1828). For \( \\left| q\\right| < 1 \) and \( w \\in {\\mathbb{C}}^{ * } \) , \[ \\mathop{\\sum }\\limits_{\\mathbb{Z}}{q}^{{m}^{2}}{w}^{m} = \\mathop{\\prod }\\limits_{{n \\geq 1}}\\left( {1 - {q}^{2n}}\\right) \\left( {1 + {q}^{{2n} - 1}w}\\right) \\left( {1 + {q}^{{2n} - 1}{w}^{-1}}\\right) . \] | In what follows, \( J\\left( {q, w}\\right) \) will denote the Jacobi series and \( A\\left( {q, w}\\right) \) the Abel infinite product like in Remmert, Funktionentheorie 2, pp. 22-27, which presents the classic proof that I will not follow, except for its quasi-trivial part. Mine (?) will show the reader getting to g... | Yes |
\[ \mathop{\prod }\limits_{{n \geq 1}}\left( {1 - {q}^{n}}\right) = \mathop{\sum }\limits_{\mathbb{Z}}{\left( -1\right) }^{m}{q}^{\left( {{3m} + 1}\right) m/2}. \] | Replace \( q \) by \( {q}^{3/2} \) and \( w \) by \( - {q}^{\frac{1}{2}} \) in (2). | No |
Corollary 2 (Jacobi).\n\n\[ \mathop{\prod }\limits_{{n \geq 1}}{\left( 1 - {q}^{n}\right) }^{3} = \mathop{\sum }\limits_{\mathbb{Z}}{\left( -1\right) }^{m}m{q}^{m\left( {m + 1}\right) /2} = \] | \[ = \mathop{\sum }\limits_{{m \geq 0}}{\left( -1\right) }^{m}\left( {{2m} + 1}\right) {q}^{m\left( {m + 1}\right) /2}. \] | Yes |
Lemma 1. Homomorphism (2) is injective if and only if \( \left( {{m}^{\prime },{m}^{\prime \prime }}\right) = 1 \) ; it is then bijective. | The injectivity of (2) means that any common multiple of \( {m}^{\prime } \) and \( {m}^{\prime \prime } \) is a multiple of \( {m}^{\prime }{m}^{\prime \prime } \), hence that \( {m}^{\prime }{m}^{\prime \prime } \) is the \( {lcm} \) of \( {m}^{\prime } \) and \( {m}^{\prime \prime } \) . As this \( {lcm} \) is alway... | Yes |
Lemma 2. Let \( m \) and \( {m}^{\prime } \) be two integers such that \( {m}^{\prime } \mid m \) . The canonical homomorphism \( G\left( m\right) \rightarrow G\left( {m}^{\prime }\right) \) is surjective. | The proof reduces to showing that any \( {a}^{\prime } \) such that \( \left( {{m}^{\prime },{a}^{\prime }}\right) = 1 \) is the class \( {\;\operatorname{mod}\;{m}^{\prime }} \) of some \( a \) such that \( \left( {m, a}\right) = 1 \), i.e. that there exists \( x \) such that \( \left( {m,{a}^{\prime } + }\right. \) \... | Yes |
Lemma 3. Let \( {m}^{\prime } \) and \( {m}^{\prime \prime } \) be two coprimes and \( m = {m}^{\prime }{m}^{\prime \prime } \) . Let \( {G}^{\prime }\left( m\right) \) (resp. \( \left. {{G}^{\prime \prime }\left( m\right) }\right) \) be the subgroup of \( x \equiv 1\left( {\;\operatorname{mod}\;{m}^{\prime \prime }}\r... | This enables us to calculate the number \( \varphi \left( m\right) \) of elements of \( G\left( m\right) \), known as the Euler indicator of \( m \) . To start with, it is clear that \( \varphi \left( {{m}^{\prime }{m}^{\prime \prime }}\right) = \varphi \left( {m}^{\prime }\right) \varphi \left( {m}^{\prime \prime }\ri... | No |
Lemma 4. Let \( \chi \) be a character \( {\;\operatorname{mod}\;m} \) and \( {m}^{\prime },{m}^{\prime \prime } \) divisors of \( m \) such that \( \chi \) is both a character \( {\;\operatorname{mod}\;{m}^{\prime }} \) and \( {\;\operatorname{mod}\;{m}^{\prime \prime }} \) . Let \( d \) be the gcd of \( {m}^{\prime }... | Let \( \mu \) be the \( {lcm} \) of \( {m}^{\prime } \) and \( {m}^{\prime \prime } \) ; it divides \( m \), so that \( \chi \) follows from the composition either of maps\n\n\[ G\left( m\right) \rightarrow G\left( \mu \right) \rightarrow G\left( {m}^{\prime }\right) \rightarrow {\chi }^{\prime } \]\n\nor of analogous ... | Yes |
Theorem 5. Let \( \chi \) be a character \( {\;\operatorname{mod}\;m} \) and \( n \) an integer. Set\n\n(9.16)\n\n\[ d = \left( {m, n}\right) ,\;m = d{m}^{\prime },\;n = d{n}^{\prime }.\]\n\n(a) In the three following cases\n\n(9.17)\n\n\[ {\Gamma }_{m}\left( {n,\chi }\right) = {\Gamma }_{m}\left( \chi \right) \overlin... | (14) shows that first of all\n\n\[ {\Gamma }_{m}\left( {n,\chi }\right) \chi \left( y\right) = \sum \chi \left( {xy}\right) \mathbf{e}\left( {{xn}/m}\right) \;\text{ for all }y \in \mathbb{Z}. \]\n\n\( y \) may be supposed \( {}^{24} \) to be in \( G\left( m\right) \) for otherwise the previous relation would reduce to... | No |
Theorem 7 (Liouville, Hermite). All entire elliptic functions are constants. For any non-constant elliptic function \( f \) , | \[ \sum {v}_{a}\left( f\right) = 0,\;\sum \operatorname{Res}\left( {f, a}\right) = 0. \] Needless to say that summations are extended to all classes \( {\;\operatorname{mod}\;L} \) . The former result is obvious even without knowing Cauchy theory: it suffices to expand the given function into a Fourier series with resp... | No |
Corollary 1. Let \( f \) and \( g \) be two elliptic functions such that \( {v}_{a}\left( f\right) = {v}_{a}\left( g\right) \) for all \( a \in \mathbb{C} \) . Then \( f \) and \( g \) are proportional. | For \( f/g \) is an elliptic function without any zeros or poles. | No |
Corollary 2. There are no elliptic functions of order 1. | For such a function would have a unique simple pole with non-trivial residue. | No |
Corollary 3. An elliptic function has as many zeros as poles. | The zeros and poles are obviously counted \( {\;\operatorname{mod}\;L} \) and their multiplicities taken account of. The total number of zeros or poles, or more generally of the equation \( f\left( u\right) = c \), is called the order of \( f \) . Applying this corollary to \( f - c \), where \( c \) is a constant, sho... | Yes |
Theorem 8. Let \( {a}_{1},\ldots ,{a}_{n} \) be the zeros and \( {b}_{1},\ldots ,{b}_{n} \) the poles \( {\;\operatorname{mod}\;L} \) of an elliptic function \( f \), counted with their multiplicities. Then\n\n\[ \sum {a}_{k} \equiv \sum {b}_{k}{\;\operatorname{mod}\;L} \] | Necessity of the condition. Let us set \( g = {f}^{\prime }/f \) and \( h\left( u\right) = {ug}\left( u\right) \) . The function \( g \) being elliptic, \( h\left( {u + \omega }\right) = h\left( u\right) + {\omega g}\left( u\right) \) . Consider the period \ | No |
Theorem 9. Let \( f \) be a rational function such that \( f\left( u\right) \asymp 1/{u}^{k} \) at infinity, with \( k \geq 3 \). Then the series \[ {f}^{L}\left( u\right) = \sum f\left( {u - \omega }\right) \] converges normally in every compact set and its sum is an elliptic function whose poles are, \( {\;\operatorn... | This is in particular the case of the functions \( {\wp }_{k} \) defined by (4): they have a pole of order \( k \) at lattice points and are holomorphic elsewhere. But the theorem applies to several other rational functions other than \( 1/{u}^{k} \), for example to \( 1/(u - a)\left( {u - b}\right) \left( {u - c}\righ... | Yes |
Theorem 10. The series \( {\wp }_{L}\left( u\right) = 1/{u}^{2} + \sum \left\lbrack {1/{\left( u - \omega \right) }^{2} - 1/{\omega }^{2}}\right\rbrack \) converges normally in every compact set and its sum is an elliptic function whose only singularities are double poles at lattice points. Any other elliptic function ... | As was seen in Chap. II, \( {\mathrm{n}}^{ \circ }{23} \), and as is now obvious because of Weierstrass’ theorem on holomorphic function series (term by term differentiation), for \( k \geq 3 \) , the functions \( {\wp }_{k}\left( u\right) = \sum 1/{\left( u - \omega \right) }^{k} \) are proportional to the derivatives... | Yes |
Theorem 11. Every elliptic function \( f \) is a rational function of \( \wp \) and \( {\wp }^{\prime } \). | This means that there is a function \( R\left( {X, Y}\right) = P\left( {X, Y}\right) /Q\left( {X, Y}\right) \) of two variables, where \( P \) and \( Q \) are polynomial, such that\n\n(14.1)\n\n\[ f\left( u\right) = R\left( {\wp \left( u\right) ,{\wp }^{\prime }\left( u\right) }\right) \]\n\nfor all \( u \in \mathbb{C}... | Yes |
Corollary 1. Let \( f \) and \( g \) be two elliptic functions having the same periods. There is a non-trivial polynomial \( P \in \mathbb{C}\left\lbrack {X, Y}\right\rbrack \) such that \( P\left( {f, g}\right) = 0 \) . | It could be tempting to prove this by constructing a polynomial \( P \) for which the elliptic function \( P\left\lbrack {f\left( u\right), g\left( u\right) }\right\rbrack \) does not have poles. This would require intricate calculations since \( f \) and \( g \) cannot have an arbitrary number of poles randomly distri... | No |
Lemma 1. For all functions \( f \in {\mathcal{H}}_{r}^{p}\left( P\right), p < + \infty \) , \[ \mathop{\lim }\limits_{{\left| x\right| \propto }}f\left( {x + {iy}}\right) = 0 \] uniformly on every compact subset of \( {\mathbb{R}}_{ + }^{ * } \) . | Indeed let us suppose that \( z \) stays in a strip \( B : 0 < a \leq y \leq b < + \infty \) and consider a strip \( {B}^{\prime } : 0 < {a}^{\prime } \leq y \leq {b}^{\prime } < + \infty \), with \( {a}^{\prime } < a \) and \( \bar{b} < \overline{{b}^{\prime }} \) . Let \( {K}^{\prime } \subset {B}^{\prime } \) be the... | Yes |
For all reals \( r > 1 \), the representation \( {L}_{r} \) of \( {B}_{ + } \) on \( {\mathcal{H}}_{r}^{2}\left( P\right) \) is irreducible. For integer \( r \), the unitary representation \( {L}_{r} \) of \( S{L}_{2}\left( \mathbb{R}\right) \) on \( {\mathcal{H}}_{r}^{2}\left( P\right) \) is irreducible and square int... | Let us first suppose that \( r \) is an integer. To prove irreducibility, it suffices to show that any operator \( A \) in \( {\mathcal{H}}_{r}^{2}\left( G\right) \) commuting with left translations \( L\left( x\right) \) is a scalar. It then amounts to showing that the function \( {\omega }^{\prime } = A{\omega }_{r} ... | Yes |
Lemma 2 (integer \( r \geq 2 \) ). Let \( f \) be a locally integrable function such that \( f * {\omega }_{r} = \) \( g \) is defined. The \( {n}^{61}g \in {\mathcal{H}}_{r}\left( G\right) \), and \( f \) is in \( {\mathcal{H}}_{r}\left( G\right) \) if and only if \( f * {\omega }_{r} = f \) . | We first recall [Chap. XI, \( {\mathrm{n}}^{ \circ }{25} \) ,(iv)] that, \( f * {\omega }_{r} \) is defined if and only if, for all \( p \in L\left( G\right) \) ,\n\n(16.36)\n\n\[ \iint \left| {p\left( {xy}\right) f\left( x\right) {\omega }_{r}\left( y\right) }\right| {dxdy} < + \infty . \]\n\nThe function\n\n(16.37)\n... | No |
Lemma 3. Let \( \mu \) and \( \nu \) be two measures such that \( \mu * \nu \) exists. Then\n\n\[ \lambda * \left( {\mu * \nu }\right) = \left( {\lambda * \mu }\right) * \nu \text{ and }\mu * \left( {\nu * \lambda }\right) = \left( {\mu * \nu }\right) * \lambda \] \n\nfor all measures \( \lambda \) with compact support... | We showed in Chap. XI, \( {\mathrm{n}}^{ \circ }{25} \) ,(ii) that, generally speaking, the previous relation only involves absolute values of the measures considered and that, for positive measures, it holds provided\n\n\[ \iiint p\left( {xyz}\right) {d\lambda }\left( x\right) {d\mu }\left( y\right) {d\nu }\left( z\ri... | Yes |
Lemma 4. Let \( U \) be an open subset of \( \mathbb{C} \) and \( \left( {g}_{n}\right) \) a sequence of holomorphic functions on \( U \) . Suppose that \( \lim {g}_{n}\left( z\right) = g\left( z\right) \) exists almost everywhere and that there is a locally integrable function \( q \geq 0 \) on \( U \) such that \( \l... | Compact convergence being a local property, using a translation, it is possible to work in a disc \( D : \left| z\right| \leq r \) contained in \( U \) . Choosing numbers \( {r}^{\prime } \) and \( {r}^{\prime \prime } \) such that \( r < {r}^{\prime } < {r}^{\prime \prime } \) assuming the closed disc \( \left| z\righ... | Yes |
Theorem 17. The modular group is generated by matrices \( S \) and \( T \) . | This means that any \( \gamma \in \Gamma \) is for the form \( {T}^{p}{S}^{q}{T}^{r}\ldots \) with exponents in \( \mathbb{Z} \) . As \( {S}^{2} = - 1 \), powers of \( S \) can be omitted, if need be by replacing \( \gamma \) with \( - \gamma \) .\n\nThe proof if simple. Let \( a, b, c, d \) be the entries of \( \gamma... | Yes |
A function \( f\left( z\right) \) satisfies\n\n\[ f\left( {\gamma z}\right) = J{\left( \gamma ;z\right) }^{r}f\left( z\right) \;\text{ for all }\gamma \in G\left( \mathbb{Z}\right) \]\n\nfor some even integer \( r \) if and only if\n\n(17.2)\n\n\[ f\left( {z + 1}\right) = f\left( z\right) ,\;f\left( {-1/z}\right) = {z}... | It goes without saying that for \( r \) odd, \( f \) is necessarily 0 since \( - 1 \in \Gamma \) . | No |
Theorem 18. Let \( F \) be the subset of the half-plane \( P \) defined by inequalities\n\n(17.5)\n\n\[ \left| z\right| \geq 1,\;\left| x\right| \leq \frac{1}{2}. \]\n\n\( {}^{62} \) In my youth, the notation \( j = \exp \left( {{2\pi i}/3}\right) \) was standard in France. Freitag and Busam use the letter \( \varrho \... | The arguments that have led us to case (b) show a bit more. We will say that a point \( z \in P \) is a fixed point of \( \Gamma \) if there exists \( \gamma \neq \pm 1 \) in \( \Gamma \) such that \( {\gamma z} = z \) . The proof of the theorem shows that the only fixed points in \( F \) are (a) \( z = i = {Sz} \) , (... | Yes |
Theorem 19. (a) Every entire modular form of weight \( r < 0 \) (resp. \( r = 0 \) ) is trivial (resp. constant). | (a) Let \( f \) be an entire modular form of weight \( r \) . Then, as was seen above, \( f\left( z\right) \) is bounded on the fundamental domain \( F \) . If \( r = 0 \), it is bounded on \( P \) since it is \( \Gamma \) -invariant. Removing the constant term of its Fourier series, it may be supposed to be trivial, i... | Yes |
Theorem 20. For any meromorphic modular form \( f \) of weight \( r \) ,\n\n\[ \nu \left( f\right) = \mathop{\sum }\limits_{{a{\;\operatorname{mod}\;\Gamma }}}{v}_{a}\left( f\right) /n\left( a\right) = \]\n\n(18.6)\n\n\[ = {v}_{i}\left( f\right) /2 + {v}_{j}\left( f\right) /3 + {v}_{\infty }\left( f\right) + \mathop{\s... | If \( r = 0 \), as shown above, \( \nu \left( f\right) \) is the sum of the orders of zeros and poles of the meromorphic function which corresponds to \( f \) on \( \widehat{X}\left( \Gamma \right) \) . Hence \( \nu \left( f\right) = 0 \) in this case (Chap. X, \( {\mathrm{n}}^{ \circ }1 \), theorem 1).\n\nIn the gener... | Yes |
Theorem 21. Every entire modular form of weight \( r \) is a linear combination of functions \( {E}_{r},{E}_{r - {12}}\Delta ,{E}_{r - {24}}{\Delta }^{2},\ldots \) | The sequence ends when it gives nonexistent forms of weight \( \leq 2 \) . These functions are linearly independent because their power series in \( q = \mathbf{e}\left( z\right) \) start respectively with the term \( 1, q,{q}^{2} \), etc. So they form a basis for the finite dimensional vector space of entire modular f... | Yes |
Theorem 23. Every modular function is a rational function of \( J\left( z\right) \) . The function \( J \) is an isomorphism from the Riemann surface \( \widehat{X}\left( \Gamma \right) \) onto the Riemann sphere C. | To see this, we again consider the relation \( \sum {\nu }_{a}\left( f\right) = 0 \) which holds for every modular form of weight 0, where \( {\nu }_{a}\left( f\right) \) is an integer obtained by dividing the usual order \( {v}_{a}\left( f\right) \) by 2 if \( a = \gamma \left( i\right) \), by 3 if \( a = \gamma \left... | Yes |
Lemma 1. Let \( \Gamma \) be a discrete subgroup of \( G \) such that \( \Gamma \cap U \neq \{ e\} \). a) For all compact sets \( M \subset P \), there exists \( T < + \infty \) such that \[ \Gamma \cdot M \subset \{ \operatorname{Im}\left( z\right) \leq T\} \] b) For sufficiently large \( T \), \[ \operatorname{Im}\le... | Let us first show that --- \( {}^{71}G \) and more generally \( G{L}_{2}\left( \mathbb{R}\right) \) are made to act on \( {P}_{1}\left( \mathbb{R}\right) \) by considering \( G \) as a group of linear transformations on \( {\mathbb{R}}^{2} \). This group acts on the set of 1-dimensional vector subspaces, which by defin... | Yes |
Theorem 25. The Riemann surface \( \widehat{X}\left( \mathit{Γ}\right) \) of every Fuchsian group \( \mathit{Γ} \) is compact. The total mass of the invariant measure on \( \Gamma \smallsetminus G \) is finite. | The converse holds and is mostly interesting because of its proof. The easiest is to show that, if \( \widehat{X}\left( \Gamma \right) \) is compact, then \( \Gamma \) is a Fuchsian group. Indeed, we saw that the set of parabolic fixed points of \( \Gamma \) is discrete in \( \widehat{P}\left( \Gamma \right) \) . It is... | No |
Theorem 26. Let \( \Gamma \) be a Fuchsian group.\n\n(a) Every entire automorphic form of weight \( r \) is trivial (resp. constant) if \( r < 0 \) (resp. \( r = 0 \) ). | (a) If \( r = 0 \), the function \( f\left( z\right) \) is holomorphic everywhere in \( \widehat{X}\left( \Gamma \right) \), hence constant. If \( r < 0 \) and if \( \infty \) is a parabolic fixed point, \( f\left( z\right) \) is bounded for \( y > T \) , hence also \( {y}^{r/2}\left| {f\left( z\right) }\right| = \left... | Yes |
Theorem 28. For all discrete subgroups \( \Gamma \subset G \), all open \( \Gamma \) -invariant subsets \( \Omega \subset P \), all \( r > 2 \) and all holomorphic functions such that\n\n\[ \n{\int }_{\Omega }{y}^{r/2}\left| {f\left( z\right) }\right| {dxdy} < + \infty \n\]\n\nthe Poincaré series \( {P}_{r, f}\left( z\... | In fact, initially Poincaré only considered series \( {P}_{r, f}\left( z\right) \) for functions \( f \) which, on the unit disc \( D \), correspond to polynomials in \( z \) . His arguments are a bit different. The conformal representation\n\n\[ \nz \mapsto \zeta = \left( {z - i}\right) /\left( {z + i}\right) = {s}^{-... | Yes |
Theorem 29. Let \( \Gamma \) be a Fuchsian group. For all \( r \in \mathbb{Z} \), even if \( - 1 \in \Gamma \), there exist non-trivial automorphic forms of weight \( r \) for \( \Gamma \) . | the previous argument proves the theorem for all \( r > 2 \) : choose \( f\left( z\right) = {\left( z - w\right) }^{-p} \) with \( p > r \) . If \( {w}^{\prime } \notin {\Gamma w} \) is not an elliptic fixed point, the function \( {f}^{\prime }\left( z\right) = {\left( z - {w}^{\prime }\right) }^{-{p}^{\prime }} \) is ... | No |
Theorem 30 (H. Petersson). Let \( \Gamma \) be a Fuchsian group. (i) For all \( r > 2 \), the space \( {\mathcal{H}}_{r}^{2}\left( {\Gamma \smallsetminus G}\right) \) of parabolic forms of weight \( r \) for \( \Gamma \) is the set of Poincaré series \( {P}_{r, f} \) associated to \( K \) -finite functions \( f \in {\m... | If \( \Gamma \) contains the matrix \( - 1, r \) is even and the general term of (11) only depends on the coset of \( \gamma \) modulo the subgroup \[ {\Gamma }_{\infty } = \Gamma \cap B = {U}_{\infty } \cup - {U}_{\infty } \] of matrices for which \( c = 0, d = \pm 1 \) . So the sum is twice that obtained by summing \... | No |
Theorem 31. For every discrete subgroup \( \Gamma \) of \( G \) having a parabolic fixed point at infinity, the series\n\n(21.18)\n\n\[ \n{M}_{\infty }\left( {g;s}\right) = \mathop{\sum }\limits_{{{\Gamma }_{\infty } \smallsetminus \Gamma }}\alpha {\left( \gamma g\right) }^{s} = \sum \operatorname{Im}{\left( \gamma z\r... | Again, the proof amounts to comparing the series with an integral. | No |
Lemma 1. Normal convergence in \( \mathfrak{S}\left( \xi \right) \) is equivalent to convergence at \( x \in G \) . | To prove this it suffices to find an upper bound\n\n\[\n\alpha \left( {\gamma xg}\right) \leq {M\alpha }\left( {\gamma x}\right)\n\]\n\nvalid for \( \gamma \notin {\Gamma }_{\infty } \) and \( g \in \mathfrak{S}\left( T\right) \) .\n\nLet us first show that there is a constant \( M \) such that\n\n(21.19)\n\n\[\n\alpha... | Yes |
Lemma 2. Let \( \varphi \) be a continuous function with values \( > 0 \) on \( G \) such that\n\n(21.21)\n\n\[ \varphi \left( {uhg}\right) = \alpha {\left( h\right) }^{s}\varphi \left( g\right) \;\text{ for some }s \in \mathbb{R}. \]\n\nFor all compact sets \( C \subset G \) with measure \( > 0 \), there exist constan... | Let \( {\varphi }^{\prime }\left( x\right) \) denote the integral. The function \( {\varphi }^{\prime } \) is continuous, satisfies (21) like \( \varphi \) and is everywhere \( > 0 \) since \( m\left( C\right) > 0 \) . The ratio \( \varphi \left( x\right) /{\varphi }^{\prime }\left( x\right) > 0 \) is invariant under \... | Yes |
Theorem 32. Let \( \Gamma \) be a Fuchsian group. Every entire automorphic form of weight \( r > 2 \) is, in a unique way, the sum of a parabolic form and of a linear combination of Eisenstein series associated to parabolic fixed points of \( \Gamma \) . | For \( r \) even, the dimension of the vector space generated by the Eisenstein series is the number of classes of parabolic fixed points. For \( r \) odd, it is zero if \( - 1 \in \Gamma \) and equal to the number of regular fixed points otherwise. | No |
Theorem 33 \( \left\lbrack {\Gamma = S{L}_{2}\left( \mathbb{Z}\right) ,\text{ Maaß }}\right\rbrack \) . The function \( \xi \left( {2s}\right) M\left( {z;s}\right) \) extends analytically to a meromorphic function of \( s \) invariant under \( s \mapsto 1 - s \) and whose only singularities are simple poles at \( s = 1... | Since \( 1/\xi \left( {2s}\right) \) is zero at \( s = 0 \), this point is not a pole of \( M\left( {z;s}\right) \) . All the zeros of \( \xi \left( {2s}\right) \) being located in \( \operatorname{Re}\left( s\right) < \frac{1}{2} \), the only pole of \( M\left( {z;s}\right) \) in \( \operatorname{Re}\left( s\right) \g... | Yes |
Theorem 35 (A. Weil). Let \( f\left( z\right) = \mathop{\sum }\limits_{{n \geq 0}}{a}_{n}\mathbf{e}\left( {nz}\right) \) be an entire modular form of weight \( r \) and \( \chi \) a proper character modulo \( m > 1 \) . The function \[ f\left( {z;\chi }\right) = \mathop{\sum }\limits_{\substack{{a{\;\operatorname{mod}\... | The integral \[ {\Lambda }_{f}\left( {s;\chi }\right) = \int f\left( {{iy};\chi }\right) {y}^{s}{d}^{ * }y = \] \[ = {\Gamma }_{m}\left( \bar{\chi }\right) {\left( 2\pi /m\right) }^{-s}\Gamma \left( s\right) \mathop{\sum }\limits_{{n \geq 1}}{a}_{n}\chi \left( n\right) /{n}^{s} \] converges for all \( s \in \mathbb{C} ... | Yes |
Lemma 1. \( {\Gamma }^{\prime } \) is a subgroup of \( G \) . | Clearly, \( x \in {\Gamma }^{\prime } \) implies \( {x}^{-1} \in {\Gamma }^{\prime } \) . On the other hand,\n\n\[ \Gamma \left( {xy}\right) = {xy\Gamma }{y}^{-1}{x}^{-1} \cap \Gamma = x\left( {{y\Gamma }{y}^{-1} \cap {x}^{-1}{\Gamma x}}\right) {x}^{-1}. \]\n\nBut if \( \Gamma \left( x\right) \) has finite index in \( ... | Yes |
For all \( x, y \in {\Gamma }^{\prime } \), the operator \( T\left( x\right) T\left( y\right) \) is a linear combination with integer coefficients \( \geq 0 \) of \( T\left( z\right), z \in {\Gamma }^{\prime } \) . | By (5), \[ T\left( x\right) T\left( y\right) \varphi \left( g\right) = \mathop{\sum }\limits_{{\Gamma \smallsetminus C\left( x\right) }}T\left( y\right) \varphi \left( {ug}\right) = \mathop{\sum }\limits_{{\Gamma \smallsetminus C\left( x\right) }}\mathop{\sum }\limits_{{\Gamma \smallsetminus C\left( x\right) }}\varphi ... | Yes |
Lemma 3. For all \( x \in {\Gamma }^{\prime } \), the operator \( T\left( x\right) \) maps \( L\left( {\Gamma \smallsetminus G}\right) \) to \( L\left( {\Gamma \smallsetminus G}\right) \) . | For \( \varphi \in L\left( {\Gamma \smallsetminus G}\right) \), let \( M \) be a compact set such that \( \varphi \) is zero outside \( {\Gamma M} \) . \( T\left( x\right) \varphi \left( g\right) \neq 0 \) only if \( {x}^{-1}{\Gamma g}\# {\Gamma M} \), hence only if \( g \in {\Gamma x\Gamma M} = {\Gamma x\Gamma }{x}^{-... | Yes |
Theorem 36. For all \( x \in {\Gamma }^{\prime } \) and all \( p \geq 1 \), the operator \( T\left( x\right) : L\left( {\Gamma \smallsetminus G}\right) \rightarrow \) \( L\left( {\Gamma \smallsetminus G}\right) \) has a continuous extension to \( {L}^{p}\left( {\Gamma \smallsetminus G}\right) \) and\n\n\[ \left( {24.10... | Let us start by remarking that the automorphism \( g \mapsto {x}^{-1}{gx} \) preserves the measure \( {dg} \) and that its inverse transforms \( \Gamma \) into \( {x\Gamma }{x}^{-1} \) . Hence for all \( p \geq 1 \) we get an isomorphism from \( {L}^{p}\left( {\Gamma \smallsetminus G}\right) \) onto \( {L}^{p}\left( {{... | Yes |
Theorem 37 (Hecke-Petersson). The operators \( T\left( x\right) \) commute and are Hermitian on \( {L}^{2}\left( {\Gamma \smallsetminus G}\right) \) . | The latter point is obvious: \( T{\left( x\right) }^{ * } = T\left( {x}^{-1}\right) \) (theorem 36) for all \( x \in {\Gamma }^{\prime } \), and \( T\left( {x}^{-1}\right) = T\left( x\right) \) .\n\nTo prove the former, we start from (9)\n\n\[ T\left( x\right) T\left( y\right) = \sum c\left( {x, y;z}\right) T\left( {z}... | Yes |
Lemma 4. Every function \( \lambda \left( p\right) \) defined on the set of prime numbers extends to a unique function \( \lambda \left( n\right) \) defined for \( n \geq 1 \) and such that \( \lambda \left( 1\right) = 1 \), Moreover, \( \lambda \left( {mn}\right) = \lambda \left( m\right) \lambda \left( n\right) \) if... | The proof is obvious. Since \( \lambda \left( 1\right) = 1 \) ,(26) holds for \( n = 1 \) for all \( p \) . Because of \[ \lambda \left( {p}^{k + 1}\right) = \lambda \left( p\right) \lambda \left( {p}^{k}\right) - {p\lambda }\left( {p}^{k - 1}\right) , \] it is possible to calculate the numbers \( \lambda \left( {p}^{k... | No |
Lemma 5. Relations (26) and (26') are equivalent to the identity between formal series | Setting\n\n\[ {F}_{p}\left( X\right) = \sum \lambda \left( {p}^{k}\right) {X}^{k} \]\n\nit is clear that the series \( F\left( X\right) = \sum \lambda \left( n\right) {X}^{n} \) satisfies\n\n\[ F\left( X\right) = \prod {F}_{p}\left( X\right) \]\n\nBut (26’) for \( n = {p}^{k + 1} \) implies that\n\n\[ {F}_{p}\left( X\r... | Yes |
Theorem 38. For every even integer \( r > 2 \), the space of parabolic forms of weight \( r \) for \( S{L}_{2}\left( \mathbb{Z}\right) \) has a basis whose elements satisfy the following properties:\n\n(i) \( {T}_{p}f = {\lambda }_{p}f \) for all prime \( p \) ;\n\n(ii) \( f\left( z\right) = \sum {\lambda }_{n}\mathbf{... | These results complete Weil’s theorem \( \left\lbrack {{\mathrm{n}}^{ \circ }{23}\text{,(ii)}}\right\rbrack \) . As \( \parallel T\left( p\right) \parallel = \begin{Vmatrix}{T\left( {h}_{p}\right) }\end{Vmatrix} \leq \lbrack \Gamma \) : \( \left. {\Gamma \left( {h}_{p}\right) }\right\rbrack = p + 1,\left| {\lambda }_{p... | No |
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