Mode-Restriction Links between a Fine and a Coarse Noise Level

In the setting of The Real Numbers: Standing Notation and Background and Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation, let levels 1 and 2 be as in Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels, with the level-indexed notation of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §notation. Orthonormal sequences, convergent series in $X_{1}$ and in $X_{2}$ and their sums, and convergent series of real numbers are those of the definitions cited.

1. (Link)¶ A mode-restriction link from level 1 to level 2 is an injective map $\kappa:\mathbb{N}\to\mathbb{N}$ such that

$$a^{(2)}_{m}=a^{(1)}_{\kappa(m)}\qquad\text{for every }m\in\mathbb{N}.$$

2. (Orthonormality of the selected modes)¶ For a mode-restriction link $\kappa$ from level 1 to level 2, the sequence $(e^{(1)}_{\kappa(m)})_{m\in\mathbb{N}}$ is an orthonormal sequence in $X_{1}$: each of its terms has norm $1$, and for $m,m'\in\mathbb{N}$ with $m\neq m'$ one has $\kappa(m)\neq\kappa(m')$ by injectivity, so that $\langle e^{(1)}_{\kappa(m)},e^{(1)}_{\kappa(m')}\rangle_{1}=0$ by the orthonormality of $(e^{(1)}_{k})_{k\in\mathbb{N}}$.

3. (Mode restriction)¶ Let $\kappa$ be a mode-restriction link from level 1 to level 2. Its mode restriction is the map $p:X_{1}\to X_{2}$,

$$p(x)=\sum_{m=1}^{\infty}\langle x,e^{(1)}_{\kappa(m)}\rangle_{1}\,e^{(2)}_{m}\qquad(x\in X_{1}).$$

The series converges in $X_{2}$, and $p(x)$ denotes its sum: by Orthonormal Expansions in a Real Hilbert Space §bessel, applied in $X_{1}$ to the orthonormal sequence $(e^{(1)}_{\kappa(m)})_{m\in\mathbb{N}}$ of clause 2, the series $\sum_{m=1}^{\infty}\langle x,e^{(1)}_{\kappa(m)}\rangle_{1}^{2}$ converges, so that Orthonormal Expansions in a Real Hilbert Space §riesz-fischer, applied in $X_{2}$ to the orthonormal sequence $(e^{(2)}_{m})_{m\in\mathbb{N}}$ with $c_{m}=\langle x,e^{(1)}_{\kappa(m)}\rangle_{1}$, gives the convergence of the series in $X_{2}$.

4. (Mode embedding)¶ With $\kappa$ as in clause 3, its mode embedding is the map $j:X_{2}\to X_{1}$,

$$j(y)=\sum_{m=1}^{\infty}\langle y,e^{(2)}_{m}\rangle_{2}\,e^{(1)}_{\kappa(m)}\qquad(y\in X_{2}).$$

The series converges in $X_{1}$, and $j(y)$ denotes its sum: by Orthonormal Expansions in a Real Hilbert Space §bessel, applied in $X_{2}$ to $(e^{(2)}_{m})_{m\in\mathbb{N}}$, the series $\sum_{m=1}^{\infty}\langle y,e^{(2)}_{m}\rangle_{2}^{2}$ converges, so that Orthonormal Expansions in a Real Hilbert Space §riesz-fischer, applied in $X_{1}$ to the orthonormal sequence $(e^{(1)}_{\kappa(m)})_{m\in\mathbb{N}}$ of clause 2 with $c_{m}=\langle y,e^{(2)}_{m}\rangle_{2}$, gives the convergence of the series in $X_{1}$. A mode-restriction link is written $(\kappa,p,j)$, together with its mode restriction and its mode embedding.