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A spectrograph records the optical spectrum dispersed by an instrument. Its spectral resolving power describes which nearby wavelengths it separates.

Fourier transform spectroscopy measures an interferogram as a function of optical path length difference. Its Fourier transform recovers the spectrum. An imaging implementation obtains an interferogram at every detector pixel, but a scanned cube is not simultaneous at all wavelengths.

For a small sinusoidal optical path length error $h_0\cos (2\pi x/\Lambda )$, expand its phase factor to first order. The two exponential harmonics each have field amplitude $i\pi h_0/\lambda$ relative to the unperturbed field, so each speckle has contrast $C=(\pi h_0/\lambda {)}^2$. Its angular displacement is $\lambda /\Lambda$. Here $h_0$ is optical path error, not physical mirror displacement; a reflecting mirror introduces twice its surface displacement in the near-normal limit.

Compressed sensing reconstructs a sparse vector, or an approximately sparse one, from fewer linear measurements than its ambient dimension. Basis pursuit replaces counting nonzero coordinates by convex ${\ell }^1$ minimization. A null space property characterizes exact uniform recovery, while a robust null space property controls errors from noise and nonsparse tails.

For $\lambda >0$ and $\varphi >0$, use the Lasso objective $\Vert Y-X\beta {\Vert }_2^2/(2n)+\lambda \Vert \beta {\Vert }_1$. Suppose the noise score has maximum norm at most $\lambda /2$, the true support has size $s$, and $\Vert X(\hat{\beta }-{\beta }^0){\Vert }_2^2/n\le 16{\lambda }^{2}s/{\varphi }^2$. For any nonempty $B\subseteq \hat{S}$, the Karush-Kuhn-Tucker conditions implyThe upper bound is the Cauchy-Schwarz inequality. Hence $|B|\le 64{\kappa }_{|B|}^{2}s/{\varphi }^2$. A finite first index $m_{\ast }$ violating this inequality must exceed $|\hat{S}|$. Its minimality also gives $|\hat{S}|\le 64{\kappa }_{m_{\ast }}^{2}s/{\varphi }^2$. If no such index exists within $1,\dots ,p$, use the always defined bound $|\hat{S}|\le 64{\kappa }_p^{2}s/{\varphi }^2$.

The ordinary topology on $[\mathbb{N}{]}^{\omega }$ has basic sets $[s]=\{X:s⊏X\}$ for finite $s$. It is the product topology on increasing enumerations, and also the subspace topology on infinite-subset indicator functions in $\{0,1{\}}^{\mathbb{N}}$.

For a finite set $s$ and an infinite set $A$ lying above $\max s$, putThese sets form the basic neighborhoods of the Ellentuck topology, also called the star topology. It is finer than the ordinary topology on infinite subsets. Every basic neighborhood is a clopen set in the Ellentuck topology, and a closed set in the ordinary topology.

A set $E\subseteq [\mathbb{N}{]}^{\omega }$ is Ramsey if every infinite $A$ has an infinite subset $B$ such that $[B{]}^{\omega }\subseteq E$ or $[B{]}^{\omega }\cap E=\varnothing$.

For a compact Hausdorff space $K$, $C(K)$ consists of continuous scalar-valued functions with the supremum norm. It is a Banach space and, with complex scalars, pointwise multiplication and conjugation make it a commutative unital C-star algebra.

Why does completeness hold? A Cauchy sequence in the supremum norm is Cauchy at every point, hence has a pointwise limit. The uniform Cauchy estimate passes to that limit, giving uniform convergence; the uniform limit theorem makes the limit continuous. Thus the norm and the chosen regularity of the functions cooperate to keep every limit inside $C(K)$.

Changing to the L2 norm on $C([0,1])$ loses this property. For integers $n\ge 2$, the continuous functions $h_n(x)=\min (1,\max (0,n(x-1/2)))$ approximate the step function $h(x)=0$ for $x\le 1/2$ and $h(x)=1$ for $x>1/2$. Direct integration givesThe triangle inequality makes $(h_n)$ a Cauchy sequence, but no continuous function agrees with $h$ almost everywhere: continuity would force value zero on the whole left open interval and one on the whole right open interval. Hence this normed space is incomplete. The missing limit exists in L2 space, its completion of a normed space, showing exactly what the larger space adds.

The continuous dual space of $H^s({\mathbb{R}}^d)$ is $H^{-s}({\mathbb{R}}^d)$ under the extension of the L2 inner product. The Fourier transform definition $\Vert f{\Vert }_{H^s}^2=\int (1+|\xi {|}^2{)}^s|\hat{f}(\xi ){|}^{2}d\xi$ proves boundedness of the pairing by the Cauchy-Schwarz inequality, and the Riesz representation theorem proves that every continuous functional has this form. Consequently a bounded linear operator $K:H^r\to H^{r+a}$ has an adjoint operator, relative to this pairing, from $H^{-r-a}$ to $H^{-r}$. For an operator that smooths by two orders for every $r$, this yields $K^{\ast }:H^2\to H^4$ by taking $r=-4$.

Multiplication by a test function is bounded on every real-index Sobolev space. Its Fourier transform acts by convolution with a Schwartz function; frequency-weight estimates and Young's convolution inequality give the norm bound.

For $1\le p<\infty$, an inhomogeneous Sobolev space norm is $\Vert u{\Vert }_{W^{k,p}(D)}=({\sum }_{|\alpha |\le k}\Vert D^{\alpha }u{\Vert }_{L^p(D)}^p{)}^{1/p}$, using weak derivatives. In particular, $\Vert u{\Vert }_{H^1(D)}^2={\int }_D(|u{|}^2+|\nabla u{|}^2)$. The gradient seminorm alone defines a different completion, the Dirichlet energy space, unless a suitable Poincaré inequality makes the two norms equivalent on the chosen zero-boundary space.

A smooth manifold with boundary has charts in the half-space $\{x_n\ge 0\}\subseteq {\mathbb{R}}^n$, with smoothly compatible transition maps. The boundary is the locus modeled on $x_n=0$.

A closed manifold is a compact manifold with boundary whose boundary is empty. Here “closed” includes compactness.

A piece is a nontrivial combinatorial path that occurs in two essentially distinct positions on oriented cell attaching paths. These may be different cells or distinct admissible occurrences on one attaching path; occurrences identified by the defining attaching-path symmetry are not counted twice. In a symmetrized group presentation, it is a common initial segment of distinct relator words. Internal arcs of a reduced disc diagram are pieces, because otherwise the neighboring cells form a cancellable pair.

The condition $C^{\prime }(\lambda )$ requires that every piece in small cancellation theory occurring in a cell boundary have length strictly less than $\lambda$ times that cell's perimeter. Length counts edges. For $\lambda =1/6$, an inner path made of at most three pieces has length strictly less than half the perimeter, so the complementary outer path is strictly longer. The attaching paths are combinatorial immersions and the piece convention is part of the chosen presentation or polygonal complex.

For the metric small cancellation condition $C^{\prime }(1/6)$, every nonempty freely reduced null-homotopic combinatorial word contains a segment of a cell boundary longer than half its perimeter. Such a segment can be replaced by the shorter complementary segment. The diagram explanation uses shells with at most three internal pieces: their internal length is less than half the perimeter. Degenerate diagrams require the disc diagram spur alternative.

A CW subcomplex has no missing shells if a cell with a shell-length boundary arc contained in the CW subcomplex is itself contained in the CW subcomplex. For $C^{\prime }(1/6)$ it suffices that every excluded cell have a complementary boundary arc of length at least half its perimeter with no edge in the CW subcomplex: an arc lying in the CW subcomplex then has length at most half and cannot be a shell's exterior arc.

A crosscut of a planar domain is a simple open arc inside the domain whose two endpoints are distinct boundary points, interpreted as prime ends when necessary. In a simply connected domain it divides the domain into two components. In the half-plane, a bounded crosscut joining two real points separates the intervening real interval from infinity; a Loewner chain completing that crosscut swallows this interval.