Definitions for Stochastic processes induced by singular operators

QAlgorithms.Papers.Independent.Intakethird100.P38.AntiDual

For every natural number $n$, including $0$, $\operatorname{AntiDual}(n)$ is the space of continuous conjugate-linear maps from the complex-valued Schwartz functions on Euclidean $\mathbb R^n$ to $\mathbb C$. Thus an element $T$ is continuous in the Schwartz topology and satisfies $T(c\psi+d\phi)=\overline c\,T(\psi)+\overline d\,T(\phi)$ for all complex scalars $c,d$ and all complex-valued Schwartz functions $\psi,\phi$.

QAlgorithms.Papers.Independent.Intakethird100.P38.CharacteristicLaw

For any natural number $n$, including $0$, let $\sigma$ be a measure on Euclidean $\mathbb R^n$, and let $\mu$ be a measure on the space $\Omega$ of continuous real-linear functionals on real-valued Schwartz functions on $\mathbb R^n$, equipped with the sigma-algebra generated by the evaluations $\omega\mapsto\omega(\psi)$. The property $\operatorname{CharacteristicLaw}(\sigma,\mu)$ means that $\mu$ is a probability measure and, for every real-valued Schwartz function $\psi$, the extended nonnegative integral

$$E_\sigma(\psi)=\int_{\mathbb R^n}\left|\int_{\mathbb R^n}e^{-i\,u\cdot x}\psi(x)\,dx\right|^2\,d\sigma(u)$$

is finite and satisfies

$$\int_\Omega e^{i\omega(\psi)}\,d\mu(\omega)=e^{-E_\sigma(\psi)/2}.$$

Here $dx$ is Lebesgue measure, $\psi$ is regarded as complex-valued inside the Fourier integral, and the finite energy on the right is regarded as a real number.

QAlgorithms.Papers.Independent.Intakethird100.P38.Closable

For every natural number $n$, including $0$, and every function $q$ from the complex-valued Schwartz functions on $\mathbb R^n$ to $\mathbb R$, this defines the following property: for every sequence $(s_k)_{k\in\mathbb N}$ of such Schwartz functions, if

$$\|v_k\|_{L^2(dx)}\longrightarrow 0$$

and

$$\forall\varepsilon>0\;\exists N\in\mathbb N\;\forall k\ge N\;\forall l\ge N, \qquad q(s_k-s_l)<\varepsilon,$$

then $q(s_k)\to0$ in $\mathbb R$ as $k\to\infty$. Here $dx$ is Lebesgue measure, and $v_k$ is the $L^2(dx)$ equivalence class of $s_k$ if it is almost everywhere strongly measurable and square-integrable, or the zero vector otherwise. The difference $s_k-s_l$ is pointwise, and the quantifiers include $k=l$. No positivity, quadratic identity, or continuity assumption on $q$ is included; the displayed bound is the strict one-sided inequality $q(s_k-s_l)<\varepsilon$, without an absolute value.

QAlgorithms.Papers.Independent.Intakethird100.P38.ComplexTest

For each natural number $n$, including $n=0$, the complex test space is the space $\mathcal S(\mathbb R^n;\mathbb C)$ of complex-valued Schwartz functions on real Euclidean $n$-space: smooth functions whose derivatives of every order decrease rapidly. For $n=0$, the domain is a single point.

QAlgorithms.Papers.Independent.Intakethird100.P38.HilbertFactor

For every natural number $n$, including $0$, every measure $\sigma$ on $\mathbb R^n$, and every continuous complex-linear map $Q$ from the complex-valued Schwartz functions on $\mathbb R^n$ to $L^2(\mathbb R^n,dx;\mathbb C)$, this defines the property that, for every pair of Schwartz functions $\psi,\phi$,

$$\langle Q\phi,Q\psi\rangle_{L^2(dx)} =\int_{\mathbb R^n}\widehat\psi(u)\,\overline{\widehat\phi(u)}\,d\sigma(u), \qquad \widehat f(u)=\int_{\mathbb R^n}e^{-i\langle u,x\rangle}f(x)\,dx.$$

The $L^2$ inner product is conjugate-linear in its first argument and linear in its second, so its left side is $\int\overline{Q\phi(x)}\,Q\psi(x)\,dx$. The inner product inside the Fourier exponential is the real Euclidean inner product, and $dx$ is Lebesgue measure. No regularity or finiteness assumption on $\sigma$ is included; the integral on the right is defined as $0$ if its integrand is not Bochner integrable.

QAlgorithms.Papers.Independent.Intakethird100.P38.InC

For every natural number $n$, including $n=0$, and every measure $\sigma$ on $\mathbb R^n$, the predicate $\operatorname{InC}(\sigma)$ means that $\sigma$ is regular and that there exists a natural number $p$ such that $\int_{\mathbb R^n}^{-}(1+\|u\|^2)^{-p}\,d\sigma(u)<+\infty$. The exponent $p=0$ is permitted, in which case this finiteness condition is finiteness of the total mass. For $n=0$, the domain is a single point and every weight in this condition is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P38.InCb

For any natural number $n$, including $0$, and any measure $\sigma$ on Euclidean $\mathbb R^n$, the property $\operatorname{InCb}(\sigma)$ means that $\sigma$ is regular and that, for every natural number $p\ge1$, there exist a natural number $q\ge1$ and a real constant $C>0$ such that, for every $v\in\mathbb R^n$,

$$\int_{\mathbb R^n}\frac{1}{(1+\|u+v\|^2)^q}\,d\sigma(u) \ \le\ \frac{C}{(1+\|v\|^2)^p}.$$

This is an inequality in the extended nonnegative reals: the left side is the nonnegative Lebesgue integral, and the right side is the displayed nonnegative real embedded into that space. The choices $q,C$ may depend on $p$ but must work for all $v$; there is no requirement for $p=0$.

QAlgorithms.Papers.Independent.Intakethird100.P38.IsKoopman

For any natural number $n$, including $0$, let $\Omega$ be the space of continuous real-linear functionals on the real Schwartz space over Euclidean $\mathbb R^n$, equipped with its evaluation-generated sigma-algebra; let $\mu$ be any measure on $\Omega$; and let $U_t$, for each $t\in\mathbb R^n$, be a bijective complex-linear isometry of $L^2(\Omega,\mu;\mathbb C)$. Define $S_t\omega$ by $(S_t\omega)(\psi)=\omega(x\mapsto\psi(x+t))$. The property $\operatorname{IsKoopman}(\mu,U)$ requires that every $S_t$ is measurable and preserves $\mu$, that $U_0$ is the identity, that $U_{t+s}=U_s\circ U_t$ for every $t,s\in\mathbb R^n$, and that, for every $t\in\mathbb R^n$ and every $f\in L^2(\Omega,\mu;\mathbb C)$,

$$(U_tf)(\omega)=f(S_t\omega)\qquad\text{for }\mu\text{-almost every }\omega.$$

The exceptional null set may depend on $t$ and $f$; no probability assumption on $\mu$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P38.KoopmanFamily

For any natural number $n$, including $0$, let $\Omega$ be the space of continuous real-linear functionals on real-valued Schwartz functions on Euclidean $\mathbb R^n$, with its evaluation-generated sigma-algebra, and let $\mu$ be any measure on $\Omega$. A $\operatorname{KoopmanFamily}(\mu)$ is an arbitrary function assigning to each $t\in\mathbb R^n$ a bijective complex-linear isometry $U_t$ of $L^2(\Omega,\mu;\mathbb C)$ onto itself. This type imposes no continuity, group law, or relationship to translations on the assignment $t\mapsto U_t$.

QAlgorithms.Papers.Independent.Intakethird100.P38.L2

For every natural number $n$, including $0$, and every measure $\sigma$ on the Euclidean space $\mathbb R^n$, this is an abbreviation for $L^2(\mathbb R^n,\sigma;\mathbb C)$: the space of equivalence classes, modulo equality $\sigma$-almost everywhere, of almost everywhere strongly measurable complex-valued functions $f$ satisfying

$$\int_{\mathbb R^n}|f(u)|^2\,d\sigma(u)<\infty.$$

No finiteness, regularity, or $\sigma$-finiteness assumption on the measure is required.

QAlgorithms.Papers.Independent.Intakethird100.P38.LocallyTotalExponentials

For every natural number $n$, including $0$, and every measure $\sigma$ on $\mathbb R^n$, this defines the property that, for every compact set $K\subseteq\mathbb R^n$, the complex linear span of the vectors $v_{K,t}$, indexed by all $t\in\mathbb R^n$, is dense in $L^2(\mathbb R^n,\sigma|_K;\mathbb C)$. Here $\sigma|_K$ is the restriction of $\sigma$ to $K$, and $v_{K,t}$ is the almost-everywhere equivalence class of

$$u\longmapsto e^{i\langle t,u\rangle}$$

if that function is almost everywhere strongly measurable and square-integrable for $\sigma|_K$, and is zero otherwise; $\langle t,u\rangle$ is the real Euclidean inner product. Density is in the $L^2$ norm. The empty compact set is included and gives the zero $L^2$ space. No assumption that compact sets have finite measure is imposed: if $\sigma(K)=\infty$, every $v_{K,t}$ is zero because the exponential has modulus $1$, so the required density for that $K$ means that $L^2(\sigma|_K)$ is the zero space.

QAlgorithms.Papers.Independent.Intakethird100.P38.Omega

For each natural number $n$, including $n=0$, $\Omega(n)$ is the space of continuous real-linear maps from the real Schwartz space $\mathcal S(\mathbb R^n;\mathbb R)$, equipped with its Schwartz topology, to $\mathbb R$. Thus an element $\omega$ assigns a real number $\omega(\psi)$ to every real-valued Schwartz function $\psi$, continuously and real-linearly. When $n=0$, the test functions have a one-point domain.

QAlgorithms.Papers.Independent.Intakethird100.P38.PositiveQuadratic

For every natural number $n$, including $0$, and every function $q$ from the complex-valued Schwartz functions on $\mathbb R^n$ to $\mathbb R$, this defines a property requiring all three conditions

$$\begin{aligned} &q(\psi)\ge 0 &&\text{for every }\psi,\\ &q(c\psi)=|c|^2q(\psi) &&\text{for every }c\in\mathbb C\text{ and every }\psi,\\ &q(\psi+\phi)+q(\psi-\phi)=2\bigl(q(\psi)+q(\phi)\bigr) &&\text{for every }\psi,\phi. \end{aligned}$$

Scalar multiplication, addition, and subtraction of Schwartz functions are pointwise. The scalar $c=0$ and zero test functions are included; no continuity assumption on $q$ is part of this property.

QAlgorithms.Papers.Independent.Intakethird100.P38.RealTest

For each natural number $n$, including $n=0$, the real test space is the space $\mathcal S(\mathbb R^n;\mathbb R)$ of real-valued Schwartz functions on real Euclidean $n$-space: smooth functions whose derivatives of every order decrease rapidly. For $n=0$, the domain is a single point.

QAlgorithms.Papers.Independent.Intakethird100.P38.SequenceL2

The abbreviated space is the complex $L^2$ space of the integers equipped with counting measure: its elements are equivalence classes, modulo equality almost everywhere for counting measure, of complex sequences $a=(a_m)_{m\in\mathbb Z}$ satisfying $\sum_{m\in\mathbb Z}|a_m|^2<\infty$, with norm

$$\|a\|_2=\left(\sum_{m\in\mathbb Z}|a_m|^2\right)^{1/2}.$$

For counting measure on $\mathbb Z$, equality almost everywhere is equality at every integer.

QAlgorithms.Papers.Independent.Intakethird100.P38.ShiftErgodic

For any natural number $n$, including $0$, let $\Omega$ be the space of continuous real-linear functionals on real-valued Schwartz functions on Euclidean $\mathbb R^n$, with its evaluation-generated sigma-algebra, and let $\mu$ be any measure on $\Omega$. Define $S_t\omega$ by $(S_t\omega)(\psi)=\omega(x\mapsto\psi(x+t))$. The property $\operatorname{ShiftErgodic}(\mu)$ means that every measurable set $A\subseteq\Omega$ satisfying

$$\text{for every }t\in\mathbb R^n,\quad S_t\omega\in A\ \Longleftrightarrow\ \omega\in A \quad\text{for }\mu\text{-almost every }\omega$$

has $\mu(A)=0$ or $\mu(A)=1$. The exceptional null set may depend on $t$. Neither probability normalization nor preservation of $\mu$ by the shifts is part of this definition.

QAlgorithms.Papers.Independent.Intakethird100.P38.SpectralDomain

For each natural number $n$, including $n=0$, the spectral domain is the real Euclidean space $\mathbb R^n$, with coordinates indexed by $\{0,\ldots,n-1\}$ and its usual Euclidean norm and inner product. When $n=0$, this space consists of a single point.

QAlgorithms.Papers.Independent.Intakethird100.P38.TrivialPointSpectrum

For any natural number $n$, including $0$, let $\Omega$ be the space of continuous real-linear functionals on the real Schwartz space over Euclidean $\mathbb R^n$, with its evaluation-generated sigma-algebra; let $\mu$ be any measure on $\Omega$; and let $U_t$, indexed by $t\in\mathbb R^n$, be bijective complex-linear isometries of $L^2(\Omega,\mu;\mathbb C)$. The property $\operatorname{TrivialPointSpectrum}(\mu,U)$ is the conjunction of two statements: for every $a\in\mathbb R^n$ and every nonzero $f\in L^2(\Omega,\mu;\mathbb C)$, if $U_tf=e^{i\,a\cdot t}f$ for every $t\in\mathbb R^n$, then $a=0$; and, for every $f\in L^2(\Omega,\mu;\mathbb C)$,

$$\bigl(\forall t\in\mathbb R^n,\ U_tf=f\bigr) \quad\Longleftrightarrow\quad \exists c\in\mathbb C,\ f=C_c.$$

Here $C_c$ is the $L^2$ class of the constant function $\omega\mapsto c$ if that function belongs to $L^2$, and is the zero $L^2$ element otherwise. The first statement requires no existence of nonzero eigenvectors and is vacuous when there are none. No probability assumption or shift-action property is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P38.adjointProduct

For any natural number $n$, including $0$, let $Q$ be a continuous complex-linear map from the complex-valued Schwartz functions on Euclidean $\mathbb R^n$ to $L^2(\mathbb R^n,dx;\mathbb C)$, and let $\psi$ be a complex-valued Schwartz function. Then $\operatorname{adjointProduct}(Q,\psi)$ is the continuous conjugate-linear functional whose value at every complex-valued Schwartz function $\phi$ is

$$\bigl(\operatorname{adjointProduct}(Q,\psi)\bigr)(\phi) =\langle Q\phi,Q\psi\rangle_{L^2} =\int_{\mathbb R^n}\overline{(Q\phi)(x)}(Q\psi)(x)\,dx,$$

where the inner product is conjugate-linear in its first argument and $L^2$ functions are understood up to almost-everywhere equality.

QAlgorithms.Papers.Independent.Intakethird100.P38.combSamples

For every complex-valued Schwartz function $\psi$ on the one-dimensional real Euclidean space and every integer $m$, the defined sample is its Fourier integral at the vector with sole coordinate $m$:

$$\operatorname{combSamples}(\psi,m) =\int_{\mathbb R}e^{-imx}\psi(x)\,dx.$$

Here $dx$ is Euclidean volume measure, and the Fourier integral has no normalization factor.

QAlgorithms.Papers.Independent.Intakethird100.P38.covarianceKernel

For any natural number $n$, including $0$, and any measure $\sigma$ on Euclidean $\mathbb R^n$, $\operatorname{covarianceKernel}(\sigma)$ is a matrix indexed in both coordinates by all real-valued Schwartz functions, whose entries are continuous complex-linear maps $\mathbb C\to\mathbb C$. For indices $\psi,\phi$, put

$$E_{\psi,\phi}=\int_{\mathbb R^n} \left|\int_{\mathbb R^n}e^{-i\,u\cdot x}\bigl(\psi(x)-\phi(x)\bigr)\,dx\right|^2\,d\sigma(u)\in[0,\infty].$$

The $(\psi,\phi)$-entry sends $z\in\mathbb C$ to $e^{-r_{\psi,\phi}/2}z$, where $r_{\psi,\phi}$ is the real value of $E_{\psi,\phi}$ when it is finite and is $0$ when $E_{\psi,\phi}=\infty$. In particular, an infinite energy gives the identity map. No regularity or energy-finiteness assumption is made on $\sigma$.

QAlgorithms.Papers.Independent.Intakethird100.P38.decayWeight

For every natural number $n$, every natural number $p$, and every $u\in\mathbb R^n$, define the real weight $w_p(u)=1/(1+\|u\|^2)^p$, using the Euclidean norm. Both $n=0$ and $p=0$ are allowed: $w_0(u)=1$ everywhere, and on the one-point space $\mathbb R^0$ the weight is $1$ for every $p$. The denominator is always strictly positive.

QAlgorithms.Papers.Independent.Intakethird100.P38.diracComb

The defined measure on the one-dimensional real Euclidean space is the sum of unit point masses at all integer-coordinate points:

$$\sigma_{\mathrm{comb}}=\sum_{m\in\mathbb Z}\delta_{(m)}.$$

Thus, for every measurable set $A$, its measure is the extended nonnegative sum $\sum_{m\in\mathbb Z}\mathbf 1_A((m))$, which may be infinite.

QAlgorithms.Papers.Independent.Intakethird100.P38.distributionShift

For any natural number $n$, including $0$, any $t\in\mathbb R^n$, and any continuous real-linear functional $\omega$ on the real-valued Schwartz functions on Euclidean $\mathbb R^n$, $\operatorname{distributionShift}(t,\omega)$ is the continuous real-linear functional given, for every real-valued Schwartz function $\psi$, by

$$\bigl(\operatorname{distributionShift}(t,\omega)\bigr)(\psi) =\omega\bigl(x\mapsto\psi(x+t)\bigr).$$

QAlgorithms.Papers.Independent.Intakethird100.P38.factorFromW

For every natural number $n$, including $0$, every measure $\sigma$ on $\mathbb R^n$, every complex-linear isometric bijection

$$W:L^2(\mathbb R^n,\sigma;\mathbb C)\longrightarrow L^2(\mathbb R^n,dx;\mathbb C),$$

and every complex-valued Schwartz function $\psi$, this defines the vector $W(v_\psi)$, where

$$\widehat\psi(u)=\int_{\mathbb R^n}e^{-i\langle u,x\rangle}\psi(x)\,dx$$

and $v_\psi=[\widehat\psi]_\sigma$ if this Fourier transform is almost everywhere strongly measurable and square-integrable with respect to $\sigma$, while $v_\psi=0$ otherwise. Here $dx$ is Lebesgue measure and $\langle u,x\rangle$ is the real Euclidean inner product. In the latter case the resulting vector is $W(0)=0$. The bijection $W$ is supplied as an argument; its existence is not asserted.

QAlgorithms.Papers.Independent.Intakethird100.P38.fourierClass

For every natural number $n$, including $0$, every measure $\sigma$ on $\mathbb R^n$, and every complex-valued Schwartz function $\psi$ on $\mathbb R^n$, define

$$\widehat\psi(u)=\int_{\mathbb R^n}e^{-i\langle u,x\rangle}\psi(x)\,dx.$$

This declaration assigns to $\psi$ the equivalence class $[\widehat\psi]_\sigma$ in $L^2(\mathbb R^n,\sigma;\mathbb C)$ if $\widehat\psi$ is almost everywhere strongly measurable and has finite integral of its squared modulus with respect to $\sigma$; otherwise it assigns the zero element of that space. The Fourier integral uses Lebesgue measure and the real Euclidean inner product. No assumption on $\sigma$ ensures that the square-integrability condition holds.

QAlgorithms.Papers.Independent.Intakethird100.P38.fourierEnergy

For every natural number $n$, including $n=0$, every measure $\sigma$ on $\mathbb R^n$, and every function $f:\mathbb R^n\to\mathbb C$, define the Fourier energy as the extended nonnegative number $E_\sigma(f)=\int_{\mathbb R^n}^{-}\left|\int_{\mathbb R^n}e^{-i\langle u,x\rangle}f(x)\,dx\right|^2\,d\sigma(u)$, where the inner integral uses Euclidean volume measure. The squared complex norm is embedded into the extended nonnegative reals by the map $r\mapsto\max(r,0)$. No measurability, integrability, or finiteness hypotheses are imposed: each inner Bochner integral is zero if its integrand is not integrable, and the outer integral may be $+\infty$. For $n=0$, the Euclidean domain is a single point.

QAlgorithms.Papers.Independent.Intakethird100.P38.l2Class

For every type $\alpha$ equipped with a measurable-space structure, every measure $\mu$ on $\alpha$, and every function $f:\alpha\to\mathbb C$, this defines an element of $L^2(\alpha,\mu;\mathbb C)$ by

$$\operatorname{l2Class}_\mu(f)= \begin{cases} [f]_\mu,&\text{if }f\text{ is almost everywhere strongly measurable and } \displaystyle\int_\alpha |f|^2\,d\mu<\infty,\\ 0,&\text{otherwise}. \end{cases}$$

Here $[f]_\mu$ denotes the equivalence class of $f$ modulo equality $\mu$-almost everywhere, and $0$ is the zero element of $L^2$. Thus the definition applies even to functions that fail the measurability or square-integrability requirement, and it imposes no additional assumptions on $\alpha$ or $\mu$.

QAlgorithms.Papers.Independent.Intakethird100.P38.lawFourier

For any natural number $n$, including $0$, let $\Omega$ be the space of continuous real-linear functionals on the real Schwartz space over Euclidean $\mathbb R^n$, with the sigma-algebra generated by evaluations. For any measure $\mu$ on $\Omega$, any $f\in L^2(\Omega,\mu;\mathbb C)$, and any real-valued Schwartz function $\psi$, the definition is

$$\operatorname{lawFourier}(\mu,f,\psi) =\int_\Omega e^{-i\omega(\psi)}f(\omega)\,d\mu(\omega),$$

using the representative supplied for the $L^2$ class $f$. No probability or finite-measure hypothesis is imposed on $\mu$; if the integrand is not integrable, the total Bochner integral in this definition equals zero.

QAlgorithms.Papers.Independent.Intakethird100.P38.measureConvolution

For every natural number $n$, including $n=0$, and every pair of measures $\sigma,\tau$ on the real Euclidean space $\mathbb R^n$, their defined convolution is the pushforward of the product measure $\sigma\otimes\tau$ under addition:

$$(\sigma*\tau)(A)=(\sigma\otimes\tau)\bigl(\{(u,v):u+v\in A\}\bigr)$$

for every measurable set $A\subseteq\mathbb R^n$. No finiteness, regularity, or probability assumption is imposed on either measure.

QAlgorithms.Papers.Independent.Intakethird100.P38.omegaMeasurableSpace

For each natural number $n$, including $n=0$, equip the space $\Omega(n)$ of continuous real-linear functionals on $\mathcal S(\mathbb R^n;\mathbb R)$ with the smallest $\sigma$-algebra making every evaluation map $\omega\mapsto\omega(\psi)$ measurable into $\mathbb R$ with its Borel $\sigma$-algebra, as $\psi$ ranges over all real-valued Schwartz functions. Equivalently, this $\sigma$-algebra is generated by the sets $\{\omega:\omega(\psi)\in B\}$ for all such $\psi$ and all Borel sets $B\subseteq\mathbb R$.

QAlgorithms.Papers.Independent.Intakethird100.P38.paperFourier

For every natural number $n$, including $n=0$, every function $f:\mathbb R^n\to\mathbb C$, and every $u\in\mathbb R^n$, define its specified Fourier value by $\widehat f(u)=\int_{\mathbb R^n}e^{-i\langle u,x\rangle}f(x)\,dx$, where the inner product is the real Euclidean inner product and integration uses Euclidean volume measure. There are no measurability or integrability hypotheses on $f$; the integral is the total Bochner integral, which is zero when the integrand is not integrable. For $n=0$, the domain is a single point.

QAlgorithms.Papers.Independent.Intakethird100.P38.periodization

For every complex-valued Schwartz function $\psi$ on the one-dimensional real Euclidean space and every point $x$ in that space, the defined periodization is the integer-indexed sum

$$P_\psi(x)=\sum_{m\in\mathbb Z}\psi\bigl(x+(2\pi m)\bigr),$$

where $(2\pi m)$ denotes the vector with sole coordinate $2\pi m$. The sum uses the total infinite-sum operation, which returns $0$ if the indexed family is not summable; the definition itself contains no summability hypothesis.

QAlgorithms.Papers.Independent.Intakethird100.P38.point1

For every real number $x$, the defined point $\operatorname{point}_1(x)$ is the vector in the one-dimensional real Euclidean space whose sole coordinate is $x$.

QAlgorithms.Papers.Independent.Intakethird100.P38.realFunction

For every natural number $n$, including $n=0$, and every real-valued Schwartz function $\psi\in\mathcal S(\mathbb R^n;\mathbb R)$, define a complex-valued function on $\mathbb R^n$ by $x\mapsto\psi(x)+0i$. Thus each value is the usual embedding of the real number $\psi(x)$ into $\mathbb C$. For $n=0$, this is a function on a single point.

QAlgorithms.Papers.Independent.Intakethird100.P38.spectralCompletion

For every natural number $n$, including $0$, and every measure $\sigma$ on $\mathbb R^n$, this defines the complex vector subspace

$$H_\sigma=\overline{\operatorname{span}_{\mathbb C}\{v_\psi:\psi\text{ is a complex-valued Schwartz function on }\mathbb R^n\}} \subseteq L^2(\mathbb R^n,\sigma;\mathbb C),$$

where the closure is taken in the $L^2$ norm. For each $\psi$, set $\widehat\psi(u)=\int_{\mathbb R^n}e^{-i\langle u,x\rangle}\psi(x)\,dx$, using Lebesgue measure and the real Euclidean inner product; $v_\psi$ is the almost-everywhere equivalence class of $\widehat\psi$ if it is almost everywhere strongly measurable and square-integrable with respect to $\sigma$, and is the zero vector otherwise. No additional assumptions on $\sigma$ are imposed, and the resulting subspace may be the zero subspace.

QAlgorithms.Papers.Independent.Intakethird100.P38.spectralDistribution

For any natural number $n$, including $0$, any measure $\sigma$ on Euclidean $\mathbb R^n$, and any complex-valued Schwartz function $\psi$, define $\widehat\eta(u)=\int_{\mathbb R^n}e^{-i\,u\cdot x}\eta(x)\,dx$ for a complex-valued Schwartz function $\eta$. If there exists a continuous conjugate-linear functional $T$ on the complex Schwartz space such that

$$T(\phi)=\int_{\mathbb R^n}\widehat\psi(u)\,\overline{\widehat\phi(u)}\,d\sigma(u) \quad\text{for every complex-valued Schwartz function }\phi,$$

then $\operatorname{spectralDistribution}(\sigma,\psi)$ is a classically chosen such functional; if none exists, it is the zero functional. No regularity or finiteness hypothesis on $\sigma$, or existence hypothesis on $T$, is imposed. The displayed complex integral uses the total Bochner-integral convention, giving zero when its integrand is not integrable.

QAlgorithms.Papers.Independent.Intakethird100.P38.spectralPairing

For every natural number $n$, including $0$, every measure $\sigma$ on the Euclidean space $\mathbb R^n$, and every pair of complex-valued Schwartz functions $\psi,\phi$ on $\mathbb R^n$, this defines the complex number

$$\operatorname{spectralPairing}_\sigma(\psi,\phi) =\int_{\mathbb R^n}\widehat\psi(u)\,\overline{\widehat\phi(u)}\,d\sigma(u), \qquad \widehat f(u)=\int_{\mathbb R^n}e^{-i\langle u,x\rangle}f(x)\,dx.$$

Here $dx$ is Lebesgue measure and $\langle u,x\rangle$ is the real Euclidean inner product. No regularity or finiteness assumption is imposed on $\sigma$. The outer integral is the Bochner integral, whose value is defined to be $0$ if its integrand is not integrable.

QAlgorithms.Papers.Independent.Intakethird100.P38.spectralQ

For every natural number $n$, including $n=0$, every measure $\sigma$ on $\mathbb R^n$, and every complex-valued Schwartz function $\psi\in\mathcal S(\mathbb R^n;\mathbb C)$, define $Q_\sigma(\psi)$ by converting the extended nonnegative quantity $\int_{\mathbb R^n}^{-}\left|\int_{\mathbb R^n}e^{-i\langle u,x\rangle}\psi(x)\,dx\right|^2\,d\sigma(u)$ to a real number. This conversion preserves every finite value and sends $+\infty$ to $0$. The inner integral uses Euclidean volume measure and has the total Bochner-integral convention of being zero when its integrand is not integrable. No hypothesis on $\sigma$ or finiteness of the energy is required; $n=0$ gives a one-point domain.

QAlgorithms.Papers.Independent.Intakethird100.P38.spectralVector

For every natural number $n$, including $0$, every measure $\sigma$ on $\mathbb R^n$, and every complex-valued Schwartz function $\psi$, this defines a vector in

$$H_\sigma=\overline{\operatorname{span}_{\mathbb C}\{v_\eta:\eta\text{ is a complex-valued Schwartz function on }\mathbb R^n\}} \subseteq L^2(\mathbb R^n,\sigma;\mathbb C).$$

Here $\widehat\eta(u)=\int_{\mathbb R^n}e^{-i\langle u,x\rangle}\eta(x)\,dx$, with Lebesgue measure and the real Euclidean inner product, and $v_\eta$ is its $L^2(\sigma)$ equivalence class when the transform is almost everywhere strongly measurable and square-integrable, or zero otherwise. The vector defined is exactly $v_\psi$, regarded as an element of $H_\sigma$; its membership follows because it belongs to the generating set, hence to its complex span and its norm closure. In particular, failure of the square-integrability or measurability condition gives the zero vector in $H_\sigma$.

QAlgorithms.Papers.Independent.Intakethird100.P38.testTranslation

For every natural number $n$, including $0$, and every $t\in\mathbb R^n$, this defines the continuous complex-linear operator on complex-valued Schwartz functions on $\mathbb R^n$ given by

$$(T_t\psi)(x)=\psi(x-t).$$

The zero translation is included. When $n=0$, the Euclidean space has one point and this operator is the identity.

QAlgorithms.Papers.Independent.Intakethird100.P38.weightedMass

For every natural number $n$, including $n=0$, every measure $\sigma$ on $\mathbb R^n$, and every natural number $p$, define the weighted mass as the extended nonnegative number $M_p(\sigma)=\int_{\mathbb R^n}^{-}(1+\|u\|^2)^{-p}\,d\sigma(u)$. The integrand is regarded as an extended nonnegative real by the map sending a real number $r$ to $\max(r,0)$; here it is already positive. No finiteness assumption is imposed, so the value may be $+\infty$. The choice $p=0$ is allowed and gives $\sigma(\mathbb R^n)$.