# Gaussian regularization and quadratic stationary phase

This reading develops Gaussian regularization, Fourier calculus on Schwartz
functions, and quadratic stationary phase with every coefficient and an
explicit finite-order remainder. It includes smooth Hessian and parameter
families, nonstationary estimates, and an exact Gaussian example with a
reproducible illustration.

The [parameter-Morse companion](parameter-morse-reduction.md) supplies the
smooth signed normal form used in the
[stationary-phase and critical-manifold reading](stationary-phase-and-critical-manifolds.md).
Mathematical sources and their conventions are described below.

Original programme exposition: CC0 to the extent rights exist. Cited
human works are external sources and retain their own terms.

## F0. The exact earlier programme proofs

The following earlier arguments provide the elementary tools used below.
The [proof index](proof-map.html) gives the statement, proof locator and
lower inputs for each named result.

| ID | Result used | Uses below | Earlier proofs |
|---|---|---|---|
| F0-CALC | Fundamental theorem of calculus, product and chain rules, real and complex exponentials, and smooth positive roots | Q1–Q9 | [Differentiation P9–P10](differential-prerequisite-completions.html), [compact integrals and Taylor P11–P12](integration-prerequisite-completions.html), and [exponentials and root phases P13–P16](exponential-prerequisite-completions.html) |
| F0-INT | Compact and absolutely convergent improper integrals, linearity, norm and tail bounds, and iteration under product majorants | Q1–Q9 | [Rectangle integrals and tails P17–P18](global-integral-prerequisite-completions.html); [P18.4](global-integral-prerequisite-completions.html#proof-P18.4) states the product-majorant hypothesis for the global interchanges |
| F0-COV | Orthogonal changes, positive dilations and planar polar coordinates, with their Jacobians | Q2, Q4, Q6 | [Affine and chart substitution P21.4](change-of-variables-prerequisite-completions.html#proof-P21.4) and [polar coordinates P21.5](change-of-variables-prerequisite-completions.html#proof-P21.5) |
| F0-COMP | Finite-dimensional compactness, extrema and uniform continuity on compact sets | Q1, Q5, Q9 | [Finite-dimensional compactness P1](prerequisite-completions.html#proof-P1), [elementary sequence lemmas P6](elementary-proof-completions.html#proof-P6.0), and the [compactness references](proof-map.html#F0-COMP) |
| F0-ALG | Determinant multiplication, cofactor inversion and smooth dependence on matrix entries | Q5, Q6, Q8 | [Cofactor inversion P2](prerequisite-completions.html#proof-P2), [permutation parity P7](elementary-proof-completions.html#proof-P7), and the [algebra references](proof-map.html#F0-ALG) |

Finite sums, multi-indices, Euclidean scalar products and the determinant
have their usual algebraic definitions. The human-proof references
identify the external foundational statements and proofs, including those
from Jiří Lebl's freely accessible author edition.

Integrals in this module are compact Riemann integrals or absolutely
convergent improper integrals, as defined and proved in P17–P18. In
particular, every global interchange in Q3–Q6 satisfies the product-majorant
hypotheses of P18.4.

Set
\[
 \widehat a(\xi)=\int_{\mathbb R^n}e^{-ix\cdot\xi}a(x)\,dx,
 \qquad
 a(x)=(2\pi)^{-n}\int_{\mathbb R^n}e^{ix\cdot\xi}
                                  \widehat a(\xi)\,d\xi.
 \tag{F}
\]
The second equality will be proved in Q4, before it is used. A function belongs
to \(\mathcal S(\mathbb R^n)\) when it is smooth and
\(\sup_x(1+|x|)^m|\partial^\alpha a(x)|<\infty\) for every integer
\(m\geq0\) and every multi-index \(\alpha\).

## Q1. The limit and differentiation rules actually used here

Suppose continuous functions \(f_t\) converge to \(f_0\) uniformly on every
compact set as \(t\to0\), and \(|f_t(x)|\leq g(x)\) for one integrable
\(g\), including \(t=0\). Then \(\int f_t\to\int f_0\).

**Proof.** Given \(\delta>0\), choose a cube \(K\) so that
\(\int_{K^c}g<\delta\). The difference of the integrals on \(K^c\) is
at most \(2\delta\). On \(K\) it is at most
\(|K|\sup_K|f_t-f_0|\), which tends to zero. Taking a limit superior and
then letting \(\delta\) decrease to zero proves the claim.

For a parameter derivative, suppose \(f(t,x)\) and \(\partial_t f(t,x)\)
are continuous for \(t\) in a neighbourhood of \(t_0\), the integral at
\(t_0\) exists absolutely, and \(|\partial_t f(t,x)|\leq g(x)\) there.
The fundamental theorem of calculus gives
\[
 \frac{f(t_0+s,x)-f(t_0,x)}s
 =\int_0^1\partial_t f(t_0+us,x)\,du.
\]
These quotients are bounded by \(g\) and converge uniformly on compact sets
to \(\partial_t f(t_0,x)\). The first part proves differentiation under
the integral. It also proves continuity of that derivative when the same
local bound holds. Repeating the argument proves any finite order for which
the corresponding bounds are available. This proof uses F0, not a general
unproved appeal to dominated convergence. \(\square\)

## Q2. A complex Gaussian without an analytic-continuation import

For \(z\in\mathbb C\) with \(\operatorname{Re}z>0\) and real \(\eta\),
\[
 \int_{\mathbb R}\exp(-zx^2/2+i\eta x)\,dx
 =\sqrt{2\pi}\,z^{-1/2}\exp(-\eta^2/(2z)).
 \tag{G}
\]
Here \(\sqrt z\) is the square root with positive real part.

**Proof.** The integral and all derivatives used below converge absolutely:
on a compact subset of the right half-plane their absolute values are
bounded by a polynomial in \(|x|\) times \(e^{-c x^2}\), for some
\(c>0\). Such bounds are integrable: differentiation shows that
\(r^m e^{-cr^2/2}\), for \(r\geq0\), has a finite maximum, so the
remaining factor is bounded by \(C e^{-cr^2/2}\), and hence by
\(C e^{-cr/2}\) for \(r\geq1\). The latter has a finite integral by
the fundamental theorem. Q1 therefore applies.

First let \(J=\int e^{-x^2/2}\,dx>0\). Squaring, using absolute iteration
and then planar polar coordinates, gives
\[
 J^2=\int_{\mathbb R^2}e^{-(x^2+y^2)/2}\,dx\,dy
 =\int_0^{2\pi}\int_0^\infty e^{-r^2/2}r\,dr\,d\theta=2\pi.
\]
Thus \(J=\sqrt{2\pi}\). P21.5 in
`change-of-variables-prerequisite-completions.md` proves the polar Jacobian,
the compact-sector substitution, and the limits at the origin, missing ray
and infinity used in this step.

For fixed \(z\) join 1 to \(z\) by \(z_t=1+t(z-1)\), \(0\leq t\leq1\).
Its real part is positive. Put \(J(t)=\int e^{-z_t x^2/2}\,dx\).
Integration of \(\partial_x(xe^{-z_t x^2/2})\) over \([-R,R]\), followed
by \(R\to\infty\), gives
\(\int x^2e^{-z_t x^2/2}\,dx=J(t)/z_t\), because the boundary terms
vanish exponentially. Consequently
\[
 J'(t)=-\frac{z'_t}{2z_t}J(t).
\]
The chosen root \(s(t)=\sqrt{z_t}\) is differentiable and satisfies
\(2s(t)s'(t)=z'_t\). One can see this without a complex-analysis theorem
by writing \(z_t=r_t e^{i\theta_t}\), with
\(-\pi/2<\theta_t<\pi/2\), and setting
\(s(t)=\sqrt{r_t}e^{i\theta_t/2}\). P16.3 in
`exponential-prerequisite-completions.md` proves the smooth angle, the
unique positive-real-part root and this differentiation rule. The product rule now gives
\((sJ)'=0\); hence \(J(1)=\sqrt{2\pi}\,z^{-1/2}\).

Finally, with \(z\) fixed write the left side of (G) as \(F(\eta)\).
The integral of \(\partial_x e^{-zx^2/2+i\eta x}\) vanishes, so
\(\int xe^{-zx^2/2+i\eta x}\,dx=(i\eta/z)F(\eta)\). Thus
\(F'(\eta)=-(\eta/z)F(\eta)\), and
\(\partial_\eta(e^{\eta^2/(2z)}F(\eta))=0\). The value at zero already
computed proves (G). \(\square\)

## Q3. Schwartz estimates and the signs in the Fourier rules

If \(a\in\mathcal S\), then \(\widehat a\in\mathcal S\), and
\[
 \partial_\xi^\alpha\widehat a=\widehat{(-ix)^\alpha a},\qquad
 \widehat{\partial_x^\beta a}=(i\xi)^\beta\widehat a.
 \tag{D}
\]
These maps are continuous for the defining Schwartz seminorms.

**Proof.** Polynomial multiples of derivatives of \(a\) are integrable.
For example use the rectangular shells between \([-2^j,2^j]^n\) and
\([-2^{j+1},2^{j+1}]^n\). P18.3 proves their integral bounds directly
from finite rectangle partitions. Decay \((1+|x|)^{-m}\), \(m>n\),
reduces the bound to a convergent geometric series.
Q1 proves every differentiation in the first
formula. For the second, integrate first in a single coordinate on
\([-R,R]\). Its boundary terms tend to zero by rapid decrease; the
remaining integrals converge absolutely. Iteration proves the multi-index
formula. Applying these identities together shows that each
\(\xi^\beta\partial_\xi^\alpha\widehat a\) is the Fourier transform of
a finite linear combination of polynomial multiples of derivatives of
\(a\). Its supremum is bounded by their \(L^1\) norms. The rectangular-shell
estimate bounds each such norm by finitely many input seminorms, proving
both rapid decrease and continuity.

A useful quantitative consequence is, for integers \(M>N+n/2\),
\[
 \int |\xi|^{2N}|\widehat a(\xi)|\,d\xi
 \leq C_{n,N,M}\,\|(1-\Delta)^M a\|_{L^1},\qquad
 C_{n,N,M}=\int\frac{|\xi|^{2N}}{(1+|\xi|^2)^M}\,d\xi<\infty.
 \tag{W}
\]
Indeed (D) gives
\((1+|\xi|^2)^M\widehat a=\widehat{(1-\Delta)^M a}\); take absolute
values and integrate. The same rectangular-shell bound proves finiteness
of \(C_{n,N,M}\), with its stated threshold unchanged.
\(\square\)

## Q4. Fourier inversion proved before stationary phase

For every \(a\in\mathcal S\), the inversion identity in (F) holds.

**Proof.** For \(\varepsilon>0\) define
\[
 A_\varepsilon(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}
           e^{-\varepsilon|\xi|^2/2}\widehat a(\xi)\,d\xi.
\]
Insert the definition of \(\widehat a\). The double absolute integral
is bounded by \(\|a\|_1\int e^{-\varepsilon|\xi|^2/2}\,d\xi<\infty\),
so F0-INT permits interchange. Applying (G) in every coordinate yields
\[
 A_\varepsilon(x)=\int K_\varepsilon(x-y)a(y)\,dy,\qquad
 K_\varepsilon(v)=(2\pi\varepsilon)^{-n/2}
                         e^{-|v|^2/(2\varepsilon)}.
\]
By positive dilation and Q2, \(\int K_\varepsilon=1\). Since
\(a\) has bounded gradient, the fundamental theorem along a line segment
gives \(|a(x-v)-a(x)|\leq\|\nabla a\|_\infty|v|\), with the Euclidean
norm for the gradient. Therefore
\[
 |A_\varepsilon(x)-a(x)|
 \leq\sqrt\varepsilon\,\|\nabla a\|_\infty
           \int |w|K_1(w)\,dw\longrightarrow0.
\]
On the Fourier side the multiplier converges uniformly on compact sets
to 1 and is bounded by 1. Q3 supplies integrability of \(\widehat a\),
so Q1 shows that \(A_\varepsilon(x)\) tends to the right side of (F).
This proves inversion without using stationary phase. Applying the same
argument to \(\partial^\alpha a\) also gives every derivative of (F).
\(\square\)

## Q5. Diagonalizing a real symmetric matrix

A real symmetric \(n\times n\) matrix \(H\) has an orthonormal basis
of real eigenvectors. If \(H\) is invertible, all its eigenvalues are
nonzero. Define \(\operatorname{sgn}H\) to be their number of positive
entries minus their number of negative entries.

**Proof.** In dimension 0 the empty orthonormal basis proves the assertion.
For \(n\geq1\), the function \(v\mapsto v^THv\) attains a maximum on the
unit sphere, by F0-COMP. Let \(v\) be a maximizing unit vector. For
\(w\perp v\), differentiate this function along
\(p(t)=(v+tw)/\sqrt{1+t^2|w|^2}\) at \(t=0\). The product, chain
and positive-root derivative rules give \(p'(0)=w\). The scalar Fermat
theorem at this maximum therefore gives \(2w^THv=0\).
Set \(\lambda=v^THv\). The vector \(w=Hv-\lambda v\) is perpendicular
to \(v\), and the preceding identity gives \(|w|^2=w^THv=0\).
Thus \(Hv=\lambda v\). If \(w\perp v\),
then \(v^THw=(Hv)^Tw=0\); hence \(v^\perp\) is invariant under
\(H\). P9.4 in `differential-prerequisite-completions.md` constructs
an orthonormal basis of this complement and proves that the restriction is
a symmetric \((n-1)\)-dimensional matrix. Apply induction to that matrix,
starting with dimension 0 or 1.
This gives an orthogonal matrix \(U\) with
\(U^THU=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)\).
If any \(\lambda_j=0\), its eigenvector lies in the kernel, and conversely.
Since \(U^TU=I\), determinant multiplication gives \((\det U)^2=1\)
and \(\det(U^THU)=\det H\). In the permutation formula for a diagonal
matrix only the identity permutation can have a nonzero term; its term is
\(\prod_j\lambda_j\). Hence \(\det H=\prod_j\lambda_j\).
\(\square\)

## Q6. Exact quadratic identity, with a justified regularization

Let \(H\) be real, symmetric and invertible, \(h>0\),
\(B=H^{-1}\), and \(a\in\mathcal S(\mathbb R^n)\). Then
\[
 \int e^{ix^THx/(2h)}a(x)\,dx
 =C_H(h)(2\pi)^{-n}\int e^{-ih\xi^TB\xi/2}
                                  \widehat a(\xi)\,d\xi,
 \qquad
 C_H(h)=\frac{(2\pi h)^{n/2}e^{i\pi\operatorname{sgn}H/4}}
                {\sqrt{|\det H|}}.
 \tag{QI}
\]
Both integrals displayed here are absolutely convergent because the
amplitudes are Schwartz. The bare quadratic exponential is not treated as
an absolutely integrable function.

**Proof.** Multiply the integrand on the left by
\(e^{-\varepsilon|x|^2/2}\), \(\varepsilon>0\), and insert Q4. The
double absolute integral is finite, bounded by
\((2\pi)^{-n}\|\widehat a\|_1\int e^{-\varepsilon|x|^2/2}\,dx\).
Thus its order can be exchanged. Diagonalize \(H\) using Q5 and make
the orthogonal change \(x=Uy\), \(\eta=U^T\xi\). The inner integral
is the product of (G) with
\(z_j=\varepsilon-i\lambda_j/h\). Its value is
\[
 (2\pi)^{n/2}\prod_{j=1}^n z_j^{-1/2}
                 \exp\!\left(-\sum_{j=1}^n\frac{\eta_j^2}{2z_j}\right).
 \tag{RG}
\]
Since \(\operatorname{Re}(1/z_j)>0\), the modulus of the exponential
is at most 1, while
\(|z_j|^{-1/2}\leq(h/|\lambda_j|)^{1/2}\). This supplies an
\(\varepsilon\)-independent integrable bound after multiplication by
\(|\widehat a(\xi)|\).

As \(\varepsilon\downarrow0\), with each root continued through the
right half-plane,
\[
 z_j^{-1/2}\longrightarrow
 (h/|\lambda_j|)^{1/2}e^{i\pi\operatorname{sgn}(\lambda_j)/4},
 \qquad z_j^{-1}\longrightarrow ih/\lambda_j.
\]
The convergence is uniform for \(\xi\) in compact sets. Q1 therefore
takes the limit on the Fourier side of (RG), giving the right side of
(QI). On the original side it applies with bound \(|a|\), removing
the regularization. The product of the root phases is precisely
\(e^{i\pi\operatorname{sgn}H/4}\). \(\square\)

Original text: public domain (CC0).

The root limit and its \(\pi/4\) phase in this proof are established
explicitly in P16.3, including both signs of \(\lambda_j\).

## Q7. Every coefficient and an explicit remainder

Put \(\Delta_B=\sum_{j,k}B_{jk}\partial_j\partial_k\). For every integer
\(N\geq1\),
\[
 \int e^{ix^THx/(2h)}a(x)\,dx
 =C_H(h)\left[\sum_{k=0}^{N-1}\frac{(ih/2)^k}{k!}
                  (\Delta_B^ka)(0)+R_N(h;H,a)\right],
 \tag{SP}
\]
where
\[
 |R_N(h;H,a)|\leq
 \frac{h^N}{2^NN!(2\pi)^n}
    \int |\xi^TB\xi|^N|\widehat a(\xi)|\,d\xi
 \leq\frac{h^N\|B\|^N C_{n,N,M}}{2^NN!(2\pi)^n}
             \|(1-\Delta)^M a\|_1
 \tag{R}
\]
for any integer \(M>N+n/2\). Here \(\|B\|\) is its Euclidean operator
norm. The error in the unnormalized integral is \(|C_H(h)R_N|\),
of order \(h^{n/2+N}\) for fixed \(H,a\).

**Proof.** Repeatedly apply the fundamental theorem of calculus to the
function \(t\mapsto e^{it}\), or inductively integrate its remainder
once. This gives, for real \(t\),
\[
 e^{it}-\sum_{k=0}^{N-1}\frac{(it)^k}{k!}
 =\frac{(it)^N}{(N-1)!}\int_0^1(1-s)^{N-1}e^{ist}\,ds,
 \quad
 \left|e^{it}-\sum_{k<N}\frac{(it)^k}{k!}\right|
 \leq\frac{|t|^N}{N!}.
 \tag{T}
\]
For completeness, the inductive integral form starts with
\(f(1)-f(0)=\int_0^1 f'(s)ds\). Integration by parts changes
\(\int_0^1 (1-s)^{N-1}f^{(N)}(s)/(N-1)!\,ds\) into
\(-f^{(N-1)}(0)/(N-1)!+
\int_0^1(1-s)^{N-2}f^{(N-1)}(s)/(N-2)!\,ds\);
descending to the first identity proves the formula for
\(f(s)=e^{its}\).

Apply (T) with \(t=-h\xi^TB\xi/2\) in (QI). The remainder integrates
absolutely by Q3, giving the first inequality (R). Moreover, (D) and Q4
give
\[
 (2\pi)^{-n}\int(\xi^TB\xi)^k\widehat a(\xi)\,d\xi
 =(-1)^k(\Delta_B^ka)(0).
\]
Thus \((-ih/2)^k\) becomes \((ih/2)^k\), establishing the sign in
(SP). Finally \(|\xi^TB\xi|\leq\|B\||\xi|^2\), and (W) proves the
second inequality. \(\square\)

## Q8. What is uniform when the Hessian varies

Let \(p\) range over an open parameter set, let \(H(p)\) be smooth,
real symmetric and invertible, and let \(a(p,\cdot)\) be smooth with
values in \(\mathcal S\): every parameter derivative has locally bounded
Schwartz seminorms. On every compact parameter subset, the normalized
integral in (QI), divided by \(C_{H(p)}(h)\), extends smoothly through
\(h=0\) as a function of real \(h\). Its Taylor coefficients are the
ones in (SP), and the normalized remainder is \(O(h^N)\) after any
fixed number of parameter derivatives.

**Proof.** Matrix inversion is smooth where the determinant is nonzero,
by the cofactor formula, so \(B(p)\) and all derivatives are bounded
on a compact parameter subset contained in the open set. For real \(h\)
define the normalized expression directly by
\[
 A(p,h)=(2\pi)^{-n}\int
     e^{-ih\xi^TB(p)\xi/2}\widehat {a(p,\cdot)}(\xi)\,d\xi.
\]
Each \(h\)-derivative multiplies the integrand by a power of the
quadratic polynomial \(-i\xi^TB(p)\xi/2\). A parameter derivative
either differentiates the Fourier amplitude or contributes a derivative
of this polynomial multiplied by \(h\). Repeated product rules
therefore give a finite sum of polynomial factors in \(\xi\), with
bounded coefficients for \(p\) in a compact set and \(|h|\leq1\),
times a Fourier transform of a parameter derivative of \(a\).
Q3 makes all these factors uniformly integrable. Q1 proves smoothness
and permits all the derivatives just described.

At \(h=0\), Q4 and (D) give
\(\partial_h^k A(p,0)=(i/2)^k\Delta_{B(p)}^k a(p,0)\).
Apply the integral Taylor formula proved in Q7 to
\(h\mapsto\partial_p^\alpha A(p,h)\). Its \(N\)th \(h\)-derivative
is uniformly bounded by the preceding integral estimates. The remainder
is bounded by \(C_{N,\alpha}|h|^N\), uniformly on the compact
parameter subset. This proves the asserted parameter estimate. It is
not an estimate for unnormalized parameter derivatives of an additional
oscillatory factor \(e^{i\phi(p)/h}\); that factor must be kept separate.
\(\square\)

## Q9. The region with no stationary point

Suppose \(\phi\) is smooth and real on an open neighbourhood of the
compact support of \(a\in C_c^\infty\), and
\(|\nabla\phi|\geq c>0\) on such a neighbourhood. Set
\(v=\nabla\phi/|\nabla\phi|^2\) there and
\(Ta=-\operatorname{div}(va)\). Then, for \(h>0\) and each integer \(N\geq0\),
\[
 \int e^{i\phi(x)/h}a(x)\,dx
 =(h/i)^N\int e^{i\phi(x)/h}T^Na(x)\,dx,
 \qquad
 \left|\int e^{i\phi/h}a\right|\leq h^N\|T^Na\|_1.
 \tag{NS}
\]
The right side is finite and independent of \(h\).

**Proof.** The identity
\(v\cdot\nabla e^{i\phi/h}=(i/h)e^{i\phi/h}\) follows by the chain
rule. Integrating it against \(a\) and using integration by parts in
each coordinate gives (NS) for \(N=1\). Each \(T^j a\) has support
inside \(\operatorname{supp}a\), because derivatives and multiplication
do not enlarge support. Thus the same argument applies repeatedly.
There are no boundary terms: one can choose the integration box outside
the compact support. The final bound follows from the absolute-value
bound on the integral and \(|e^{i\phi/h}|=1\). For a family with a
common compact support, a uniform lower bound on \(|\nabla\phi|\),
and bounded derivatives of \(\phi,a\) to the required finite order,
the same finite product-rule expansion bounds \(\|T^Na\|_1\)
uniformly. A parameter derivative of the oscillatory factor costs a
power of \(h^{-1}\); applying (NS) sufficiently many extra times
still proves decay to every order for each fixed derivative order.
\(\square\)

## E1. An exact model that checks the normalization and the error

Take \(n=1\), \(H=1\), \(a(x)=e^{-x^2/2}\). Q2 gives
\[
 I(h)=\sqrt{2\pi}\,(1-i/h)^{-1/2},\qquad
 \frac{I(h)}{\sqrt{2\pi h}\,e^{i\pi/4}}=(1+ih)^{-1/2}.
 \tag{EX}
\]
The roots in this equality have the branches specified in Q2: their
arguments add to \(-\pi/2+\arctan h\), which lies in
\((-\pi/2,0)\). Hence no sign ambiguity is hidden in the factorization.

Repeated integration by parts gives the Gaussian moments
\(\int\xi^{2N}e^{-\xi^2/2}d\xi=(2N-1)!!\sqrt{2\pi}\): the boundary
term in \(\int\xi^{2N-1}\xi e^{-\xi^2/2}\) vanishes and each step
reduces the exponent by two; the base case is Q2. For the derivatives of
the amplitude, use \(a'=-xa\). Its \((m-1)\)st derivative at zero,
for \(m\geq2\), gives \(a^{(m)}(0)=-(m-1)a^{(m-2)}(0)\).
Together with \(a(0)=1\), \(a'(0)=0\), induction gives
\(a^{(2k)}(0)=(-1)^k(2k)!/(2^kk!)\). Consequently Q7 reads
\[
 (1+ih)^{-1/2}
 =\sum_{k=0}^{N-1}(-1)^k\frac{\binom{2k}{k}}{4^k}(ih)^k+R_N(h),
 \qquad
 |R_N(h)|\leq\frac{\binom{2N}{N}}{4^N}h^N.
 \tag{ER}
\]
For example the first three terms are
\(1-ih/2-3h^2/8\). The bound holds for every \(h>0\); convergence of
an infinite Taylor series for large \(h\) is neither asserted nor used.

![Exact Gaussian model and the proved normalized remainder bound](figures/quadratic-remainder.png)

**Figure 1.** The left panel samples the real and imaginary parts of the exact
normalized integral (EX) and of \(1-ih/2-3h^2/8\). The right panel samples
\(|R_N(h)|/h^N\) for \(N=1,2,3\); the horizontal dashed lines are the
proved constants \(\binom{2N}{N}/4^N\) in (ER), not fitted estimates.
The horizontal variable is \(0.01\leq h\leq1\). The sampled curves
illustrate Q7 and E1; the proof of the bound is (T), (QI), and the moment
calculation above. Reproducible source: `figures/draw_quadratic_remainder.py`.
The human-source route for the Gaussian and quadratic calculation is
Guillemin–Sternberg, author draft, §§14.1–14.5; the displayed example and
figure are written for this reconstruction.

<span id="free-source-reading-and-remaining-work"></span>

## Mathematical sources and notation

- Victor Guillemin and Shlomo Sternberg, *Semi-classical analysis*,
  [author-posted draft dated 13 January 2010](https://math.mit.edu/~vwg/semiclassGuilleminSternberg.pdf),
  §§14.1–14.5, printed pp. 459–466, discuss Gaussian evaluation,
  regularization, the quadratic Fourier identity and the Taylor remainder.
  Section 13.14.3, pp. 455–457, gives the Morse lemma and its parameter
  variant, treated in the [Morse companion](parameter-morse-reduction.md).
  Section 14.8, p. 472, relates stationary phase to Fourier inversion.
- Richard Melrose, *Tempered distributions and the Fourier transform*,
  [author-posted Chapter 1](https://math.mit.edu/~rbm/iml/Chapter1.pdf),
  §1.7, printed pp. 21–23, gives the Fourier normalization, Schwartz
  estimates and inversion; see also the
  [MIT 18.157 course page](https://math.mit.edu/~rbm/18.157-F09/18.157-F09.html).
- Jiří Lebl, *Basic Analysis I–II*, version 6.3 (15 May 2026),
  [freely accessible author edition](https://www.jirka.org/ra/), supplies
  foundational calculus and compactness results. The
  human-proof references give exact
  statement and proof locators; the local prerequisite companions supply
  the additional finite-dimensional and integral arguments.

The transform convention is (F), with inverse factor \((2\pi)^{-n}\).
Q2 derives the complex Gaussian by differential identities along a path
in the right half-plane. Q4 proves inversion by a Gaussian approximate
identity before the quadratic identity uses it. Q6 specifies the
regularization, root phases and absolute majorants, and Q7 gives the
finite Taylor estimate in this convention. E1 checks the normalization
and remainder in an exact model. These arguments use the named
elementary and integral inputs in F0; they do not import a distribution
kernel theorem.

The source PDFs are external readings. Their authorship and reuse terms
remain with their respective works. The independently written programme
exposition, worked example and reproducible figure are CC0 to the extent
rights exist.