# Learning with singular measurements

*Introductory calculations and foundational reading routes by GPT-6.1 Sol (OpenAI). Advanced reading routes and programme integration by GPT-6 Astra (OpenAI). October 2026. Text: CC0.*

A distribution records what a localized smooth measurement sees. This course follows that idea through three questions: how a singular object acts on tests, how an operator transports those measurements, and how support or frequency information constrains the result. The following two calculations give a concrete starting point. The routes below then identify the proofs needed for each question.

## Start with an impulse and its derivative

Choose a nonnegative smooth function \(\rho\) supported in \([-1,1]\), with integral one and \(\rho(0)>0\). For \(\varepsilon>0\), its rescaling \(\rho_\varepsilon(x)=\varepsilon^{-1}\rho(x/\varepsilon)\) has height of order \(1/\varepsilon\). Its value at the origin tends to infinity. Nevertheless every smooth compactly supported measurement has a limit:

\[
\begin{gathered}
\int\rho_\varepsilon(x)\phi(x)\,dx\\
=\int\rho(t)\phi(\varepsilon t)\,dt,\\
\left|\int\rho_\varepsilon\phi-\phi(0)\right|\\
\le\varepsilon\|\phi'\|_\infty\int|t|\rho(t)\,dt.
\end{gathered}
\tag{G1}
\]

The equality is the substitution \(x=\varepsilon t\). The bound is the fundamental theorem of calculus between zero and \(\varepsilon t\), followed by integration against the nonnegative \(\rho\). Thus the measurements approach evaluation at zero, the distribution \(\delta_0\).

Integration by parts gives the derivative measurement and its sign:

\[
\begin{gathered}
\int\rho'_\varepsilon(x)\phi(x)\,dx\\
=-\int\rho(t)\phi'(\varepsilon t)\,dt,\\
\left|\int\rho'_\varepsilon\phi+\phi'(0)\right|\\
\le\varepsilon\|\phi''\|_\infty\int|t|\rho(t)\,dt.
\end{gathered}
\tag{G2}
\]

The limiting functional is \(\delta'_0(\phi)=-\phi'(0)\). The negative sign is part of the definition of a distributional derivative. Both errors are uniform over test families with the displayed derivative norm bounded. The support and bounded-family lemma in [*Distributions as kernels of continuous operators*](src/distributions-as-kernels.md), “What continuity means,” explains why this also gives strong distributional convergence on bounded test families. [*Order, positivity and distributional limits*](src/order-positivity-and-limits.md) develops the convergence and order arguments further.

These estimates explain what “order” measures: evaluation needs a zeroth derivative bound; its derivative needs a first derivative bound. They do not bound the height of an approximating function. [*Jets, supported distributions and local operators*](src/jets-supported-distributions-and-local-operators.md), Theorems 3.1, 3.3 and 3.4, identifies all finite-order distributions concentrated on a plane and proves how the orders of their transverse and tangential measurements combine.

## Watch an operator move a measurement

Consider \(T_h\phi(x)=\phi(x+h)\). Its kernel is concentrated on the graph \(y=x+h\). On a smooth compactly supported test \(H(x,y)\) of two real variables, define

\[
\begin{gathered}
K_h(H)=\int H(x,x+h)\,dx,\\
K_h(\psi\otimes\phi)\\
=\int\psi(x)\phi(x+h)\,dx.
\end{gathered}
\tag{G3}
\]

This is a distribution: for a fixed compact support of \(H\), its projection on the \(x\) axis lies in a bounded interval \(I\), and the pairing has absolute value at most \(|I|\|H\|_\infty\). A test whose support misses the graph gives zero. This particular kernel therefore represents translation and exhibits its support directly.

For \(h\ne0\), the difference quotient tends to
\(L(H)=\int\partial_yH(x,x)\,dx\). In fact Taylor's integral remainder gives

\[
\begin{gathered}
\left|\frac{K_h(H)-K_0(H)}h-L(H)\right|\\
\le \frac{|h|}{2}|I|\|\partial_y^2H\|_\infty.
\end{gathered}
\tag{G4}
\]

The same interval \(I\) contains the projection of the compact test support at every height in this calculation. Outside it, \(H(x,y)\) and its derivatives vanish for every \(y\). The factor \(1/2\) comes from \(\int_0^1(1-t)\,dt\). On product tests, \(L(\psi\otimes\phi)=\int\psi\phi'\). Hence the kernel of differentiation is \(L=-\partial_yK_0\): differentiating a distribution introduces a negative sign, while the operator differentiates its input with a positive sign.

The kernel theorem in [*Distributions as kernels of continuous operators*](src/distributions-as-kernels.md), Theorem 4.1, proves that every operator satisfying the stated continuity conditions has one unique kernel; the example above supplies a kernel directly. [*When a kernel is smooth*](src/when-a-kernel-is-smooth.md) identifies when such a distribution is a smooth function. [*Tensor products and parameter-dependent distributions*](src/tensor-products-and-parameters.md) supplies the complete product and parameter-pairing proofs. [*Jets, supported distributions and local operators*](src/jets-supported-distributions-and-local-operators.md), Theorem 4.1, turns concentration on the diagonal into a locally finite differential operator.

The full [integral Taylor proof](src/when-a-kernel-is-smooth.md#taylor-estimates-on-compact-neighborhoods) justifies the remainder estimate above, including either sign of the increment.

## Choose a route by the question

The lesson numbers identify stable files. Within a route, follow the linked prerequisite sections and the complete providers named by each lesson.

- **Can local measurements be joined or multiplied?** Begin with [*Local data and compatible products*](src/local-data-and-compatible-products.md) and [*Weak equations and classical functions*](src/weak-equations-and-classical-functions.md). The gluing proofs explain local definitions; the product conditions explain why two individually meaningful distributions cannot always be multiplied.
- **What does support geometry permit?** The normal-jet theorem leads to [*Compatible jets on closed sets*](src/compatible-jets-on-closed-sets.md), [*Paths control supported distributions*](src/paths-control-supported-distributions.md) and [*Locality forces continuity*](src/locality-forces-continuity.md). For a physical boundary, continue to [*Boundary flux and weak identities*](src/boundary-flux-and-weak-identities.md) and [*Lipschitz graphs and surface measures*](src/lipschitz-graphs-and-surface-measures.md). The normal direction and the sign of a boundary source are part of the mathematics.
- **How do sources propagate?** [*Convolution as addition of supports*](src/convolution-as-addition-of-supports.md) proves the proper-support condition before a convolution is defined. [*Causal integration of complex order*](src/causal-integration-of-complex-order.md) and [*Causal point sources and characteristic cones*](src/causal-point-sources-and-characteristic-cones.md) then connect support constraints to evolution kernels. [*Convex supports and convolution cancellation*](src/convex-supports-and-convolution-cancellation.md) studies which support information survives cancellation.
- **How does a complex boundary encode a singularity?** Start with [*Cauchy kernels and distributional boundary limits*](src/cauchy-kernels-and-boundary-limits.md), then [*Gluing holomorphic sides*](src/gluing-holomorphic-sides.md) and [*Holomorphic boundaries in convex cones*](src/holomorphic-boundaries-in-convex-cones.md). The finite-part, complex-power and homogeneous-extension lessons apply these boundary and jet proofs to explicit singularities. [Holomorphic boundaries and analytic directions](src/holomorphic-boundaries-and-analytic-directions.md) also treats local wedges with angularly varying polynomial bounds and the effect of regular remainders.
- **What can frequency data determine?** [*Tempered growth and spectral cutoffs*](src/tempered-growth-and-spectral-cutoffs.md) sets the global growth framework. The finite-spectrum, periodic-sampling and compact-spectrum lessons distinguish discrete frequencies, spectral gaps and support bounds. [*Fourier–Laplace slices and boundary poles*](src/fourier-laplace-slices-and-boundary-poles.md) recovers weighted inputs from analytic observations. The Gaussian, quadratic-transform and oscillatory lessons then examine how those observations change under weights, phases and differential operators.

- **How can a finite-order zero be divided uniformly?** After [*Zero hypersurfaces as curvature measures*](src/zero-hypersurfaces-as-curvature-measures.md), read [*Classical finite-order preparation and division*](src/classical-finite-order-preparation-and-division.md). It separates the holomorphic uniqueness theorem from smooth existence and follows coefficients through root collisions, derivative losses and actual input domains.

- **What remains after solving several smooth complex equations?** After [*Classical finite-order preparation and division*](src/classical-finite-order-preparation-and-division.md), read [*Smooth complex equations and flat remainders*](src/smooth-complex-equations-and-flat-remainders.md). Follow common neighborhoods, smooth complex graphs and remainders that belong to every ideal power.
- **Can a real finite-order zero become a polynomial by changing coordinates?** Continue with [*An oriented coordinate normal form for a real finite-order zero*](src/oriented-coordinate-normal-form.md). Its local flow keeps the function value and proves the distinguished coordinate's orientation directly.

- **Which derivative bounds describe a singular direction?** Read [*Derivative growth and finite-order localization*](src/derivative-growth-and-finite-order-localization.md), [*Cones and derivative-growth wavefronts*](src/cones-and-derivative-growth-wavefronts.md) and [*Analytic maps and derivative growth*](src/analytic-maps-and-derivative-growth.md). The proofs track the growing derivative order, compact cutoffs and changes of variables. The smooth-wavefront results also identify compact frequency projections and the exact conormal set of a smooth weighted density, including its zeros.
- **How do holomorphic functions recover directional singularities?** [*Holomorphic boundaries and analytic directions*](src/holomorphic-boundaries-and-analytic-directions.md) proves the forward cone bound. Continue through [*Holomorphic decompositions of tempered distributions*](src/holomorphic-decompositions-of-tempered-distributions.md), [*Continuing the spherical kernel*](src/continuing-the-spherical-kernel.md) and [*Boundary regularity and conic decomposition*](src/boundary-regularity-and-conic-decomposition.md). [*From wavefront cones to holomorphic wedges*](src/from-wavefront-cones-to-holomorphic-wedges.md) supplies the converse and states exactly where a regular remainder is local.
- **Can singular directions be prescribed or transformed?** [*Prescribing every singular direction*](src/prescribing-every-singular-direction.md) constructs distributions with an assigned closed conic wavefront. [*Fourier rotation and directions at infinity*](src/fourier-rotation-and-directions-at-infinity.md) treats homogeneous Fourier correspondence and origin fibers. [*Analytic pullback and cotangent directions*](src/analytic-pullback-and-cotangent-directions.md) and [*Keeping zero covectors in kernel operations*](src/keeping-zero-covectors-in-kernel-operations.md) state the actual hypotheses for substitution, products, integration and composition. The density formulas retain the full intersection angle factor and distinguish a weight vanishing at a point from one vanishing on the entire intersection. Compact packets also realize every smooth direction set with any finite differentiability, and real-valued realizations exist exactly for sign-symmetric sets. Tangential coefficients, analytic densities and transverse intersections have exact wavefront formulas, including both normal signs at zeros within the coefficient support.
- **What forces analytic uniqueness or characteristic propagation?** Start with [*Both analytic directions at support boundaries*](src/both-analytic-directions-at-support-boundaries.md) and [*Characteristic motion and closed-set barriers*](src/characteristic-motion-and-closed-set-barriers.md). [*Analytic equations and characteristic directions*](src/analytic-equations-and-characteristic-directions.md) connects differential estimates to uniqueness. The [half-space](src/characteristic-half-spaces-and-ramified-roots.md) and [convex-domain](src/characteristic-planes-and-convex-domain-uniqueness.md) lessons give exact geometric examples. [*Analytic parametrices and straight characteristics*](src/analytic-parametrices-and-straight-characteristics.md) proves propagation, while [*A whole characteristic line with smooth forcing*](src/a-whole-characteristic-line-with-smooth-forcing.md) constructs a global singular ray with smooth forcing. The parametrix lesson also proves a uniform bound for arbitrary imaginary perturbations, with explicit linear and nonlinear examples.
- **When does a reciprocal control analytic regularity?** [*Analytic regularity for convolution equations*](src/analytic-regularity-for-convolution-equations.md) keeps the reciprocal identity distributional and proves the estimate on proper support sums. [*Imaginary displacements and microhyperbolic cones*](src/imaginary-displacements-and-microhyperbolic-cones.md) starts from imaginary zero avoidance, proves the general cone theorem and distinguishes actual wavefront fibers from their optimal convex bounds. [*Exact waves on a single light ray*](src/exact-waves-on-a-single-light-ray.md) constructs a homogeneous wave with one covector sign and relates it to smooth Gaussian packets.

- **How can a point measure infinitely many derivatives?** [*Analytic tests and real carriers*](src/analytic-tests-and-real-carriers.md) proves the moment criterion, Gaussian approximation near arbitrary real compact sets, the extension to local holomorphic tests, and the precise inclusion of compact distributions. Its carrier operations lead to the Fourier–Laplace estimate and uniqueness. Regular simple arcs are minimal complex carriers, with a full polynomial-approximation proof and a regular looped-path counterexample.

- **Can an exterior harmonic field reveal an analytic source?** [*Recovering analytic data from harmonic fields*](src/recovering-analytic-data-from-harmonic-fields.md) proves the entire odd harmonic lift, the Poisson half-jump normalization and recovery by compact cutoff tests. Its converse identifies the entire harmonic ambiguity and gives the smallest real carrier from harmonic continuation.

- **How can analytic sources be divided without analytic cutoffs?** [*Localizing analytic sources without analytic cutoffs*](src/localizing-analytic-sources-without-analytic-cutoffs.md) splits a functional over arbitrary real compact carriers by correcting a smooth cutoff in its harmonic field. It proves the exact overlap ambiguity and completeness when differences eventually concentrate near a smaller carrier, including initially complex carriers and sequences with unbounded size.

- **How do analytic sources define local objects on a manifold?** [*Hyperfunctions from local analytic sources*](src/hyperfunctions-from-local-analytic-sources.md) constructs boundary quotients, proves restriction and gluing for arbitrary covers, and identifies compact sections with analytic functionals. It gives coordinate changes and atlas independence, then extends every section with support in the closure of its domain. Examples distinguish these extensions from distributional ones and describe their full boundary ambiguity.

- **Can a holomorphic function have a boundary even when it grows faster than every power?** [*Holomorphic boundary values without growth bounds*](src/holomorphic-boundary-values-without-growth-bounds.md) constructs that boundary as a hyperfunction, reads its analytic singular directions from a spherical kernel, and proves the positive-dual bound and its converse. It proves uniqueness and analytic joining of opposite sides. An essential singularity gives a boundary beyond distributions and an explicit infinite point-supported jump.

- **How do different holomorphic sides represent the same singular object?** [*Directional splitting and holomorphic compatibility*](src/directional-splitting-and-holomorphic-compatibility.md) separates hyperfunctions by any finite closed conic cover. It proves that the exact ambiguity consists of antisymmetric exchanges on cone intersections, represented by holomorphic functions on sums of approach cones. Worked examples distinguish the intrinsic hyperfunction from its noncanonical angular pieces.

- **When can a singular object be substituted into an analytic map?** [*Analytic substitution for hyperfunctions*](src/analytic-substitution-for-hyperfunctions.md) constructs noncharacteristic pullback, proves its precise covector bound and representation independence, and compares it with the existing distribution and coordinate definitions. It includes exact submersion equality, the inverse absolute Jacobian, and examples reaching beyond distributions.

- **When can singular objects be multiplied?** [*Tensor directions and admissible hyperfunction products*](src/tensor-directions-and-admissible-hyperfunction-products.md) retains every zero covector component, constructs products by diagonal restriction and proves their chain and Leibniz rules. It gives the joint condition for finite products and examples showing why individual or pairwise conditions can be insufficient.

- **What survives integration over a fiber?** [*Integrating hyperfunctions over proper fibers*](src/integrating-hyperfunctions-over-proper-fibers.md) constructs the proper integral, proves that only zero fiber covectors survive and checks its derivative, coordinate and iterated-integration rules. A parabolic fold shows exactly how an integrable singularity appears at a critical output value.

- **How can a kernel act on a singular input?** [*Singular inputs and zero covectors in hyperfunction kernels*](src/singular-inputs-and-zero-covectors-in-hyperfunction-kernels.md) gives the normal exclusion, output-axis terms and full relation for contracting two kernels. It constructs proper convolution and proves associativity using compact analytic representatives, with examples reaching beyond distributions.

- **What does the boundary of a hyperfunction support force?** [*Support normals force both hyperfunction directions*](src/support-normals-force-both-hyperfunction-directions.md) detects compact fibers through all analytic weights, using a uniform Baire estimate and Cauchy reconstruction. It proves both normal signs, rough and convex contact, and uniqueness from a nonempty open zero set, without growth bounds.

- **When does a contracted kernel give the successive actions?** [*Interchanging hyperfunction integration and analytic substitution*](src/interchanging-hyperfunction-integration-and-analytic-substitution.md) proves the noncharacteristic interchange theorem, the singular projection formula and finite kernel associativity. Fixed fiber boundaries, exact carrier comparisons and canceling exponential tails handle hyperfunctions of arbitrary growth.

- **How can analytic iteration survive a loss of derivatives?** [*Analytic Cauchy data on small polydiscs*](src/analytic-cauchy-data-on-small-polydiscs.md) pairs Cauchy derivative costs with weighted zero-data primitives. It proves quantitative local existence, an explicit iteration remainder, uniqueness of analytic germs and the thin rescaling that permits arbitrary analytic normal data.

- **How far does a local analytic Cauchy solution continue?** [*Continuing analytic Cauchy solutions through convex domains*](src/continuing-analytic-cauchy-solutions-through-convex-domains.md) follows noncharacteristic boundaries through expanding convex domains. It proves compatibility of local extensions, global existence, complex-cone solvability and continuation through the cone vertex when the forcing extends.

- **What does an analytic equation control for a hyperfunction?** [*Noncharacteristic equations for hyperfunctions*](src/noncharacteristic-equations-for-hyperfunctions.md) proves equality of solution and forcing singular directions off the characteristic set. It constructs solutions modulo a directionally regular error, explains their quotient uniqueness, defines traces of solutions already given across a noncharacteristic surface and proves Holmgren uniqueness.

- **How does a solution on one side acquire Cauchy data?** [*Supported division and one-sided Cauchy data*](src/supported-division-and-one-sided-cauchy-data.md) divides arbitrary sources supported on a noncharacteristic surface, selects a unique extension by its finite boundary source, and recovers the normal jets through an exact triangular matrix. It proves agreement with classical, extendible distributional and two-sided hyperfunction traces, including coordinate changes.

- **How does a Gaussian detect an analytic singularity?** [*Gaussian windows and analytic directions*](src/gaussian-windows-and-analytic-directions.md) proves the exact exponential criterion for arbitrary compact hyperfunctions. It connects carrier displacement, a corrected inversion formula, recovery on the imaginary sphere and local representative independence.

- **How does physical support constrain a singular fiber?** [*Tangent support and analytic directions*](src/tangent-support-and-analytic-directions.md) places both signs of every fiber normal in the tangent cone of physical support. It treats affine sections through closure after projection, propagates regularity through convex fiber regions, and proves analytic boundary regularity at every normal frequency. An explicit integral solution shows why smooth Cauchy data do not give that conclusion.

- **Which entire functions represent a compact real source?** [*Entire growth and convex real carriers*](src/entire-growth-and-convex-real-carriers.md) constructs the Fourier–Laplace inverse for every compact convex real carrier using rotated Laplace rays and small Cauchy contours. It includes infinite point derivatives and explains exactly which information a nonconvex set loses in its support function.

- **Can regular germs have global representatives?** [*Global regular differences*](src/global-regular-differences.md) constructs a global function of the same derivative-growth class up to an analytic difference, using Gaussian centers in a small ball. It proves the global wavefront intersection formula and simultaneous representation at finitely many points.

- **How can smooth data cross a boundary or descend through a symmetry?** [*Smooth extension and symmetry coordinates*](src/smooth-extension-and-symmetry-coordinates.md) constructs one extension operator preserving every finite derivative order, then resolves coordinate parity and exchange symmetry. It also gives smooth positive majorants with a flat boundary.

- **How can oscillation permit a singular function to cross an endpoint?** [*Oscillation and extension order*](src/oscillation-and-extension-order.md) gives the exact finite-order threshold, excludes equality, and constructs a real phase for an arbitrary continuously differentiable amplitude.

- **What survives a concentrating or rapidly oscillating limit?** [*Concentration and oscillatory limits*](src/concentration-and-oscillatory-limits.md) computes point masses, periodic limits and a radial center contribution, resolves two boundary parameters, and proves when weak convergence becomes locally uniform.

- **How do crossings and moving poles affect products?** [*Products, crossings and singular limits*](src/products-crossings-and-singular-limits.md) proves product principal values, quartic subtractions, the logarithmic axis-crossing limit and the exact separation law for overlapping kernels.

- **When does a distribution extend across an endpoint?** [*Primitives and endpoint extension*](src/primitives-and-endpoint-extension.md) proves interval primitive estimates, the finite-measure extension criterion, explicit weighted subtractions and the exact order allowed by an exponential phase.

- **What remains when singular poles cancel?** [*Reciprocal scaling and cancellation at a pole*](src/reciprocal-scaling-and-pole-cancellation.md) proves reciprocal-tail compression, convergent finite-difference formulas for every derivative, and the full smooth-coefficient boundary limits with their point terms.

- **When can a smooth function divide a distribution?** [*Jumps, point jets and smooth division*](src/jumps-jets-and-smooth-division.md) calculates reflected endpoint terms, classifies division by smooth functions on intervals, and proves the obstruction at a flat zero.

- **What happens when a differential coefficient vanishes?** [*Equations across degenerate points*](src/equations-across-degenerate-points.md) classifies flat and growing branches, point-forced solutions, and the exact piecewise constants and point terms at locally finite zeros.

- **How do multipliers constrain singular distributions?** [*Normal jets and parabolic poles*](src/normal-jets-and-parabolic-poles.md) describes finite normal jets with unrestricted tangential coefficients, solves simultaneous polynomial constraints, and proves a directional uniqueness criterion and exact parabolic pole orders.

- **When do singular families converge to points or curves?** [*Concentration and logarithmic boundaries*](src/concentration-and-logarithmic-boundaries.md) proves the full Hessian criterion for planar concentration, a logarithmic condition for cut boundary values, and convergence of moving logarithmic sources to circular arclength.

- **How do boundaries determine derivatives and division?** [*Boundary derivatives, division and moving jumps*](src/boundary-derivatives-division-and-moving-jumps.md) proves exact graph derivative orders, the complete criterion for division by a quadratic smooth multiplier, all interfaces in a weak-solution family, and logarithmic branch jumps.

- **How do cancellation and damping combine?** [*Weighted cancellation for complex phases*](src/weighted-cancellation-for-complex-phases.md) proves a finite-regularity integral bound with precise gradient and imaginary-part weights, a nonnegative derivative estimate, and the uniform passage through degenerate points.

- **Which finite jets determine a stationary integral?** [*Finite stationary jets and quadratic exchange*](src/finite-stationary-jets-and-quadratic-exchange.md) derives the complex quadratic remainder, the full finite-derivative stationary expansion and the exact algebra of nilpotent quadratic exchange, with its sharp order loss.

- **What does a Fourier pairing do to a nonlinear phase?** [*Paired variables and stationary parameters*](src/paired-variables-and-stationary-parameters.md) follows real critical points through parameters and proves the paired-phase expansion with its exact derivative count, factorials and sharp half-order mechanism.

- **How does a small boundary value control the interior?** [*Boundary smallness and harmonic weights*](src/boundary-smallness-and-harmonic-weights.md) constructs a positive harmonic weight from disk comparison, proves quantitative boundary propagation on general upper domains, and recovers full decay orders.

- **What replaces a missing real critical point?** [*Virtual critical points and explicit coefficients*](src/virtual-critical-points-and-explicit-coefficients.md) derives the explicit Gaussian coefficients from a smooth complex critical graph, with its positive critical value, oriented Jacobian and independence of representatives.

- **How does curvature control lattice counts?** [*Curvature, two boundary points and lattice counts*](src/curvature-two-boundary-points-and-lattice-counts.md) computes the two normal phases of a convex body, proves a smooth Fourier summation identity and balances the two smoothing errors.

- **How does a boundary change stationary phase?** [*Half-space phases and the Fresnel transition*](src/half-space-phases-and-the-fresnel-transition.md) proves the three local boundary regimes, their complete coefficients and remainders, and the Fresnel transition between an exterior and an interior critical point.

- **Which powers survive a degenerate critical point?** [*Finite-order phases and missing powers*](src/finite-order-phases-and-missing-powers.md) derives every finite-order coefficient with its exact remainder, identifies the missing powers, and gives the two canonical Airy profiles through a cubic collision.

- **How do two critical points merge?** [*Cubic collisions and uniform amplitudes*](src/cubic-collisions-and-uniform-amplitudes.md) constructs smooth polynomial coordinates, expands every compact amplitude in two Airy profiles, and treats one cubic direction together with transverse quadratic directions.

- **When does cancellation define a distribution?** [*Complex phases and oscillatory distributions*](src/complex-phases-and-oscillatory-distributions.md) constructs complex-phase distributions across the full symbol range, locates their stationary covector directions, and determines the wave kernel and both causal halves. Quadratic damping gives an exact example with one singular covector ray.

- **How do differences measure fractional smoothness?** [*Fractional energy and sharp Fourier embeddings*](src/fractional-energy-and-sharp-fourier-embeddings.md) proves the exact difference-norm identity, its two limiting constants, compact Hölder inclusions and the sharp weighted Fourier thresholds.

- **Which derivative estimates survive a rough symbol?** [*Annular multipliers and elliptic endpoints*](src/annular-multipliers-and-elliptic-endpoints.md) proves the averaged multiplier theorem, Hölder bounds and interior elliptic estimates, and constructs continuous forcing with unbounded highest derivatives.

- **Which directions survive at a quadratic vertex?** [*Quadratic cones, point sources and vertex directions*](src/quadratic-cones-point-sources-and-vertex-directions.md) derives the determinant phases, one-sided source coefficients and exact conormal and vertex wavefronts, including the logarithmic case in two dimensions. General moving-map limits retain the same boundary distribution at real submersions; examples distinguish flat heights, simple level zeros and critical concentration. The quadratic and causal classifications hold in smooth, analytic and every stated derivative-growth class, including curved analytic coordinates and full vertex fibers.

- **When does one compact cutoff detect regularity at every order?** [*Summable smoothing and fixed Fourier tests*](src/summable-smoothing-and-fixed-fourier-tests.md) constructs quantitative cutoffs in arbitrary norms, proves the compact-cutoff equivalence for nonquasianalytic classes and gives the fixed Fourier criterion with Gevrey examples.

- **When do derivative bounds determine a function from its infinite jet?** [*Derivative budgets and unique jets*](src/derivative-budgets-and-unique-jets.md) proves the full Denjoy–Carleman criterion, including irregular weights, the logarithmic borderline and the exact numerical condition for compact Fourier tests.

- **Equations in Banach spaces.** [*Differentiation and local inverses in Banach spaces*](src/differentiation-and-local-inverses-in-banach-spaces.md) proves two-sided and one-sided inverse theorems, dense-direction differentiation and Taylor formulas with eight complete exercises.

- **The geometry of a norm.** [*Ellipsoids, contact points and sharp norm comparison*](src/ellipsoids-contact-points-and-sharp-norm-comparison.md) proves the symmetric ellipsoid theorem, uniqueness, dual estimates and sharp examples, with six complete exercises.

- **Local measurements and separation.** [*Partitions for moving norms and smooth separation*](src/partitions-for-moving-norms-and-smooth-separation.md) constructs locally finite covers and partitions in arbitrary norms, controls fixed and varying derivative budgets, and separates closed sets, with eight complete exercises.

- **Prescribing boundary derivatives.** [*Smoothing scales and prescribed normal jets*](src/smoothing-scales-and-prescribed-normal-jets.md) treats both signs of a smoothing scale, finite jets with exact local regularity, and infinite smooth jets with fixed support or local bounds, with eight complete exercises.

- **Localizing and averaging functions.** [*Local data and compatible products*](src/local-data-and-compatible-products.md#one-compact-profile-for-every-lattice-translate) constructs a lattice partition in any prescribed neighborhood of its fundamental cube. [*Convolution as addition of supports*](src/convolution-as-addition-of-supports.md#norm-convergence-of-varying-averaging-kernels) gives finite-regularity derivative allocation, uniform and finite-p norm approximation, and almost-everywhere determination by tests.

- **Averaging and analytic continuation.** [*Summable smoothing and fixed Fourier tests*](src/summable-smoothing-and-fixed-fourier-tests.md#one-sided-averages-and-their-full-derivative-bounds) gives the full one-sided convolution bounds and local decay when the width sum diverges. [*Derivative budgets and unique jets*](src/derivative-budgets-and-unique-jets.md#factorial-bounds-and-complex-neighborhoods) proves holomorphic extension from factorial derivative estimates, including both interval endpoints and compatible extensions in several variables.

- **Measuring complex data.** [*Order, positivity and distributional limits*](src/order-positivity-and-limits.md#distributions-are-locally-finite-derivatives-of-measures) identifies total variation with the exact norm on smooth compact tests and on continuous functions vanishing at infinity. It also characterizes the order of every locally finite family of discrete jets and the dual of a summable array of continuous-function spaces.

- **Concentration at points and endpoints.** [*Concentration and oscillatory limits*](src/concentration-and-oscillatory-limits.md#compact-profiles-and-their-surviving-moments) gives the full compact-profile moment expansion through both scale signs, the one-sided oscillatory endpoint expansion, and the exact point mass for every monomial phase degree and frequency sign, with uniform remainder bounds and twelve solved exercises.

- **Compact components and support norms.** [*Compatible jets on closed sets*](src/compatible-jets-on-closed-sets.md#infinitely-many-components-defeat-every-finite-jet-bound) proves the full finite-jet extension theorem, a uniform criterion for omitting Taylor remainders, and an order-one obstruction on every compact set with infinitely many connected components. An interval model and nine solved exercises explain why even arbitrarily many pointwise derivatives may fail to control a supported distribution.

- **Continuous primitives at a boundary.** [*Gluing holomorphic sides*](src/gluing-holomorphic-sides.md#one-more-primitive-has-a-continuous-trace) constructs continuous traces from integrable derivative bounds, proves quantitative convergence of distributional measurements, and derives the signed pole jump from two continuous primitives. Thirteen solved exercises include sharp Hölder bounds and a bounded holomorphic function with no pointwise boundary limit.

- **Mixed Gaussian moments.** [*Point sources and complex Gaussian kernels*](src/point-sources-and-complex-gaussian-kernels.md#cross-terms-and-gaussian-moments) evaluates polynomial Gaussian tests for complex symmetric matrices, distinguishes total parity from coordinate parity, and proves concentration by Gaussian Taylor subtraction at the finite derivative count. Nine solved exercises include second and fourth mixed moments and a purely oscillatory cross term.

- **Finite measurements and zero measures.** [*Convolution as addition of supports*](src/convolution-as-addition-of-supports.md#finite-measurements-and-the-weak-topology) identifies the continuous weak dual and gives exact smooth interpolation of finitely many tests, with seventeen solved exercises. [*Compactness and complex-line positivity*](src/compactness-and-complex-line-positivity.md#equal-weights-and-polynomial-zero-measures) constructs integer-weighted atomic approximations to positive Radon measures and realizes them as normalized polynomial-zero measures, with ten solved exercises.

- **Families of compact supports.** [*Convex supports and convolution cancellation*](src/convex-supports-and-convolution-cancellation.md#families-of-supports-and-finite-approximations) characterizes uniformly bounded families by their support functions and proves uniform finite approximation of the convex envelope. A compact zero-mass kernel confines convolution with a domain indicator to its boundary neighborhood. Twelve solved exercises include escaping negative supports, the sharp polygon error and two signed boundary intervals.

- **Finite order and continuous primitives.** [*Fundamental solutions, continuation and approximation*](src/fundamental-solutions-continuation-and-approximation.md#finite-families-and-prescribed-coefficient-regularity) represents every distribution by locally finite derivatives of coefficients with any prescribed finite regularity. A finite coefficient family exists exactly for distributions of globally finite order. Eleven solved exercises include a periodic source without compact support, unbounded local jet orders and the sharp derivative cost of a point source.

- **Convolution endpoints and cancellation.** [*Convolution estimates and weak-gradient embeddings*](src/convolution-estimates-and-weak-gradient-embeddings.md#endpoint-norms-and-the-reflected-kernel) proves the exact endpoint kernel norms and both maximal-function and cancellation proofs of fractional integration. Thirteen solved exercises include concentrated unit mass, sharp endpoint obstructions and countable mean-zero decompositions.

- **Regularization and kernel limits.** [*Distributions as kernels*](src/distributions-as-kernels.md#recovering-a-kernel-by-decreasing-its-smoothing-scale) gives Fourier-series and scale-Taylor constructions of the distribution kernel, with explicit finite-order bounds. Seven solved exercises include logarithmic convergence, differentiated point evaluations and the second-moment error.

- **Finite-order pullback and openness.** [*Regular level sets and transverse convolution*](src/regular-level-sets-and-transverse-convolution.md#pullback-with-a-fixed-derivative-cost) treats finite differentiability, the openness obstruction and all differential layers. Twenty-eight solved exercises include signed target cancellation, an open critical map and a weighted parabolic layer.

- **Wave kernels and initial data.** [*Wave powers and complex potentials*](src/wave-powers-and-complex-potentials.md#time-slices-and-the-full-smooth-cauchy-problem) supplies compact spatial slices, all initial jets, half-space uniqueness and the full smooth Cauchy formula. Eleven solved exercises include forcing normalization and finite propagation.

- **Scalar distributions and densities.** [*Regular level sets and transverse convolution*](src/regular-level-sets-and-transverse-convolution.md#distributions-and-densities-in-charts) constructs distributions on manifolds from an atlas and proves both test-space dualities. Twenty-eight solved exercises include nonlinear chart changes, orientation reversal and positive background densities.

- **Tangent vectors and cotangent geometry.** [*Regular level sets and transverse convolution*](src/regular-level-sets-and-transverse-convolution.md#tangent-vectors-covectors-and-bundle-gluing) constructs smooth bundles and tangent vectors from first jets, proves the canonical one-form and full conormal bundle, and includes twenty-eight solved exercises. The problems distinguish tangent and covector transport and exhibit a nontrivial real line bundle.

- **Differential forms and boundary orientation.** [*Regular level sets and transverse convolution*](src/regular-level-sets-and-transverse-convolution.md#exterior-differentiation-and-radial-primitives) proves exterior differentiation, radial primitives and Stokes’ formula, including the support conditions for submanifold distributions. Its twenty-eight solved exercises include polynomial primitives, annular periods and omitted endpoints.

- **Principal symbols and Hamiltonian motion.** [*Regular level sets and transverse convolution*](src/regular-level-sets-and-transverse-convolution.md#principal-symbols-between-vector-bundles) proves the invariant leading polynomial, the local and global potential criteria for Lagrangian graphs, and Hamiltonian conservation and symplectic preservation. Its twenty-eight solved exercises include frame changes, a nonexact characteristic graph and both conormal signs of the wave equation.

- **Flowouts and Cauchy data.** [*Regular level sets and transverse convolution*](src/regular-level-sets-and-transverse-convolution.md#embedded-flowouts-along-an-entire-initial-manifold) constructs embedded Hamiltonian flowouts, solves the noncharacteristic Hamilton–Jacobi Cauchy problem and treats two compatible equations without assuming commuting flows. Its twenty-eight solved exercises include a caustic, a repeated-bracket obstruction and an explicit compatible pair.

- **Characters and Fourier rigidity.** [*Tempered growth and spectral cutoffs*](src/tempered-growth-and-spectral-cutoffs.md#complete-test-spaces-and-exponential-characters) proves Schwartz completeness, classifies continuous characters and shows that every linear Schwartz operator commuting with all coordinates and derivatives is scalar. Its fourteen solved exercises include the tempered-character criterion and the uniqueness of the Fourier intertwiner.

- **Fourier inversion and integrable functions.** [*Tempered growth and spectral cutoffs*](src/tempered-growth-and-spectral-cutoffs.md#two-semicircles-and-the-inversion-constant) gives the two contour signs, inversion with arbitrary complex Schwartz cutoffs, almost-everywhere inversion for integrable transforms, weighted tempered measures and both dual topologies. Worked problems include a complex cutoff and a singular function below the first integrability endpoint.

- **Interpolation and Fourier integrability.** [*Convolution estimates and weak-gradient embeddings*](src/convolution-estimates-and-weak-gradient-embeddings.md#interpolating-two-strong-operator-bounds) proves interpolation between two strong operator bounds on arbitrary measure spaces, including every infinite exponent. [*Fourier–Laplace slices and boundary poles*](src/fourier-laplace-slices-and-boundary-poles.md#integrability-between-the-fourier-endpoints) identifies the Hausdorff–Young map, its dual Haar measure, its distributional meaning and its exact normalization. Six solved problems explore equality, approximation, threshold splitting and scaling.

- **Compact factors and homogeneous Fourier transforms.** [*Compact factors and homogeneous equations*](src/compact-factors-and-homogeneous-equations.md#compact-distributions-acting-on-schwartz-functions) proves compact-factor convolution on tempered distributions and the exact coordinate laws, then treats homogeneity and smoothness away from the origin. [*Complex powers at a boundary*](src/complex-powers-at-a-boundary.md#the-damped-euler-integral-at-every-parameter) gives both ray orientations for the Euler integral and the full finite-part Fourier table. Six solved problems separate global and punctured hypotheses and determine the phases explicitly.

- **Residues and elliptic point sources.** [*Annular multipliers and elliptic endpoints*](src/annular-multipliers-and-elliptic-endpoints.md#polynomial-logarithms-from-radial-residues) derives the exact logarithmic coefficient of every homogeneous elliptic fundamental solution, its obstruction to homogeneity and its full polynomial freedom. The smooth parametrix has an entire error term. Six worked problems compare anisotropic and complex kernels, odd dimensions, logarithmic length scales and cutoff choices.

- **Analytic elliptic kernels.** [*Annular multipliers and elliptic endpoints*](src/annular-multipliers-and-elliptic-endpoints.md#complex-cones-and-analytic-kernels) constructs a holomorphic parametrix kernel on a conic neighborhood of the punctured real space. Its homogeneous remainder has analytic derivative bounds at every radius. Six solved problems examine contour signs, sharp complex-pole barriers, radial sources, disconnected cones and compact forcing.

- **Polynomial zeros and smooth regularity.** [*Annular multipliers and elliptic endpoints*](src/annular-multipliers-and-elliptic-endpoints.md#polynomial-zeros-and-smooth-regularity) relates real-frequency derivatives to distance from complex zeros and gives the qualitative criterion for hypoellipticity. Six solved problems examine multiplicity, higher-order heat operators, exact nearest zeros, failure of analytic regularity and lower-order perturbations.

- **Quadratic Fourier transforms.** [*Quadratic cones, point sources and vertex directions*](src/quadratic-cones-point-sources-and-vertex-directions.md#fourier-transforms-on-the-dual-quadratic-cone) proves both boundary transforms with their signature phases, principal values and dual-cone measures through the vertex. Six solved problems treat anisotropy, definite and split signatures, fundamental solutions and exact Gaussian mass.

Every available lesson contains its own graded problems and complete solutions. The course scope describes the lessons collected in this reader.

## Sources and contributions

The course uses independently written proofs and teaching organization, drawing on the works identified in each lesson. These include Hörmander's second-edition reprint, DOI 10.1007/978-3-642-61497-2, and the named free sources. Laurent Schwartz's distribution and kernel theory is a foundational contribution. Individual bibliographies credit the actual mathematical inputs; source books do not supply an exercise sequence or replace the proofs given here.

The lesson editions and prerequisite readings identify their authors and component licences. Complete earlier programme proofs are linked at the point of use with exact lesson and section locators. Source credit identifies the mathematical contributions; an internal proof link identifies the argument a reader needs. Independently authored text in this guide is CC0.