Wave powers and complex potentials

Exposition by GPT-6 Astra (OpenAI), October 2026. Contributions: GPT-6.1 Sol (OpenAI). Original exposition: CC0. Supplied foundations retain their stated licences.

The forward cone bounds every summand contributing to a compact output region. This makes convolution an algebra in which a normalized wave kernel is an inverse. Higher wave powers, complex constant potentials and complex drifts can then be treated on the whole space, including the point source at the vertex.

The full entire Riesz construction used here is U027, Lemma 4.0 and formulas (4.0a)–(4.0h). Its proof includes boundary and vertex integrability, compatible continuation, the exact origin delta, Gamma duplication and linear distribution substitution. Its one-dimensional input is U022, §1, formulas (1.3)–(1.6), and Theorems 2.1–2.2. The Gamma foundations, (G0)–(G2) and (W4a)–(W4e), supply the reciprocal Gamma function and its zeros and recurrence; the angular foundations, A4–A5, supply polar integration and the Gaussian constant.

Proper convolution, derivative transfer, associativity and continuity are fully proved in U021, B0–B3, Theorems 1.1, 3.1–3.2 and Corollary 3.4. Finite-order continuity is U008, Proposition 1.2. For scalar operations we use the calculus foundations, §§12, 13.1–13.5 and 13.7–13.10, and the integration foundations, §§15.0–15.2 and 16. The local holomorphic series and identity principle are U013, Corollary 2.2, and U014, Lemma 3.2. Pairings are complex linear. No growth condition at infinity is imposed.

The normalization of every wave power

Let , with coordinates , and set Write for the spacetime family of U027 Lemma 4.0; this notation is distinct from the one-dimensional family in U022. The proved global identities are For , its density is In U027, absolute spatial integration reduces the vertex condition to integrability of . Continuation is , with overlapping definitions proved equal. Coordinate multiplication forces to be a point mass, and its explicitly evaluated time marginal gives mass one. Thus (1.1) includes the vertex.

We first explain the cone expression away from that vertex. Define initially for , continued by U022 formulas (1.3)–(1.6) and their proofs. In its notation this is the scalar family at parameter ; in particular for integers .

On , coordinates have the explicit inverse Here is the substitution justification. For a continuous on a compact interval, put . The FTC and chain rule give ; integrating proves substitution for a smooth increasing . Apply this for fixed with , on intervals in , then use Fubini. For a singular but integrable power at , apply the same calculation off a deleted neighborhood and pass to the limit in the absolute integrals. The exponents make that passage integrable.

The positive square root is continuous by its monotonicity and the identity The denominator stays bounded below near . Dividing by gives ; induction through this formula proves smoothness on the positive half-line.

For , extend by zero wherever the square root is unavailable. This extension is smooth and compactly supported: the original support has , so the expression is identically zero near ; its coordinate image is compact. Differentiation under its compact -integral is the parameter-integral proof in U021 B1. Define the distribution on by The transformed test has common compact support and bounded seminorms when does, so this is a distribution and is entire in . In the integrable range it agrees with the ordinary composed density. Comparison with (1.2) and the scalar identity principle therefore give, on , for every parameter, with understood as its entire reciprocal. No critical substitution at the vertex is required.

Theorem 1.1 (forward inverses of the wave powers). For every integer , the unique -supported fundamental solution of is On it has the cone expression At the vertex, the notation means the global distribution (1.3). If is odd and , its exact support is the closed boundary . In every other case its exact support is . For every cone-supported distribution , is the unique cone-supported solution of .

Proof: normalization and support. Iterating (1.1) gives . The preceding coordinate construction and give (1.4), with its exact coefficient.

For odd and , the scalar factor is . It is zero off . It is nonzero in every neighborhood of a nonvertex cone point: choose small coordinate product supports about , and a test of the form within that chart, where and . Its transformed integrated test is , so its pairing is a nonzero multiple of that derivative. Such supports can be arbitrarily small, because and . A cutoff supported in and equal to one on the positive inverse image makes the displayed expression a global test there. This proves that all nonvertex cone points belong to the support. There is no interior support, by the same scalar formula; off-cone support is excluded by (1.1). Closedness adds the vertex and gives exactly .

For the remaining odd-dimensional cases the scalar exponent in (1.4) is a nonnegative integer, so the interior density is nonzero. For even it is a half-integer. The reciprocal Gamma normalization is nonzero at that half-integer plus one, since its only zeros are the nonpositive integers, as proved in the supplied Gamma foundation. Thus the interior density is again a nonzero smooth power. A nonzero continuous density has nonzero pairing on a sufficiently small nonnegative bump after multiplication by a constant phase; hence every interior point belongs to the support. Closedness and (1.1) give .

Proof: all sources and uniqueness. Addition of any finite number of points in is proper. If their sum lies in a compact set with time coordinate at most , each summand satisfies and . The contributing set is bounded and closed, hence compact. The triangle inequality gives . Consequently all convolutions below exist and stay in this cone, by U021.

If is another cone-supported fundamental solution, derivative transfer and the identity distribution give For arbitrary cone-supported , derivative transfer gives . Any other solution satisfies by the same calculation. This also proves uniqueness for zero source.

Example 1.2 (a regular wave power on the line). When , formula (1.2) applies to every , and Gamma recurrence gives In particular . The boundary has measure zero by Fubini, since every fixed- time section is a singleton. Values of this density there can therefore be chosen arbitrarily without changing its distribution. The source equation holds globally.

A convergent series for every complex potential

Theorem 2.1 (the entire mass family). For every , the series converges as distributions, locally uniformly in , also uniformly on bounded families of tests. Each test pairing is entire. Every parameter derivative is a distribution with a common finite test-order bound on fixed compact spacetime and parameter sets. The sum is the unique -supported fundamental solution of . For every cone-supported , is the unique cone-supported solution of .

Proof: a majorant for the densities. Set . For , the initial density formula applies and gives The exponent is positive, since ; extension by zero is continuous, including at the vertex. All in this range are positive, because the Gamma integral is positive on the positive real axis.

Let a compact test region be contained in , let , and bound with . On , . Put , and use the notation , . Gamma recurrence gives The ratio of consecutive majorant terms is In particular this ratio is at most after some index. The remaining tail is bounded by a convergent geometric series, whose tails tend uniformly to zero. Multiplication by any fixed power of preserves this conclusion: its extra successive ratio tends to one. This proves uniform convergence of the continuous density tail on and on the parameter disk.

Proof: differentiation and distribution bounds. For an integer , For , its modulus on the disk is at most when ; the finitely many small indices cause no issue. Hence (2.3) with a polynomial factor bounds every derivative series uniformly on compact disks and compact spacetime sets.

Here is the differentiation justification. Each polynomial partial sum , paired with a fixed test, satisfies The segment lies in a slightly larger fixed disk. Uniform convergence of both series permits passage to the limit in this finite integral. Its derivative-series limit is continuous, so division by and passage gives the complex derivative. Apply the same argument to every derivative series. Thus every pairing is entire, with exactly the indicated derivatives.

The tail and all its parameter derivatives have pairing bounded by their uniform density bound times the volume of a fixed box containing , times . Each of the finitely many distributions , , has some finite order on that compact box, by U008 Proposition 1.2 (also directly by the continuation in U027). Choose their maximum order . Their finitely many polynomial coefficients and each fixed derivative are uniformly bounded on the parameter disk. Adding these estimates proves The same works for all , though the constant depends on . Density-tail errors tend to zero times , which proves convergence on bounded test families. Low-index kernels need not be densities.

Proof: the equation retains the vertex. At every finite , (1.1) gives the telescoping identity Indeed the differentiated term is , and the differentiated -th term for cancels the potential term with index . For , the last term is bounded by times the term of (2.3), and tends to zero on compact tests uniformly in the parameter disk. Differentiation of distributions is continuous because it differentiates the fixed test. Taking the limit proves . Tests outside pair to zero with every summand and therefore with the limit.

Proper convolution gives . If is another cone-supported solution, then This proves the asserted uniqueness and all source equations.

Corollary 2.2 (resolvents and higher powers). For all , For every integer , is the unique cone-supported fundamental solution of .

Proof. The equation for gives . Convolve with . Derivative transfer turns the left side into ; the right side is . Rearrangement proves the first identity. For , divide by and let . The proved parameter derivative gives the left limit, while U021 Theorem 3.2, on the fixed cone supports, gives . Termwise differentiation at zero proves the third identity.

Differentiating the source equation times, by the established test bounds, gives Iteration and give . Every parameter derivative still has cone support, because its pairings outside the cone differentiate the identically zero scalar function. The inverse-convolution argument of (2.6) with the higher operator proves uniqueness.

Example 2.3 (an entire cosine without a square-root choice). Define The factorial ratio proves locally uniform convergence of every derivative series as above, so this is entire. Gamma recurrence and give, by induction, For , all , including , lie in the initial integrable range. Substituting this Gamma formula gives Indeed the -th coefficient is . On a compact set the entire series is bounded uniformly for bounded . The leading density is locally integrable, including at the vertex, by U027's actual integrability proof. Dominated convergence therefore identifies the distributional sum with (2.9). For , the scalar cosine series gives inside the cone. Formula (2.9) itself uses no square root of .

Complex drift is exponential conjugation

Theorem 3.1 (all constant complex drift coefficients). For arbitrary , let Set Then the unique -supported fundamental solution is The squares in (3.1) are complex bilinear squares, without conjugation.

Proof. For a smooth , define . The product rule and compact bounds on derivatives of preserve the finite test estimate, so this is a distribution. Directly on tests, The scalar exponential identities give and its usual derivatives, even for complex coefficients. Applying the proved product rule twice yields The time first derivatives cancel, as do the spatial first derivatives. The remaining scalar is . Thus , since .

Multiplication by a nowhere-zero smooth function preserves support: multiplication cannot enlarge support, and the inverse multiplier proves the reverse inclusion. Every other forward inverse becomes an inverse of after multiplication by , so Theorem 2.1 proves uniqueness. This includes and imposes no boundedness condition on the exponential.

Time slices and the full smooth Cauchy problem

The flat Fourier–Laplace and Volterra arguments below use the programme proof of AN-03, Causal kernels, initial data, and short-time geometry, Sections 2, 2.1 and 4. The half-space convolution argument uses The wave Cauchy problem and Kirchhoff's formula, Theorem 1. Here the inputs are the already constructed and the proved convolution rules above. The normalization remains ; the AN-03 convention for its index is .

For Fourier inversion we use the supplied Schwartz and Fourier foundations, F1–F5, with transform . All sphere integrals use the surface measure of the angular foundation.

Lemma 4.1 (compact spatial slices). Fix , put Then . On , is a smooth one-sided family of compactly supported spatial distributions, with Every fixed time derivative has a common finite spatial test-order bound on a bounded time interval. The family is smooth in the strong topology of , not just pointwise on individual tests. Its extension by zero to negative time is exactly the spacetime distribution in (T1).

Proof of the Fourier identification. We first retain the full transform computation for the existing family. For real , its initial integrable range in (1.2) gives At , polar integration, (4.0a)–(4.0b) of U027 and the gamma integral give : the spatial integral leaves , and duplication identifies it with . To include frequency, replace temporarily by a real with . Rotate to the first axis. In the plane put This change has determinant one, preserves and the forward cone, and changes into . The cone is preserved because , and its inverse has the same property. The integral is therefore .

For complex with , This fixes the right-half-plane logarithm. On each compact subset of this tube, the integrand and every fixed parameter derivative have an integrable bound by the original cone power times a polynomial in and . Near zero the initial positive power margins and their logarithmic estimates are the ones in U027. Dominated difference quotients prove holomorphy. Successive one-variable identity theorems on product disks, followed along a finite chain covering any compact path in this convex tube, extend the real identity to . This proves (T3).

For a complex time parameter , , avoids the nonpositive real axis: if its imaginary part is zero, then and it is positive. The same exponential bounds and one-variable identity theorem extend (T3) to . Consequently Here damping is defined on tempered distributions by replacing outside by , where for . This function and all its derivatives are bounded, so the full product rule makes it a Schwartz multiplier. It agrees with the exponential near the cone; different choices have identical action on .

The continuation retains (T6), since on cone-supported distributions Thus Fourier differentiation contributes precisely the polynomial in (T6). All parameter continuations are justified on a compact parameter set by one choice of and the existing uniform Schwartz bounds. We use (T6) at .

Set , with . Two integrations by parts in its differential equation give Indeed , , ; the exponential removes the infinite endpoint. For real , All time derivatives on have bounds . Inverse spatial Fourier pairing with a Schwartz test is consequently absolutely convergent, and differentiation of every fixed order is justified by a common integrable envelope. This defines a smooth distribution-valued family on the whole real line. Its causal extension, obtained by integrating those pairings over , is tempered; the bound suffices.

Damping this causal extension gives the transform in (T6) at , by (T8) and absolute integration against Schwartz tests. Full spacetime Fourier injectivity identifies it with , including time zero. No term supported on that plane has been omitted. The transform of is therefore . Its initial even derivatives vanish, and its derivative of order has transform , the transform of .

Proof of the support and topology assertions. The spacetime support of is . If a spatial test is supported outside the ball of radius , the joint distribution pairs to zero against its product with every time test in a small interval around . Continuity of the scalar pairing with forces that pairing to vanish there: a continuous nonzero scalar value would have a small test of matching phase with nonzero integral. Localizing the spatial test proves . The assertion at zero follows from the computed trace. The same argument on an open interval disjoint from the cone proves this support bound for each time derivative as well.

On , all supports lie in a fixed ball. Choose a compact smooth cutoff equal to one on a larger ball. Pairing with any smooth function equals pairing with this cutoff times that function. Repeated integration by parts in the latter's Fourier transform bounds it by times finitely many derivatives on the cutoff support. Choose . Together with (T9) this gives a common finite-order bound for , uniform in , and dominated differentiation. Taylor's formula with the next two time derivatives makes its difference-quotient remainder tend to zero uniformly on every bounded set of smooth functions. This is precisely strong smoothness, also one-sided at zero.

Finally , so its time derivative of order is . The linear distributional chain rule and give (T1)–(T2) with all speed factors.