Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus
equationeq:hamilton-jacobi-white-noise-heat-torus-2026bAnalysisPDEThe discounted viscous Hamilton-Jacobi equation on the Sobolev space of order -(s+1) of the torus whose second-order term is the trace along the trigonometric white noise, whose Hamiltonian is half the squared norm of the gradient, and whose drift is minus the Laplacian, written…The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian
lemmalem:sobolev-triple-white-noise-torus-2026aAnalysisThe Sobolev spaces of orders -(s+1) and -s on the torus form a Hilbert triple, which along any enumeration of the lattice is the diagonal triple of the rescaled trigonometric basis with the Fourier weights; the rescaled trigonometric classes are eigenvectors of the form operator…The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale
lemmalem:negative-sobolev-space-torus-2026aAnalysisEach negative-order Sobolev space of the torus is a real Hilbert space isometric to the square-integrable classes through the realisation map; the Fourier coefficient families of square-integrable classes embed contractively; the realisation preimages of the trigonometric classes…The Negative-Order Sobolev Spaces of the Torus
definitiondef:negative-sobolev-space-torus-2026aAnalysisFor a natural number m, the Sobolev space of order -m on the torus is the space of coefficient families on the integer lattice whose rescaling by the m-th power of the inverse square roots of the Fourier weights is the Fourier coefficient family of a square-integrable class; its…Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families
lemmalem:fourier-coefficients-torus-2026aAnalysisThe Fourier coefficient map is linear and injective, Parseval's identity and the Fourier expansion hold along any enumeration of the lattice, a square-summable coefficient family along an enumeration comes from exactly one class, and for each natural number m the families whose r…The Fourier Coefficients of a Square-Integrable Class on the Torus
definitiondef:fourier-coefficients-torus-2026aAnalysisThe Fourier coefficient family of a square-integrable class on the torus is the real-valued map on the integer lattice whose value at a lattice point is the inner product of the class with the corresponding trigonometric system class.Transport of an Inner Product and of Hilbert Space Structure along a Linear Bijection
lemmalem:hilbert-structure-transport-2026aAnalysisPulling an inner product back along a linear bijection from a real vector space onto a real inner product space yields an inner product for which the bijection is isometric; if the target is a Hilbert space so is the source, its inverse is linear, and orthonormal bases pull back…The Trigonometric System is an Orthonormal Basis of the Square-Integrable Space of the Torus
theoremthm:trigonometric-system-complete-torus-2026aAnalysisA square-integrable class on the torus that is orthogonal to every function of the trigonometric system is zero; consequently, enumerated along any bijection of the natural numbers with the integer lattice, the trigonometric system is an orthonormal basis of the square-integrable…Uniform Convergence of the Fejer Means of a Continuous Periodic Function on the Torus
lemmalem:fejer-mean-uniform-convergence-torus-2026aAnalysisThe Fejer means of a continuous lattice-periodic function converge to it uniformly on Euclidean space.The Cell Integral of a Translated Periodic Function
lemmalem:shifted-cell-integral-periodic-torus-2026aAnalysisTranslating a measurable lattice-periodic function whose restriction to the unit cell is integrable does not change its integral over the unit cell.The Fejer Means of a Continuous Periodic Function on the Torus
lemmalem:fejer-mean-torus-2026aAnalysisThe Fejer mean of order N of a continuous periodic function is the trigonometric polynomial with the Fourier coefficients of the function damped by the Fejer weights; it is continuous and periodic, pairs with any square-integrable class through those coefficients, equals the cell…Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus
lemmalem:finite-sum-continuous-periodic-torus-2026aAnalysisA finite linear combination of continuous periodic functions is continuous and periodic, its class in the square-integrable space of the torus is the corresponding linear combination of classes, and its inner product with any class is the corresponding combination of inner produc…The Reproducing Identity for the Fejer Kernels of the Torus
lemmalem:fejer-kernel-reproducing-torus-2026aAnalysisThe Fejer kernel evaluated at a difference of two points is a finite linear combination of products of trigonometric system functions evaluated at the two points separately, in one variable and on the torus; in particular the product kernel is symmetric in its two arguments.- The map sending an index a of the initial segment [N] to N+1-a is a bijection of [N] onto itself, and a finite sum is unchanged when its summands are taken in reverse order.
The Fejer Kernel of the Torus: Nonnegativity, Periodicity, Mass and the Far-Field Integral Bound
lemmalem:fejer-kernel-product-torus-2026aAnalysisThe product of one-dimensional Fejer kernels over the coordinates is a nonnegative periodic function whose integral over the cell is one, and whose integral over the part of the cell where some coordinate stays away from the integers tends to zero as the order grows.The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay
lemmalem:fejer-kernel-torus-2026aAnalysisThe Fejer kernel is continuous, periodic and integrable over the cell, is nonnegative, has integral one over the cell, and away from the integers is bounded by a constant over the order of the kernel.- Multiplying the Dirichlet sum by the sine of pi t collapses it to a single sine of an odd multiple, and multiplying the Fejer sum by the square of that sine collapses it to the square of a sine; both are proved by induction on the number of terms.
Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine
lemmalem:sine-cosine-parity-double-angle-2026aAnalysisThe cosine is even and the sine is odd, the double-angle identities in their three standard forms, and the factorisation of a difference of two squared sines as a product of two sines.What the Hamilton-Jacobi Equation of the Heat Equation with Square-Integrable White Noise Models, and What Is Not Proved
remarkrem:white-noise-heat-equation-torus-2026aAnalysisProbabilityPDEReads the well-posedness corollary for the white-noise heat equation as the dynamic programming equation of a controlled stochastic heat equation on the torus, explains why the noise is square-integrable white noise and why the equation lives on a negative-order Sobolev space, an…Well-Posedness of the Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus
corollarycor:white-noise-heat-hamilton-jacobi-well-posed-torus-2026aAnalysisPDEOn the Sobolev space of order -(s+1) of the torus, with s at least the dimension, the discounted viscous Hamilton-Jacobi equation driven by the trigonometric white noise, with drift minus the Laplacian and a bounded uniformly continuous running cost, has a bounded uniformly conti…