Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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-2026bAnalysisProbabilityPDEReads 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-2026bAnalysisPDEOn 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…The Data of the Hamilton-Jacobi Equation of the Heat Equation with Square-Integrable White Noise on the Torus: the Noise, the Nonlinearities and the Drift
lemmalem:white-noise-heat-data-torus-2026aAnalysisPDEOn the Sobolev triple of order s of the torus, the Fourier coefficient families of the trigonometric classes, enumerated along the lattice, form a sequence that is square-summable in the order -s space with an explicit bound on the sum; the ambient space is infinite-dimensional;…The Laplacian of the Trigonometric System, and the Fourier Coefficients of a Laplacian on the Torus
lemmalem:laplacian-trigonometric-system-torus-2026aAnalysisEach member of the trigonometric system is twice continuously differentiable and is an eigenfunction of the Laplacian with eigenvalue minus four pi squared times the squared length of its frequency; consequently the Fourier coefficients of the Laplacian of a twice continuously di…Summability of the Negative Powers of the Fourier Weights of the Torus
lemmalem:fourier-weights-summable-torus-2026aAnalysisFor an exponent at least the dimension, the reciprocals of the powers of the Fourier weights of the torus, taken along any injective indexing of the integer lattice by the natural numbers, form a convergent series with a bound independent of the indexing.Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares
lemmalem:reciprocal-squares-bounded-2026aAnalysisReciprocals reverse the order on the positive reals; the partial sums of the reciprocals of the squares are bounded by two, so that series converges; and a quadratic sum taken symmetrically about its centre is bounded independently of its length.Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits
lemmalem:finite-set-indexed-sum-nonnegative-2026bAnalysisFor real-valued maps on a nonempty finite set, sums over the set preserve order and nonnegativity, grow with the index set, dominate each nonnegative term, satisfy the triangle inequality, count the elements of the set, and commute with limits.The Integer Lattice Admits an Enumeration by the Natural Numbers
lemmalem:integer-lattice-enumeration-2026aSet TheoryThe integer lattice of Euclidean space is countable and not finite, and therefore admits a bijection from the natural numbers onto it.- A subset of the natural numbers that is not finite is the set of terms of a strictly increasing sequence, which is a bijection onto it; consequently every countable set that is not finite is in bijection with the natural numbers.