Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Elementary Properties of the Weak Partial Derivative on the Torus
lemmalem:weak-derivative-torus-basic-2026aAnalysisPDEThe weak partial derivative extends the classical one on continuously differentiable periodic functions, is linear, may be tested against continuously differentiable periodic functions, commutes with mollification, is preserved by limits in the square-integrable space, and equips…- A function on the torus is an i-th weak partial derivative of another when integration against every smooth periodic test function obeys the integration-by-parts identity; the notion descends to almost-everywhere classes, where the weak derivative is unique.
Pairing an Integrable Function on the Torus with a Continuous Periodic Function
lemmalem:periodic-test-pairing-torus-2026aAnalysisThe product of an integrable function on the torus with a continuous periodic function is integrable, its integral is bounded by the supremum of the periodic factor times the integral of the absolute value of the other, and that integral is linear in each factor, depends only on…Transfer of a Partial Derivative in Periodic Convolution
lemmalem:periodic-convolution-derivative-torus-2026aAnalysisFor a smooth compactly supported kernel and a continuously differentiable periodic function, the derivative in a periodic convolution may be moved between the kernel and the function; consequently the mollifications of such a function converge to it uniformly together with their…The Lebesgue Bound for Periodic Convolution and Convergence of Mollifications
theoremthm:periodic-convolution-lp-torus-2026aAnalysisPeriodic convolution is bounded on every Lebesgue space of the torus by the mass of its kernel, the mollifications of a power-integrable function converge to it in that space, and the smooth periodic functions are dense there.- Wrapping a compactly supported kernel around the lattice gives a kernel that is continuous and periodic in each variable separately, as smooth as the original, has mass over the cell no larger than the original, is jointly measurable on the cell squared, and represents the period…
A Continuous Lattice-Periodic Function is Uniformly Continuous
lemmalem:continuous-periodic-uniformly-continuous-2026aAnalysisA continuous function that repeats over the integer lattice is uniformly continuous on the whole of Euclidean space, because it is determined by its values on a compact box.The Fundamental Lemma of the Calculus of Variations on the Torus
lemmalem:fundamental-lemma-torus-2026aAnalysisPDEA function integrable on the torus whose integral against every smooth periodic test function vanishes is zero almost everywhere.- Periodic convolution produces continuous periodic functions, is as smooth as its kernel with derivatives falling on the kernel, depends only on the class of the function, is linear in each argument, is bounded uniformly by a multiple of the seminorm of the function, and reproduce…
Periodic Convolution and Mollification on the Torus
definitiondef:periodic-convolution-torus-2026aAnalysisDefines the periodic convolution of a compactly supported continuous kernel with a power-integrable function on the torus, and the mollification obtained by taking the kernel to be a rescaled mollifier kernel.The Periodic Extension of a Function on the Unit Cell
lemmalem:periodic-extension-torus-2026aAnalysisWrapping a function on the unit cell around the lattice gives a periodic function on Euclidean space. The extension is measurable, linear, insensitive to changes on null sets, integrable on every bounded set, and has the same integral over every translate of the cell.Convolution of a Locally Integrable Function with a Compactly Supported Kernel
lemmalem:kernel-convolution-locally-integrable-2026aAnalysisMultivariable CalculusThe convolution of a function integrable on every bounded set with a continuous kernel vanishing outside a ball is defined at every point, is continuous, is linear in each argument, and is as differentiable as the kernel, the derivatives falling on the kernel.The Laplacian of a Twice Continuously Differentiable Function
definitiondef:laplacian-euclidean-2026aAnalysisMultivariable CalculusDefines the Laplacian of a twice continuously differentiable function on a Euclidean open set as the sum of its unmixed second partial derivatives.Vanishing Mean of a Derivative and Integration by Parts on the Torus
lemmalem:periodic-integration-by-parts-2026aAnalysisPDEA partial derivative of a continuously differentiable periodic function has integral zero over the torus, so integration by parts on the torus carries no boundary term.- Step functions taking rational values on the dyadic subcubes of the unit cube form a countable dense family, so every Lebesgue space of the torus is separable.
Continuous Periodic Functions are Power-Integrable and Dense on the Torus
lemmalem:continuous-periodic-dense-lp-torus-2026aAnalysisRestricting a continuous periodic function to the unit cube gives an element of every Lebesgue space of the torus, linearly in the function, and these elements are dense there.- Standing notation for analysis on the flat torus: the half-open unit cube with the restriction of Lebesgue measure, the resulting Lebesgue spaces, the periodic function classes, and the gradient and Laplacian.
Elementary Properties of Lattice-Periodic Functions
lemmalem:periodic-function-basic-2026aAnalysisMultivariable CalculusPeriodic functions are closed under sums, scalar multiples and products; the partial derivatives of a periodic function are periodic; a continuous periodic function is bounded, attaining the largest absolute value on the closed unit cube; and a continuous function vanishing outsi…The Half-Open Unit Cell Tiles Euclidean Space
lemmalem:unit-cell-tiling-2026aAnalysisMultivariable CalculusThe half-open unit cube and its integer translates partition Euclidean space. Records the resulting wrapping map, identifies the interior, closure and measure of the cube, and shows that a periodic function has the same integral over every translate of the cube.Lattice-Periodic Functions and the Periodic Function Classes
definitiondef:periodic-function-lattice-2026aAnalysisMultivariable CalculusDefines the integer lattice in R^n, what it means for a function on R^n to be periodic under it, and the classes of continuous, k-times continuously differentiable, and smooth periodic functions.