Multivariate Taylor-Measure Function Space

• The multivariate Taylor-measure function space is defined as a framework using finite signed Taylor measures to represent measurable functions, generalizing classical Taylor series.
• It establishes a robust functional-analytic setting as a Hilbert and, under integrability restrictions, a Polish space, enabling orthonormal expansions and effective approximation.
• The framework unifies diverse representations, including analytic functions, Lie series, and solutions to differential equations, under a common summation and measure-based structure.

Searching arXiv for the primary paper and closely related Taylor-expansion/function-space works. The multivariate Taylor-measure function space is a function space built from the notion of a Taylor measure, intended to generalize Taylor’s theorem beyond analytic functions and beyond differentiation-based expansions. In the formulation summarized in Micheas’s work, a multivariate Taylor-measure function (MTMF) on $\mathbb{R}^p$ is any real-valued function that, at each point, can be represented as a finite signed Taylor measure of the form $\sum_{n\in B} a_n(x)[g(x)]^n/n!$ for measurable $g$ and coefficient sequence $a(x)$, with finite total mass. The resulting space, denoted $\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$, is presented as a Hilbert space and, under an additional integrability restriction, as a Polish space; it also contains measurable functions, analytic Taylor expansions, several classical special functions, and certain series solutions of ordinary differential equations as special cases (Micheas, 14 Aug 2025).

1. Definition and basic construction

Let $\mathcal{B}(\mathbb{N})$ be the Borel $\sigma$-algebra on $\mathbb{N}$, and write $\mathcal{T}^{\mathcal{F}}$ for the collection of all signed, finite Taylor measures on $(\mathbb{N},\mathcal{B}(\mathbb{N}))$, namely those $\sum_{n\in B} a_n(x)[g(x)]^n/n!$0 of the form

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$1

with real or complex weights $\sum_{n\in B} a_n(x)[g(x)]^n/n!$2 and finite total mass $\sum_{n\in B} a_n(x)[g(x)]^n/n!$3 (Micheas, 14 Aug 2025).

Fix $\sum_{n\in B} a_n(x)[g(x)]^n/n!$4. A real-valued multivariate Taylor-measure function on $\sum_{n\in B} a_n(x)[g(x)]^n/n!$5 is any function

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$6

such that for each fixed $\sum_{n\in B} a_n(x)[g(x)]^n/n!$7,

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$8

for some measurable $\sum_{n\in B} a_n(x)[g(x)]^n/n!$9, some $g$0, and some fixed $g$1, with $g$2 (Micheas, 14 Aug 2025). The collection of all such real-valued MTMFs is denoted

$g$3

Two remarks are structurally important. First, all sums and products of measurable $g$4 and $g$5 are measurable, so $g$6 is Borel-measurable on $g$7. Second, taking $g$8, $g$9, and $a(x)$0 yields the trivial embedding of any measurable function into $a(x)$1 (Micheas, 14 Aug 2025). This makes the space extremely broad. A plausible implication is that the construction is less a regularity class in the classical sense than a representation framework that organizes heterogeneous function classes under a common summation form.

2. Relation to Taylor’s theorem

The defining representation is explicitly positioned as a generalization of Taylor’s theorem. In the univariate analytic case, if $a(x)$2 is real-analytic at $a(x)$3 with Taylor coefficients $a(x)$4, then

$a(x)$5

with $a(x)$6 and $a(x)$7, so $a(x)$8 (Micheas, 14 Aug 2025).

For analytic functions on $a(x)$9, the classical multivariate Taylor expansion can be regrouped in nested one-dimensional Taylor measures. More precisely, if $\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$0 is analytic in $\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$1 around $\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$2, then

$\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$3

can be rewritten through iterated Taylor measures of the form

$\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$4

so that

$\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$5

This provides the formal bridge between the classical multivariate Taylor series and the MTMF representation (Micheas, 14 Aug 2025).

The proposal differs substantially from other uses of “Taylor” in recent analysis. In Sobolev theory, an $\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$6-Taylor approximation is an averaged first-order expansion characterizing $\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$7 (Spector, 2014). In Wasserstein analysis, higher order Lions-Taylor expansions develop Taylor formulas on $\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$8 via Lions derivatives (Salkeld, 2023). In anisotropic approximation theory, Taylor expansions define model classes $\mathcal{F}_{\mathbb{R}}^{\mathbb{R}^p}$9 controlled by weighted coefficient decay and lower-set polynomial approximation (Bonito et al., 2019). These are conceptually adjacent but technically distinct constructions.

3. Algebraic, metric, and topological structure

The space $\mathcal{B}(\mathbb{N})$0 is endowed with an inner product. For two MTMFs

$\mathcal{B}(\mathbb{N})$1

the inner product is defined by

$\mathcal{B}(\mathbb{N})$2

This is stated to be a bona-fide inner product on $\mathcal{B}(\mathbb{N})$3, with induced norm

$\mathcal{B}(\mathbb{N})$4

Lemma 2.7 states completeness of $\mathcal{B}(\mathbb{N})$5, and Theorem 2.8 states that, equipped with $\mathcal{B}(\mathbb{N})$6, it is a Hilbert space (Micheas, 14 Aug 2025).

A corollary is the existence of an orthonormal basis $\mathcal{B}(\mathbb{N})$7 in $\mathcal{B}(\mathbb{N})$8 and the expansion

$\mathcal{B}(\mathbb{N})$9

The source describes this as a “reproducing” expansion (Micheas, 14 Aug 2025). Since no RKHS structure is separately specified in the summarized material, the safer interpretation is that the statement records Hilbert-space orthonormal expansion rather than a full reproducing-kernel theorem.

The space is also an algebra under pointwise operations: if $\sigma$0, then both $\sigma$1 and $\sigma$2 again lie in $\sigma$3. Constant functions belong to the space, and the space separates points of $\sigma$4: for $\sigma$5 there is some $\sigma$6 with $\sigma$7 (Micheas, 14 Aug 2025).

Under an additional restriction—namely, for each $\sigma$8 and some fixed $\sigma$9,

$\mathbb{N}$0

Theorem 2.10 states that $\mathbb{N}$1 is separable and complete, hence a Polish space (Micheas, 14 Aug 2025). This suggests that the full MTMF space is used as a broad ambient Hilbert space, while the restricted subclass is singled out when separability is required for probabilistic or descriptive-set-theoretic arguments.

4. Canonical examples and scope

The defining representation is flexible enough to subsume a wide range of examples already familiar from classical analysis.

Class	Representation in MTMF form	Source
Measurable real functions	$\mathbb{N}$2, $\mathbb{N}$3, $\mathbb{N}$4	(Micheas, 14 Aug 2025)
Simple functions	$\mathbb{N}$5 with $\mathbb{N}$6, $\mathbb{N}$7	(Micheas, 14 Aug 2025)
Analytic functions	$\mathbb{N}$8 at $\mathbb{N}$9	(Micheas, 14 Aug 2025)
Riemann zeta	$\mathcal{T}^{\mathcal{F}}$0 for $\mathcal{T}^{\mathcal{F}}$1, with $\mathcal{T}^{\mathcal{F}}$2	(Micheas, 14 Aug 2025)
Hypergeometric series	$\mathcal{T}^{\mathcal{F}}$3 with $\mathcal{T}^{\mathcal{F}}$4 and $\mathcal{T}^{\mathcal{F}}$5 rational in $\mathcal{T}^{\mathcal{F}}$6	(Micheas, 14 Aug 2025)
Lie series	$\mathcal{T}^{\mathcal{F}}$7 with $\mathcal{T}^{\mathcal{F}}$8, $\mathcal{T}^{\mathcal{F}}$9	(Micheas, 14 Aug 2025)

Further examples include generalized Rodrigues expansions and many classical orthogonal polynomials—Legendre, Hermite, Laguerre, Jacobi—obtained when $(\mathbb{N},\mathcal{B}(\mathbb{N}))$0 is weighted by $(\mathbb{N},\mathcal{B}(\mathbb{N}))$1 on the corresponding standard supports (Micheas, 14 Aug 2025). Monomials $(\mathbb{N},\mathcal{B}(\mathbb{N}))$2 also fit immediately by taking $(\mathbb{N},\mathcal{B}(\mathbb{N}))$3, $(\mathbb{N},\mathcal{B}(\mathbb{N}))$4, and $(\mathbb{N},\mathcal{B}(\mathbb{N}))$5 (Micheas, 14 Aug 2025).

Because every measurable real function embeds trivially, the space is not selective in the manner of Sobolev, Besov, or analytic classes. Its significance therefore lies primarily in its common encoding of functions via Taylor-measure data rather than in exclusionary regularity constraints. This also distinguishes it from the anisotropic analytic spaces $(\mathbb{N},\mathcal{B}(\mathbb{N}))$6, where weighted coefficient summability imposes nontrivial decay and yields near-optimal lower-set polynomial approximation results (Bonito et al., 2019).

5. Differentiation and differential-equation representations

If

$(\mathbb{N},\mathcal{B}(\mathbb{N}))$7

with $(\mathbb{N},\mathcal{B}(\mathbb{N}))$8 and $(\mathbb{N},\mathcal{B}(\mathbb{N}))$9 sufficiently smooth, then repeated Leibniz rules yield

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$00

and each $\sum_{n\in B} a_n(x)[g(x)]^n/n!$01 expands by the multinomial formula into products of derivatives of $\sum_{n\in B} a_n(x)[g(x)]^n/n!$02 (Micheas, 14 Aug 2025). In the finite-$\sum_{n\in B} a_n(x)[g(x)]^n/n!$03 case, with $\sum_{n\in B} a_n(x)[g(x)]^n/n!$04 and for example $\sum_{n\in B} a_n(x)[g(x)]^n/n!$05, the summary states an iterative first-derivative formula

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$06

where $\sum_{n\in B} a_n(x)[g(x)]^n/n!$07 removes the $\sum_{n\in B} a_n(x)[g(x)]^n/n!$08 term from the sequence $\sum_{n\in B} a_n(x)[g(x)]^n/n!$09 (Micheas, 14 Aug 2025). Higher partial derivatives similarly reduce the index set by one at each step.

The space is also used as an ansatz space for ODEs. For a linear differential operator

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$10

and the inhomogeneous problem $\sum_{n\in B} a_n(x)[g(x)]^n/n!$11 with linear boundary conditions, one seeks a solution of the form

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$12

If $\sum_{n\in B} a_n(x)[g(x)]^n/n!$13, then linearity forces, for each $\sum_{n\in B} a_n(x)[g(x)]^n/n!$14,

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$15

The summary then states: choose $\sum_{n\in B} a_n(x)[g(x)]^n/n!$16, and each coefficient $\sum_{n\in B} a_n(x)[g(x)]^n/n!$17 satisfies a first-order linear ODE determined by the Green-function representation of $\sum_{n\in B} a_n(x)[g(x)]^n/n!$18 (Micheas, 14 Aug 2025). Theorem 3.3 is summarized as asserting that, under mild regularity and integrability conditions, the general solution of $\sum_{n\in B} a_n(x)[g(x)]^n/n!$19 subject to linear boundary data is an MTMF (Micheas, 14 Aug 2025).

For first-order nonlinear ODEs, Lie-series solutions are likewise cast in MTMF form. If

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$20

then this is a MTMF with $\sum_{n\in B} a_n(x)[g(x)]^n/n!$21, $\sum_{n\in B} a_n(x)[g(x)]^n/n!$22, and $\sum_{n\in B} a_n(x)[g(x)]^n/n!$23 (Micheas, 14 Aug 2025). In this sense, the formalism is used not only to represent pre-existing functions, but also to encode constructive solution series.

6. Approximation, simple representatives, and neighboring notions

Lemma 2.11 states that every $\sum_{n\in B} a_n(x)[g(x)]^n/n!$24 is the pointwise limit of a double sequence of simple MTMFs $\sum_{n\in B} a_n(x)[g(x)]^n/n!$25 of the same form, with rational-valued weights (Micheas, 14 Aug 2025). Together with the separability result under the additional integrability condition, this gives the space a concrete approximation theory internal to its own representation language.

This approximation perspective should be distinguished from several other Taylor-based frameworks. In the anisotropic analytic setting, one studies functions

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$26

whose weighted coefficients satisfy $\sum_{n\in B} a_n(x)[g(x)]^n/n!$27 and approximates them by polynomial spaces $\sum_{n\in B} a_n(x)[g(x)]^n/n!$28 indexed by lower sets $\sum_{n\in B} a_n(x)[g(x)]^n/n!$29; for a budget $\sum_{n\in B} a_n(x)[g(x)]^n/n!$30, specially constructed lower sets $\sum_{n\in B} a_n(x)[g(x)]^n/n!$31 give certifiable and, in the surrogate norm, optimal approximation error bounds (Bonito et al., 2019). In Taylor-based quasi-Trefftz methods, one defines a local space

$\sum_{n\in B} a_n(x)[g(x)]^n/n!$32

so that the “Taylor measure” of PDE residual vanishes up to order $\sum_{n\in B} a_n(x)[g(x)]^n/n!$33 (Imbert-Gerard, 24 May 2025). That usage is local and polynomial, whereas the MTMF space is global and representation-theoretic.

A different neighboring use of measure-based Taylor language appears in Lions-Taylor expansions on Wasserstein space. There the expansion variable is a probability measure, the derivative is the Lions derivative $\sum_{n\in B} a_n(x)[g(x)]^n/n!$34, and higher-order terms are organized by partition-sequence combinatorics (Salkeld, 2023). The similarity lies in a generalized Taylor philosophy on nonclassical domains; the underlying spaces, differentiability notions, and remainder estimates are otherwise unrelated.

These distinctions matter because “Taylor-measure” can refer to different technical objects across the recent literature. The multivariate Taylor-measure function space of Micheas is specifically the space $\sum_{n\in B} a_n(x)[g(x)]^n/n!$35 built from pointwise Taylor-measure representations of measurable functions (Micheas, 14 Aug 2025).

7. Interpretation, significance, and limitations

The MTMF framework is presented as a unifying setting that includes analytic Taylor expansions, measurable simple and non-analytic functions, classical special functions, and Lie and Green series for ODEs as special cases (Micheas, 14 Aug 2025). Its significance comes from this unification together with its stated Hilbert-space and Polish-space structure, which the summary identifies as useful for orthonormal expansions, approximation, and the study of convergence and stability of series solutions to differential equations (Micheas, 14 Aug 2025).

At the same time, the breadth of the trivial embedding of measurable functions creates an important interpretive constraint. Since any measurable function can be represented by taking $\sum_{n\in B} a_n(x)[g(x)]^n/n!$36, $\sum_{n\in B} a_n(x)[g(x)]^n/n!$37, and $\sum_{n\in B} a_n(x)[g(x)]^n/n!$38, membership in $\sum_{n\in B} a_n(x)[g(x)]^n/n!$39 alone does not encode differentiability, analyticity, or Sobolev regularity (Micheas, 14 Aug 2025). This suggests that the primary mathematical content lies in the representational calculus and the induced functional-analytic structure, not in a restrictive smoothness criterion.

This also clarifies a potential misconception. The MTMF space is not the same object as Spector’s first-order $\sum_{n\in B} a_n(x)[g(x)]^n/n!$40-Taylor approximation space, even though both invoke “Taylor” and “measure-based” averaging. Spector’s formulation characterizes $\sum_{n\in B} a_n(x)[g(x)]^n/n!$41 through the vanishing of an averaged first-order Taylor remainder in $\sum_{n\in B} a_n(x)[g(x)]^n/n!$42 (Spector, 2014). By contrast, the MTMF construction starts from a Taylor-measure representation over $\sum_{n\in B} a_n(x)[g(x)]^n/n!$43 and then studies algebraic, Hilbert, and Polish-space properties (Micheas, 14 Aug 2025).

In that sense, the multivariate Taylor-measure function space occupies a distinctive position at the intersection of generalized series representation, measurable function theory, and functional analysis. It extends the formal reach of Taylor-type expansions while remaining compatible with non-analytic and even arbitrary measurable functions, and it packages that reach into a space with stated algebraic, orthonormal, complete, and separable structures under the hypotheses given in the source (Micheas, 14 Aug 2025).