---
title: Super Order Differentiable Functions
url: https://www.emergentmind.com/topics/super-order-differentiable-functions
type: topic
---

# Super Order Differentiable Functions

Searching arXiv for the cited papers and closely related terminology.
“Super order differentiable functions” is not a single standardized notion across mathematics; rather, it denotes several technically distinct constructions. In approximation theory, it refers to classes of \(2\pi\)-periodic functions whose \((\psi,\beta)\)-derivatives belong to unit balls of \(L_p\), with \(\psi(k)\) tending to zero faster than any power function but slower than geometric progression; these classes yield exact order estimates for best orthogonal trigonometric approximation [1812.06475]. In the differential-algebraic setting of supergeometry, closely related language concerns supercommutative algebras on which infinitely differentiable functions can be evaluated via a super Fermat theory, with differentiability encoded by divided-difference axioms and jet factorization [1211.6134]. A further, explicitly synthetic formulation appears in the theory of non-integer differentiability, where “super-order differentiable of order \(\alpha>0\)” means membership in fractional differentiability spaces such as \(C_{RL}^{\alpha}\), \(C_{C}^{\alpha}\), or equivalently \(C^{n,\alpha-n}\) for non-integer \(\alpha\) [2503.22854]. These usages are connected by a common theme: differentiability is indexed by a scale finer than classical integer-order smoothness.

## 1. Approximation-theoretic definition of super-order differentiable classes

In the approximation-theoretic literature, the basic object is the class \(C_{\beta,p}^{\psi}\) of \(2\pi\)-periodic functions whose \((\psi,\beta)\)-derivative belongs to the unit ball of \(L_p\) [1812.06475]. Here \(\psi=\psi(t)\), \(t\ge 1\), is a convex (down-ward) continuous function on \([1,\infty)\) with \(\psi(t)\to 0\) as \(t\to\infty\), and \(\psi(k)\) is its restriction to \(k\in\mathbb N\). The auxiliary quantities are
\[
n(t):=\psi^{-1}(\psi(t)/2), \qquad \mu(\psi;t):=\frac{t}{n(t)-t}.
\]
The class-defining condition is \(\psi\in\mathcal M''\), meaning that there is \(K>0\) such that for all \(t\ge 1\),
\[
n(t)-t\ge K,
\]
and \(\mu(\psi;t)\) remains bounded. This is equivalent to the pair of asymptotic conditions: \(\psi(k)\downarrow 0\) faster than \(k^{-m}\) for every \(m>0\), and \(\psi(k+1)/\psi(k)\to 1\) as \(k\to\infty\), i.e. slower than any geometric progression [1812.06475].

If \(f\) is a \(2\pi\)-periodic function with Fourier coefficients
\[
\hat f(k)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)e^{-ikx}\,dx,
\]
its \((\psi,\beta)\)-derivative \(f_\beta^\psi\) is defined by requiring that the formal series
\[
\sum_{k\in\mathbb Z\setminus\{0\}} \frac{\hat f(k)}{\psi(|k|)}\,e^{\,i\bigl(kx+\tfrac{\beta}{2}\pi\operatorname{sgn}k\bigr)}
\]
converges in \(L_1\) to some function \(\varphi\), in which case \(f_\beta^\psi(x)=\varphi(x)\) [1812.06475]. With
\[
B_p:=\left\{\varphi\in L_p(-\pi,\pi): \|\varphi\|_p\le 1,\ \int_{-\pi}^{\pi}\varphi=0\right\},
\]
the super-order differentiable class is
\[
C_{\beta,p}^{\psi}:=\{\,f\in L_p(-\pi,\pi): f_\beta^\psi \text{ exists and } f_\beta^\psi\in B_p\,\}.
\]

A closely related earlier notation is \(C_p^{\psi,\beta}\), together with \(L_{s,1}^{\psi,\beta}\), defined for \(2\pi\)-periodic functions whose \((\psi,\beta)\)-derivatives lie in \(L_p\) or \(L_1\), respectively [1304.7379]. In that setting, the requirement that \(\psi(k)\) tends to zero faster than any power is expressed as
\[
\forall m>0:\quad \lim_{k\to\infty} k^m\psi(k)=0,
\]
and the corresponding subclass is denoted \(\mathfrak M_\infty\) [1304.7379]. This suggests that the approximation-theoretic meaning of “super-order” is a smoothness scale strictly beyond Sobolev-type power decay but still below analytic geometric decay.

## 2. Exact order estimates for orthogonal trigonometric approximation

The principal quantitative objects are the best orthogonal trigonometric approximations in the uniform and \(L_s\)-metrics,
\[
E_n(C)_\infty:=\sup_{f\in C}\inf_{|\Lambda|=n}\left\|f(x)-\sum_{k\in\Lambda}\hat f(k)e^{ikx}\right\|_\infty,
\]
\[
E_n(C)_s:=\sup_{f\in C}\inf_{|\Lambda|=n}\left\|f(x)-\sum_{k\in\Lambda}\hat f(k)e^{ikx}\right\|_s.
\]
For \(\psi\in\mathcal M''\) and \(\beta\in\mathbb R\), Stepanyuk–Serdyuk establish the following exact order estimates [1812.06475].

For \(1<p<\infty\), with \(p'\) defined by \(1/p+1/p'=1\),
\[
E_n\bigl(C_{\beta,p}^{\psi}\bigr)_\infty \asymp \psi(n)\,\bigl(n(n)-n\bigr)^{1/p'}.
\]

For \(p=1\) in the uniform norm,
\[
E_n\bigl(C_{\beta,1}^{\psi}\bigr)_\infty \asymp \psi(n)\,\bigl(n(n)-n\bigr).
\]

For \(p=1\) and \(1<s<\infty\) in the \(L_s\)-norm,
\[
E_n\bigl(C_{\beta,1}^{\psi}\bigr)_s \asymp \psi(n)\,\bigl(n(n)-n\bigr)^{1-1/s}.
\]

Here \(\asymp\) means equality of order: each side is bounded above and below by the other up to absolute constants independent of \(n\) [1812.06475]. The earlier paper on best approximations and Fourier sums gives parallel two-sided estimates for \(E_n\) and for the Fourier partial-sum error \(\mathcal E_n\), including explicit constants under hypotheses such as
\[
n(\psi;n)-n\ge a>2,\qquad n(\psi;n)-n\ge b>2,
\]
and concludes that
\[
E_n \sim \mathcal E_n \sim \psi(n)\,[n(\psi;n)-n]^{1/\nu}\qquad (\nu=p',\,s')
\]
for the corresponding classes [1304.7379].

These formulas make the smoothness scale operational. The factor \(\psi(n)\) records the principal decay of Fourier coefficients, while the correction \((n(n)-n)^{\cdot}\) measures how rapidly the half-level inverse \(n(t)\) separates from \(t\). In this sense, the geometry of \(\psi\) near infinity determines the approximation order.

## 3. The intermediate smoothness regime and canonical examples

A central example is
\[
\psi(k)=\exp(-\alpha k^r),\qquad 0<r<1,\ \alpha>0.
\]
For this sequence,
\[
\psi(k+1)/\psi(k)=e^{-\alpha((k+1)^r-k^r)}\to 1
\]
and \(\psi(k)=o(k^{-m})\) for every \(m\), so \(\psi\in\mathcal M''\) [1812.06475]. Moreover,
\[
n(n)\asymp \bigl(n^r+(\ln 2)/\alpha\bigr)^{1/r}\asymp n+C\,n^{1-r},
\]
hence
\[
n(n)-n\asymp n^{1-r}.
\]
Substituting this into the approximation theorems yields the rates
\[
E_n \asymp e^{-\alpha n^r}\,n^{(1-r)/p'} \qquad \text{(uniform, }p>1\text{)},
\]
\[
E_n \asymp e^{-\alpha n^r}\,n^{1-r} \qquad \text{(uniform, }p=1\text{)},
\]
\[
E_n \asymp e^{-\alpha n^r}\,n^{(1-r)(1-1/s)} \qquad \text{(}L_s,\ p=1\text{)}
\]
[1812.06475]. The earlier analysis gives the same specialization, expressed as
\[
n(\psi;n)=\bigl(n^r+\tfrac{\ln 2}{\alpha}\bigr)^{1/r},\qquad
n(\psi;n)-n\sim \frac{\ln 2}{\alpha}\,n^{1-r},
\]
and therefore
\[
E_n\bigl(C_p^{\psi,\beta}\bigr)_C\asymp e^{-\alpha n^r}n^{(1-r)/p'},\qquad
E_n\bigl(L_{s,1}^{\psi,\beta}\bigr)_{L_s}\asymp e^{-\alpha n^r}n^{(1-r)/s'}
\]
[1304.7379].

The significance of this example is stated explicitly: these classes interpolate between classical Sobolev-type classes with power-decay \(\psi(k)=k^{-r}\) and analytic or Gevrey classes with geometric decay \(\psi(k)=e^{-\alpha k}\) [1812.06475]. The condition \(\psi\in\mathcal M''\) guarantees decay of Fourier coefficients faster than any power, yet too slow to be analytic; accordingly, the resulting approximation orders fill the gap between polynomial and exponential convergence [1812.06475]. A plausible implication is that this scale is particularly natural for stretched-exponential regularity.

## 4. Relation to Fourier sums and kernel methods

The approximation-theoretic results are not limited to abstract best approximation. For \(f\in C_p^{\psi,\beta}\) with \(f_\beta^\psi=p\in L_p\), the deviation of the Fourier partial sum admits the integral representation
\[
f(x)-S_{n-1}(f;x)=\frac1\pi\int_{-\pi}^{\pi} Y_{\psi,n}(x-t)\,p(t)\,dt,
\]
where
\[
Y_{\psi,n}(t)=\sum_{k=n}^{\infty}\psi(k)\cos\!\Bigl(kt-\tfrac{\beta\pi}{2}\Bigr)
\]
[1304.7379]. Upper bounds then follow via Young or Hölder inequalities from estimates on \(\|Y_{\psi,n}\|_{p'}\), together with a decomposition of the integral into regions \(|t|\le \delta\) and \(|t|>\delta\), where
\[
\delta=\frac{\pi}{n(\psi;n)-n}
\]
[1304.7379]. On the small-\(t\) region, Abel summation and Dirichlet-kernel estimates are used; on the complementary region, tail-sum bounds and integral estimates for \(\psi\) provide the same order factor \(\psi(n)\,[n(\psi;n)-n]^{1/p'}\) [1304.7379].

Lower bounds are obtained by constructing an extremal function
\[
f^*(x)=\sum_{k=n+1}^{n(\psi;n)}\frac{\varepsilon_k}{\psi(k)}\cos kx
\]
with carefully chosen signs \(\varepsilon_k\), and then using orthogonality against trigonometric polynomials of degree \(<n\) to prove
\[
\|f^*-t\|_C\ge C_a\,\psi(n)\,[n(\psi;n)-n]^{1/p'}
\]
[1304.7379]. This identifies not only the approximation rate but also the mechanism by which the frequency window \([n+1,n(\psi;n)]\) governs the asymptotics.

The same source states that these techniques extend to more general linear methods, including Cesàro and de la Vallée-Poussin means, via their kernels [1304.7379]. Since that statement appears as an application rather than a theorem in the extracted data, it is best read as a methodological indication rather than as a fully quantified general result.

## 5. Super-order differentiability in super Fermat theories

In a different domain, the phrase connects to supergeometry through theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Carchedi and Roytenberg call such a theory a super Fermat theory [1211.6134]. Formally, \(T\) is a 2-sorted Lawvere theory with objects \((n|m)\), \(n,m\in\mathbb N\), whose hom-sets
\[
T((n|m),(1|0))=:T(n|m)
\]
are interpreted as generalized functions of \(n\) even and \(m\) odd variables [1211.6134].

A super Fermat theory satisfies four structural conditions. First, for each object \((n|m)\), the category of \(T\)-algebras is equivalent to the usual category of supercommutative \(\mathbb K\)-algebras \(A=A_0\oplus A_1\) with
\[
a\cdot b=(-1)^{|a||b|}b\cdot a.
\]
Second, there is a canonical embedding
\[
C^\infty(\mathbb R^n)\otimes \Lambda^m \hookrightarrow T(n|m),
\]
whose image is denoted by smooth expressions in even variables \(x^1,\dots,x^n\) and odd variables \(\theta^1,\dots,\theta^m\). Third, for any \(T\)-algebra \(A\) and \(f\in C^\infty(\mathbb R^n)\otimes\Lambda^m\subseteq T(n|m)\), there is an evaluation map
\[
\operatorname{ev}_f:A_0^n\times A_1^m\to A_0.
\]
Fourth, the theory satisfies a super-Fermat divided-difference property in each even slot and a linearity property in each odd slot, encoded by unique elements \(\Delta_{x^i}F\) and \(\Delta_{\theta^j}F\) such that
\[
F(x,\ldots,x_i,\ldots;\theta)-F(y,\ldots,x_i,\ldots;\theta)
=(x-y)\cdot \Delta_{x^i}F(x,y,\ldots;\theta),
\]
\[
F(x;\ldots,\theta^j+\xi,\ldots,\theta)-F(x;\theta)
=\xi\cdot \Delta_{\theta^j}F(x;\theta,\xi)
\]
[1211.6134].

Within this framework, a morphism of superspaces or of \(C^\infty\) superalgebras,
\[
\phi:M\to N,
\]
is said to be \(k\)-times super-differentiable if, in local coordinates, all mixed partial super-derivatives up to total order \(k\) exist and are continuous [1211.6134]. Equivalently, using the \(k\)-th jet algebra
\[
J^k(A)=A\otimes_{C^\infty} C^\infty(J^k(\mathbb R^n)),
\]
the pullback \(\phi^\sharp:C^\infty(N)\to C^\infty(M)\) is \(k\)-super-differentiable if it factors through the \(k\)-jet prolongation
\[
C^\infty(N)\to J^k\bigl(C^\infty(M)\bigr)\to C^\infty(M)
\]
[1211.6134]. In this setting, “super-order” refers not to approximation order but to the calculus generated by even and odd directions in superalgebraic smoothness.

## 6. Fractional order as a synthetic notion of super-order differentiability

A third usage makes the term explicit: a function
\[
f:[t_0,t_1]\to X
\]
is said to be “super-order differentiable of order \(\alpha>0\)” if it belongs to certain spaces of continuously fractional differentiable functions [2503.22854]. Let \(m=\lceil\alpha\rceil\). The Riemann–Liouville and Caputo spaces are defined by
\[
C_{RL}^{\alpha}([t_0,t_1];X)
=
\Bigl\{\,f\in C^{m-1}([t_0,t_1];X): D_{RL}^{\alpha}f\in C^0([t_0,t_1];X)\Bigr\},
\]
\[
C_{C}^{\alpha}([t_0,t_1];X)
=
\Bigl\{\,f\in C^{m-1}([t_0,t_1];X): D_{C}^{\alpha}f\in C^0([t_0,t_1];X)\Bigr\}.
\]
If \(\alpha\in\mathbb N\), both spaces coincide with the usual \(C^\alpha([t_0,t_1];X)\) [2503.22854].

The same source states a synthetic equivalence for non-integer \(\alpha\): “super-order differentiable of order \(\alpha>0\)” means either that \(D_{RL}^{\alpha}f\) exists and is continuous, or that \(D_C^\alpha f\) exists and is continuous, or that
\[
f\in C^n([t_0,t_1];X)\quad\text{with }n=\lfloor\alpha\rfloor
\]
and \(f^{(n)}\) is Hölder continuous of order \(\alpha-n\), i.e.
\[
f\in C^{n,\alpha-n}
\]
[2503.22854].

For the regime \(0<\alpha<1\), the identification is especially sharp:
\[
C_{RL}^{\alpha}([t_0,t_1];X)=C^{0,\alpha}([t_0,t_1];X),
\qquad
C_C^{\alpha}([t_0,t_1];X)=C^{0,\alpha}([t_0,t_1];X)
\]
isomorphically [2503.22854]. The paper also provides examples of functions that are fractional-smooth but not classically integer-smooth: \(f(t)=t^\alpha\) on \([0,1]\) belongs to \(C_{RL}^\alpha([0,1])\) but not to \(C^1([0,1])\); the Weierstrass-type function
\[
W_\alpha(t)=\sum_{k=0}^{\infty}2^{-k\alpha}\cos(2^k t)
\]
lies in \(C^{0,\alpha}(\mathbb R)\) and hence in \(C_{RL}^\alpha\), while having no classical derivative anywhere [2503.22854].

This usage differs from the approximation-theoretic one in formalism, but both organize smoothness by a scale intermediate between classical categories. A plausible implication is that the phrase “super-order” functions as a label for smoothness beyond integer-order classes rather than as a universally fixed technical term.

## 7. Conceptual scope, distinctions, and common misconceptions

The approximation-theoretic, supergeometric, and fractional-differentiability usages should not be conflated. In approximation theory, a super-order differentiable class is specified by a decay function \(\psi\) and a \((\psi,\beta)\)-derivative, and its defining feature is that \(\psi(k)\) decreases faster than any power function but slower than geometric progression [1812.06475]. In supergeometry, differentiability is encoded by operations labeled by smooth functions and Grassmann variables, together with divided-difference axioms and jet factorization [1211.6134]. In fractional analysis, “super-order differentiable of order \(\alpha\)” is identified with continuity of a fractional derivative or, equivalently for non-integer \(\alpha\), Hölder regularity of the appropriate order [2503.22854].

A common misconception is to read “super-order” as meaning analytic or ultra-analytic smoothness. The approximation-theoretic papers explicitly exclude the geometric regime: \(\psi(k+1)/\psi(k)\to 1\) means decay is slower than any geometric progression, even though it is faster than any power [1812.06475]. Another possible misunderstanding is to equate “super-order” with arbitrary higher integer differentiability. In the fractional framework, the point is precisely that non-integer order regularity can be characterized independently of classical \(C^m\) smoothness; for example, \(t^\alpha\in C_{RL}^\alpha\) but \(t^\alpha\notin C^1\) when \(0<\alpha<1\) [2503.22854].

Taken together, these strands show that “super order differentiable functions” is best understood as a family of context-dependent notions indexed by refined smoothness data: by the decay law \(\psi\) in periodic approximation theory, by even-odd differential calculus in super Fermat theories, or by the order \(\alpha\) of fractional differentiation. The shared mathematical content is the systematic refinement of the classical hierarchy of differentiability.

Source: https://www.emergentmind.com/topics/super-order-differentiable-functions