Rational Chebyshev Basis Functions

• Rational Chebyshev basis functions are rational-function analogues of classical Chebyshev polynomials that retain key properties like orthogonality, recurrence, and minimax optimality.
• They are constructed via rational expansions, variable mappings, and prescribed poles, creating tailored approximation spaces for unbounded domains and multipoint Padé interpolation.
• Their versatility is demonstrated in applications such as cosmography, finite-grid multivariate approximation, and polar image analysis, offering improved accuracy and stability over traditional methods.

Rational Chebyshev basis functions are rational-function analogues or extensions of classical Chebyshev polynomial systems, used when the approximation space is built from ratios of Chebyshev expansions, from Chebyshev polynomials composed with rational maps, or from rational functions with prescribed poles that retain core Chebyshev structures such as orthogonality, recurrences, minimax approximation, and Padé-type interpolation. Across the literature, the term covers several closely related constructions: rational approximants expressed in a Chebyshev basis, mapped Chebyshev systems on semi-infinite domains, multipoint rational Chebyshev functions with prescribed poles, and generalized rational models in which numerator and denominator are linear forms in arbitrary continuous basis functions that may be chosen as Chebyshev polynomials (Capozziello et al., 2017, Millán et al., 2020, Derevyagin et al., 11 Jun 2026).

1. Classical Chebyshev structure and rationalization

The starting point is the Chebyshev polynomial basis of the first kind,

$T_n(z)=\cos(n\theta),\qquad \theta=\arccos z,$

with orthogonality

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$

and recurrence

$T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$

together with

$$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$

A generic function may be expanded as

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$

In one major rationalization, the approximation space is no longer polynomial but a ratio of two Chebyshev expansions,

$$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$

with coefficients chosen so that the Chebyshev series matches through order $n+m$. The product identity

$$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$

reduces this matching to $n+m+1$ linear equations for the $n+m+1$ unknowns $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$0 (Capozziello et al., 2017).

A second rationalization keeps the Chebyshev polynomials themselves but changes the independent variable by a rational map. On the semi-infinite interval, one standard definition is

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$1

with $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$2. This maps $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$3 onto $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$4, sending $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$5 and $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$6. In that setting, the rational Chebyshev functions inherit a weighted orthogonality framework with

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$7

The same map appears in the radial basis used for galaxy-profile decomposition,

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$8

which is orthogonal on $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$9 with respect to the weighted measure

$T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$0

(Parand et al., 2010, Jiménez-Teja et al., 2011).

A third rationalization fixes poles explicitly. For a prescribed sequence $T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$1, one sets

$T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$2

equivalently

$T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$3

forms the finite Blaschke product

$T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$4

and defines rational Chebyshev functions by

$T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$5

These functions are rational in $T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$6, not polynomial, and reduce to the classical Chebyshev objects in the limiting case $T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$7 for all $T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$8 (Derevyagin et al., 11 Jun 2026).

2. Rational basis forms and approximation spaces

Rational Chebyshev basis constructions differ according to how the rationality enters the approximation space. In the cosmographic formulation, the object is explicitly a rational approximant in the Chebyshev basis,

$T_{n+1}(z)=2zT_n(z)-T_{n-1}(z),$9

and the practical problem is coefficient determination from a target Chebyshev series (Capozziello et al., 2017).

In generalized rational approximation, the numerator and denominator are linear forms in basis functions: $$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$0 Here $$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$1 and $$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$2 are continuous basis functions on a compact domain $$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$3. The classical rational Chebyshev problem,

$$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$4

is the monomial special case. The generalized framework admits any continuous basis functions $$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$5, so Chebyshev polynomials can be used without changing the minimax objective or the positivity constraint on the denominator,

$$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$6

This is the sense in which rational Chebyshev basis functions fit naturally into a ratio-of-linear-forms framework rather than requiring a monomial basis (Millán et al., 2020).

The same structure extends to the multivariate finite-grid setting: $$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$7 subject to

$$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$8

The basis functions are again not limited to monomials. The paper explicitly notes that “one group of approaches … is dedicated to ‘nearly optimal’ solutions, whose construction is based on Chebyshev polynomials,” while also stating that the extension of that approach to non-monomial basis functions remains open. Within the optimization model, however, arbitrary basis functions are admissible, so a Chebyshev basis may be chosen in both numerator and denominator (Millán et al., 2021).

A more specialized Chebyshev-basis rational framework is developed for systems of functions expanded directly in Chebyshev polynomials,

$$T_0=1,\quad T_1=z,\quad T_2=2z^2-1,\quad T_3=4z^3-3z,\quad T_4=8z^4-8z^2+1.$$9

with denominator and numerators

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$0

satisfying

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$1

The resulting rational fractions

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$2

are called linear Hermite–Chebyshev approximations. A nonlinear variant imposes direct truncated-series matching,

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$3

and the paper emphasizes that nonlinear Hermite–Chebyshev approximations may fail to exist, and even when they exist they may differ from the linear ones (Starovoitov et al., 21 Jul 2025).

3. Recurrences, continued fractions, and Padé-type interpretation

A defining feature of many rational Chebyshev systems is that classical Chebyshev algebraic structure survives in rational form. The simplest nontrivial rational Chebyshev approximant,

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$4

can be rewritten as

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$5

which converges for

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$6

This gives an explicit convergence radius

$f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$7

which was compared directly with the $f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$8 Padé and second-order Taylor radii in cosmography (Capozziello et al., 2017).

In the prescribed-pole theory, the rational Chebyshev functions $f(z)=\sum_{k=0}^{\infty} c_k T_k(z).$9 and $$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$0 satisfy generalized three-term recurrences depending on the pole sequence $$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$1. The initial data are

$$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$2

and

$$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$3

These formulas show directly that the basis elements are rational. The recurrences imply a continued fraction representation of Thiele or $$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$4 type for the ratio $$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$5, which plays the same structural role as the ordinary continued fraction generated by classical Chebyshev recurrences (Derevyagin et al., 11 Jun 2026).

The Padé connection is exact in both the polynomial and multipoint rational settings. Classically,

$$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$6

is the $$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$7 Padé approximant to

$$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$8

at infinity, because

$$R_{n,m}(z)=\frac{\sum_{i=0}^n a_i T_i(z)}{\sum_{j=0}^m b_j T_j(z)} =\frac{\sum_{i=0}^n a_i T_i(z)}{1+\sum_{j=1}^m b_j T_j(z)},$$9

In the rational setting with interpolation nodes

$n+m$0

the same ratio becomes the multipoint Padé approximant of type $n+m$1 to $n+m$2, satisfying

$n+m$3

for $n+m$4. This makes rational Chebyshev bases the prescribed-pole, multipoint counterpart of the ordinary Chebyshev–Padé correspondence (Derevyagin et al., 11 Jun 2026).

A related asymptotic statement is available for shrinking domains $n+m$5. For fixed degrees $n+m$6, a rational Chebyshev approximant behaves asymptotically like the $n+m$7-Padé approximant at the origin. If

$n+m$8

then for suitably scaled interpolation nodes $n+m$9,

$$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$0

For interpolatory best approximants, the rescaled nodes converge to Chebyshev nodes, and the best uniform error becomes the Padé coefficient multiplied by the Chebyshev constant (Jawecki, 2024).

4. Extremal, minimax, and optimization viewpoints

Rational Chebyshev basis functions are also studied through minimax approximation and variational formulations. In the generalized ratio-of-linear-forms framework, the uniform error functional

$$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$1

is quasi-convex, and the stronger result proved is that $$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$2 is pseudo-convex in the sense of Penot and Quang. Consequently, its Clarke subdifferential $$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$3 is a pseudomonotone operator, and the first-order condition

$$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$4

implies that $$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$5 is a global minimizer. The explicit subdifferential is written in terms of active points attaining the positive and negative extrema of the pointwise deviation (Millán et al., 2020).

This structure leads to a variational-inequality formulation and to a projection-type method with a linesearch. The proposed “Algorithm F” chooses a direction from the active set, projects onto the feasible set

$$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$6

performs a backtracking linesearch, and updates by projection onto a halfspace

$$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$7

or onto $$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$8. The convergence proof shows finite termination of the linesearch whenever the current point is not already a solution; Variant 1 is Fejér convergent to the solution set, and the whole sequence converges to a point in the solution set $$T_n(z)T_m(z)=\frac{1}{2}\Big[T_{n+m}(z)+T_{|n-m|}(z)\Big]$$9 (Millán et al., 2020).

For multivariate generalized rational approximation on a finite grid, the central object is

$n+m+1$0

which is proved quasiconvex because ratios of linear forms are quasilinear and the supremum of quasilinear functions is quasiconvex. A bisection method is then applied with initial bounds

$n+m+1$1

iterating until $n+m+1$2. For fixed objective level $n+m+1$3, the feasibility subproblem reduces to linear programming on a finite grid after multiplication by the positive denominator (Millán et al., 2021).

A more classical extremal theory studies rational functions with prescribed real poles over subsets of $n+m+1$4. The rational space is

$n+m+1$5

where the poles $n+m+1$6 are real and lie outside $n+m+1$7. The extremizers satisfy a rational alternation theorem: a real $n+m+1$8 with $n+m+1$9 is extremal iff it has a maximal alternation set of size $n+m+1$0, with sign correction by the pole-parity function $n+m+1$1,

$n+m+1$2

The extremizer has only real generalized zeros, is real-valued, and has at most one generalized zero in each gap. Under weak convergence of pole-counting measures,

$n+m+1$3

the root asymptotics are

$n+m+1$4

uniformly on compact subsets of $n+m+1$5. Under stronger hypotheses, the paper proves Szegő–Widom asymptotics for these rational extremizers (Eichinger et al., 2021).

A plausible implication is that rational Chebyshev basis design can be separated into two layers: the algebraic specification of the space by poles or basis functions, and an extremal or optimization principle that selects stable approximants within that space.

5. Domain mappings, semi-infinite intervals, and separable polar systems

One of the most common computational uses of rational Chebyshev basis functions is the transfer of finite-interval Chebyshev machinery to unbounded or semi-infinite domains by algebraic mapping. For the semi-infinite interval, the mapped basis

$n+m+1$6

generates the approximation space

$n+m+1$7

with projection

$n+m+1$8

The convergence statement quoted in the collocation literature is

$n+m+1$9

which is interpreted there as showing exponential convergence of the rational Chebyshev approximation for sufficiently smooth functions (Parand et al., 2010).

In Volterra’s population model, this basis is used after converting the integro-differential equation into the nonlinear ODE

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$00

The ansatz

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$01

is substituted into the residual

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$02

and collocation is imposed at RC-Gauss-Radau points together with the two initial conditions. The paper reports very high accuracy for the maximum population $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$03, with values matching the exact solution to essentially all displayed digits for several $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$04 values (Parand et al., 2010).

In polar image analysis, the same radial map is combined with Fourier angular modes to form the CHEF basis

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$05

with

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$06

The basis is orthonormal in

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$07

with inner product

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$08

The basis is motivated by the fact that galaxy profiles fall rapidly near the center and then have extended outer wings; rational Chebyshev functions are described as fitting quickly decaying functions like galaxy profiles efficiently. The reported performance includes total flux estimates with typical relative error around $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$09 and ellipticity errors typically below $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$10 in the simulated tests described in the paper (Jiménez-Teja et al., 2011).

These mapped and separable constructions show that “rational Chebyshev basis” often denotes not a single canonical family but a transport principle: choose a rational change of variables that preserves the useful approximation properties of Chebyshev polynomials on $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$11, then build spectral or orthogonal systems on the physical domain.

6. Applications, comparisons, and current distinctions

In late-time cosmology, rational Chebyshev polynomials were introduced as a cosmographic alternative to standard Taylor and Padé expansions. The scale factor is expanded around the present time,

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$12

with cosmographic parameters

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$13

The standard Taylor expansion of the luminosity distance is reliable only for low redshift, roughly $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$14, and becomes unstable for the high-$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$15 observational domain. The Chebyshev-based rational approximation is constructed from the Chebyshev expansion of $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$16, and in the paper the $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$17 rational Chebyshev approximation is identified as especially effective: it approximates the $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$18CDM luminosity distance very accurately, behaves better at $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$19, and is preferred over higher-order alternatives in their tests. The empirical analysis combines JLA Type Ia supernovae, observational Hubble data from differential ages, and BAO measurements using Monte Python and the Metropolis–Hastings algorithm. The reported qualitative outcome is that Taylor gives very large uncertainties, Padé improves over Taylor, and rational Chebyshev gives the smallest relative errors overall (Capozziello et al., 2017).

In finite-grid multivariate approximation, the comparison between polynomial and generalized rational models with the same number of decision variables also favors the rational form in accuracy, though not in computational time. For

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$20

the degree-4 polynomial approximation with 11 decision variables attains maximum deviation $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$21 in $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$22 seconds, whereas the degree-$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$23 generalized rational approximation with the same 11 decision variables attains maximum deviation $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$24 in $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$25 seconds. The paper concludes that polynomial approximation is much faster, rational approximation gives better accuracy, and rational approximation better captures the nonsmooth behavior of the target function (Millán et al., 2021).

Another active distinction is between linear and nonlinear Hermite–Chebyshev approximations. For linear approximations, existence and uniqueness are reduced to a rank condition on the trigonometric coefficient matrix,

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$26

called weak normality. Theorem 6 states that the Chebyshev problem $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$27 has a unique solution for fixed $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$28, $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$29, iff $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$30 is weakly normal for the associated trigonometric system. In the nonlinear theory, existence is not automatic, and the paper proves by example that nonlinear Hermite–Chebyshev approximations may differ from the linear ones (Starovoitov et al., 21 Jul 2025).

A further contemporary issue is robustness under noisy data in multipoint interpolation. For exact interpolation data, rational Chebyshev multipoint Padé approximants converge locally uniformly under the stated hypothesis

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$31

With noisy data,

$\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$32

the observed behavior includes spurious pole-zero pairs, or Froissart doublets. The reported numerical evidence is that the breakdown order $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$33 grows roughly like $\int_{-1}^{1} T_n(z)T_m(z)\,w(z)\,dz= \begin{cases} \pi, & n=m=0,\[4pt] \frac{\pi}{2}\delta_{nm}, & \text{otherwise}, \end{cases} \qquad w(z)=(1-z^2)^{-1/2},$34, while unlike the series case, the slope of this logarithmic growth depends on the distribution of interpolation nodes. Symmetric node arrangements on a circle are reported as comparatively more robust to noise (Derevyagin et al., 11 Jun 2026).

Taken together, these strands show that rational Chebyshev basis functions are best understood as a family of approximation frameworks rather than a single basis in a narrow sense. They preserve the Chebyshev emphasis on uniform approximation, extremality, and structure-preserving recurrences, while extending the basis concept to rational interpolation, mapped unbounded domains, multipoint Padé theory, and generalized optimization models with continuous non-monomial basis functions.