Discrete Trigonometric B-Splines

• Discrete Trigonometric B-Splines are spline constructions that replace polynomial pieces with trigonometric functions over discrete knots, ensuring local support and prescribed smoothness.
• They are formulated in multiple settings including cubic collocation schemes, Fourier-based global representations, and h-trigonometric analogues with discrete calculus properties.
• Their applications in numerical PDEs and geometric design yield sparse algebraic systems and preserve differential structure for enhanced computational efficiency.

Searching arXiv for recent and foundational papers on discrete trigonometric B-splines and related trigonometric spline theory. Discrete trigonometric B-splines are spline constructions in which trigonometric functions replace algebraic polynomial pieces or generators on a discrete knot set or a finite equidistant grid. In the literature, the term covers at least three closely related settings: cubic trigonometric B-spline bases used on uniform meshes in collocation schemes for differential equations; finite-grid trigonometric spline classes built from discrete Fourier coefficients, convergence factors, and interpolation kernels; and the explicitly discrete $h$-trigonometric B-splines defined through discrete analogues of exponential, sine, and cosine and an $h$-trigonometric divided difference (Nazir et al., 2015, Denysiuk et al., 27 Feb 2025, Zürnacı-Yetiş et al., 6 Aug 2025). Across these settings, the central themes are locality or grid-adapted representation, prescribed smoothness, and preservation of differential structure in a trigonometric rather than purely polynomial framework (Denysiuk, 2021).

1. Terminological scope and principal formulations

In numerical PDE papers, cubic trigonometric B-splines are introduced on a uniform partition of an interval and used as spatial trial functions in collocation. They are described as $C^2$ piecewise functions with local compact support, and an approximate solution is expanded as a linear combination of basis functions with time-dependent coefficients (Nazir et al., 2015). This usage is prominent in work on the Burgers, Kuramoto–Sivashinsky, Gardner, telegraph, and time-fractional Burgers equations (Dag et al., 2014, Hepson, 2016, Hepson et al., 2017, Yaseen et al., 2017).

A second formulation treats trigonometric splines as finite trigonometric Fourier series on discrete equidistant nodes. Here the spline basis is not necessarily piecewise local in the classical sense; instead, it is given by globally defined analytic expressions built from cosine and sine modes, modified by convergence factors and normalized so that they interpolate grid data exactly (Denysiuk et al., 27 Feb 2025). In this line of work, trigonometric Riemann B-splines and related kernels are periodic analogues of classical polynomial B-splines, and interpolation splines are represented as convolutions of B-splines with kernels carrying the data (Denysiuk et al., 2023).

A third formulation is explicitly discrete. The paper on $h$-trigonometric B-splines introduces discrete analogues of the exponential, sine, and cosine functions and then defines discrete analogues of trigonometric B-splines by means of a discrete trigonometric version of a non-polynomial divided difference (Zürnacı-Yetiş et al., 6 Aug 2025). This construction is not merely a sampling of continuous trigonometric splines; it is a discrete calculus analogue with its own recurrence relation, discrete derivative, and Marsden identities (Zürnacı-Yetiş et al., 6 Aug 2025).

2. Defining constructions

In the $h$-calculus setting, the basic discrete trigonometric functions are

$e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$

$$\cos_h x=\frac{e_h^{ix}+e_h^{-ix}}{2}, \qquad \sin_h x=\frac{e_h^{ix}-e_h^{-ix}}{2i}.$$

They satisfy the limiting relations

$$\lim_{h\to 0} e_h^x=e^x,\qquad \lim_{h\to 0}\sin_h x=\sin x,\qquad \lim_{h\to 0}\cos_h x=\cos x$$

(Zürnacı-Yetiş et al., 6 Aug 2025). For a knot sequence $\{x_i\}$ with $x_{j+m}>x_j$ and $h$0, the $h$1-trigonometric B-spline of order $h$2 is defined by

$h$3

and, for general $h$4,

$h$5

where $h$6 is the $h$7-trigonometric divided difference (Zürnacı-Yetiş et al., 6 Aug 2025).

The same paper derives a two-term recurrence relation,

$h$8

which mirrors the classical recurrence structure of polynomial and trigonometric B-splines (Zürnacı-Yetiş et al., 6 Aug 2025).

In cubic collocation formulations, the basis is presented in a more classical mesh-based form. On a uniform grid, cubic trigonometric B-spline basis functions are $h$9 and have local support. The approximate solution is written as

$C^2$0

for the telegraph equation, or as

$C^2$1

for coupled systems such as the Gardner equation, where $C^2$2 is chosen from the trigonometric cubic B-spline family (Nazir et al., 2015, Hepson et al., 2017). At grid nodes, the solution and its derivatives reduce to linear combinations of neighboring coefficients, which is the algebraic mechanism behind sparse linear systems in collocation (Nazir et al., 2015, Yaseen et al., 2017).

In the finite-Fourier formulation, a spline on the grid $C^2$3 with $C^2$4 is constructed from the discrete coefficients

$C^2$5

and modified generators $C^2$6, $C^2$7 normalized by $C^2$8, yielding

$C^2$9

(Denysiuk et al., 27 Feb 2025). This representation is analytic over the whole period rather than piecewise polynomial (Denysiuk et al., 27 Feb 2025).

3. Structural properties, smoothness, and normalization

Local support is a defining property in the mesh-based cubic constructions. In the time-fractional Burgers formulation, each basis function is nonzero over four consecutive subintervals, and at a grid point $h$0 only $h$1, $h$2, and $h$3 are nonzero (Yaseen et al., 2017). In the telegraph formulation, local compact support yields sparse tridiagonal systems after collocation (Nazir et al., 2015). The cubic trigonometric bases used there are twice differentiable, which explains both their utility and one of their main limitations: only first and second derivatives are directly available at the nodes (Nazir et al., 2015, Hepson, 2016).

Smoothness in the discrete Fourier-series-based class is governed by the decay of the convergence factors. When $h$4 decays as $h$5, the basis functions are $h$6-smooth, so by increasing $h$7 one obtains splines of increasing smoothness (Denysiuk et al., 27 Feb 2025). Denysiuk’s treatment emphasizes that trigonometric splines can be considered as discrete rows whose differential properties are stored, and argues that approximation and smoothing are expedient in the discrete formulation precisely because these differential properties are preserved (Denysiuk, 2021).

A recurrent misconception is that trigonometric B-splines automatically inherit all standard polynomial B-spline design properties. This is not true in the original unnormalized setting. The normalization study states that trigonometric and hyperbolic B-splines, in their original formulation, do not form a partition of unity and consequently do not admit the notion of control polygons with the convex hull property for design purposes (Speleers, 3 Aug 2025). For odd order $h$8, normalized trigonometric B-splines are defined by

$h$9

with weights chosen so that

$h$0

on the relevant span, thereby restoring partition of unity, nonnegativity, local support, and the convex hull property (Speleers, 3 Aug 2025).

The explicitly discrete $h$1-trigonometric formulation extends the algebraic structure further. It provides a two-term formula for the discrete derivative and two variants of the Marsden identity, including an $h$2-trigonometric Marsden identity of the form

$h$3

with $h$4 (Zürnacı-Yetiş et al., 6 Aug 2025). This is the discrete analogue of a central structural identity in classical spline theory.

4. Fourier and kernel viewpoints

On a uniform grid with $h$5 equidistant nodes, the finite trigonometric Fourier system

$h$6

satisfies discrete orthogonality relations under summation over the grid (Denysiuk et al., 27 Feb 2025). This provides the algebraic basis for interpolation, but the spline literature modifies the raw Fourier representation so that additional smoothness and interpolation properties are built into the basis functions themselves (Denysiuk et al., 27 Feb 2025, Denysiuk, 2020).

The globally defined periodic viewpoint is especially clear in the Riemann B-spline construction. Trigonometric Riemann B-splines are given by

$h$7

and are described as periodic analogues of classical polynomial B-splines; on $h$8 they coincide with polynomial B-splines of degree $h$9 (Denysiuk et al., 2023). The central theorem in that work states that interpolation trigonometric splines are convolutions of trigonometric B-splines with corresponding kernels (Denysiuk et al., 2023).

The kernel viewpoint becomes more general when the stitching grid and interpolation grid are allowed to differ. The literature introduces crosslink grids and interpolation grids, which may match or may not match, and this choice changes the resulting spline family (Denysiuk, 2020). A later generalization shows that for various combinations of stitching and interpolation grids, distinct trigonometric B-splines exist, and separates first-kind and second-kind decompositions according to whether grid dependence is carried by the kernel or the B-spline (Denysiuk et al., 2023).

The spectral-analysis literature adds a further discrete interpretation. It derives relations between discrete Fourier coefficients and Fourier coefficients of trigonometric splines and studies the overlay effect in the frequency domain. In that framework,

$e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$0

but aliasing leads to relations such as

$e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$1

(Denysiuk, 2019). The same work argues for taking into account the differential properties of signals that are losing during the primary sampling, which aligns with the discrete-series viewpoint that preserves differential structure (Denysiuk, 2019, Denysiuk, 2021).

5. Discretization of differential equations

A major application area is collocation for evolution equations. In the telegraph equation study, the cubic trigonometric B-spline basis functions are utilized as an interpolating function in the space dimension, time is discretized by a finite difference weighted scheme, and the resulting method is shown to be unconditionally stable for $e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$2 by the von Neumann method; because of local support, each time step leads to a sparse tridiagonal system that can be solved by the Thomas algorithm (Nazir et al., 2015).

For Burgers’ equation, the standard pattern is Crank–Nicolson in time and cubic trigonometric B-spline collocation in space. The nonlinear term is linearized by

$e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$3

producing a tridiagonal linear system after substitution of the nodal spline formulas (Dag et al., 2014). The Kuramoto–Sivashinsky equation requires an additional reduction step: because only first and second order derivatives of the trigonometric cubic B-splines are available at the nodes, the fourth-order equation is converted to a coupled system by introducing $e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$4, after which a Crank–Nicolson discretization is applied (Hepson, 2016). The Gardner equation exhibits the same structural issue in third order: because cubic polynomial and trigonometric B-splines lack continuous third derivatives, the term $e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$5 is reduced by introducing $e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$6, leading to a coupled system and a collocation–Crank–Nicolson scheme whose stability is studied by Von Neumann analysis and whose accuracy is measured by the discrete maximum norm and by the relative absolute changes of three conservation laws (Hepson et al., 2017).

The time-fractional Burgers equation extends the same philosophy to Caputo time derivatives. There the Caputo derivative is approximated by a finite difference formula, the nonlinear advection term is replaced by the linearization

$e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$7

and the resulting method is proved to be globally unconditionally stable, with an error bound

$e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$8

under the stated smoothness assumptions (Yaseen et al., 2017). Across these PDE applications, discrete trigonometric B-splines function as a spatial discretization mechanism that couples local smoothness with sparse algebraic structure.

6. Geometric generalizations and broader spline frameworks

The geometric-design literature emphasizes the consequences of normalization and high-order smoothness. For odd order $e_h^x=(1+h)^{x/h}, \qquad e_h^{ix}=(1+h)^{ix/h},$9, normalized trigonometric B-splines can represent a full circle exactly,

$$\cos_h x=\frac{e_h^{ix}+e_h^{-ix}}{2}, \qquad \sin_h x=\frac{e_h^{ix}-e_h^{-ix}}{2i}.$$0

using a uniform knot vector and a regular polygon of control points; the construction allows a variable number of control points and yields $$\cos_h x=\frac{e_h^{ix}+e_h^{-ix}}{2}, \qquad \sin_h x=\frac{e_h^{ix}-e_h^{-ix}}{2i}.$$1 smoothness (Speleers, 3 Aug 2025). This is presented as an example application of normalization rather than as an incidental corollary.

A broader theoretical framework is provided by Tchebycheffian B-splines. Trigonometric B-splines appear there as a special case in ECT-spaces such as

$$\cos_h x=\frac{e_h^{ix}+e_h^{-ix}}{2}, \qquad \sin_h x=\frac{e_h^{ix}-e_h^{-ix}}{2i}.$$2

subject to critical-length restrictions for existence of a TB-spline basis (Raval et al., 2022). Under suitable assumptions, TB-splines retain the classical properties of non-negativity, local support, partition of unity, local linear independence, and boundary interpolation, and they are proposed as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods (Raval et al., 2022). This places discrete and trigonometric B-spline theory within a larger non-polynomial spline program.

Another point of ambiguity concerns representation. Some trigonometric spline systems are piecewise local bases, but others are single analytic expressions over the whole period; the literature explicitly contrasts representation by coefficients of the interpolation trigonometric polynomial, by trigonometric B-splines, and by fundamental trigonometric splines, and generalizes the first and third forms to non-periodic functions (Denysiuk et al., 2024). A plausible implication is that “discrete trigonometric B-spline” should be read as a family name rather than a single universally standardized object.

The current state of the subject therefore combines discrete calculus, finite Fourier analysis, collocation technology, normalization theory, and generalized spline spaces. The available results show that many of the standard results for classical polynomial B-splines extend naturally both to trigonometric B-splines and to discrete trigonometric B-splines (Zürnacı-Yetiş et al., 6 Aug 2025), while the class of trigonometric splines also suggests generalizations in several directions and certainly requires further research (Denysiuk et al., 27 Feb 2025).