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Interpolations: a linear algebra approach

This paper introduces a unified linear-algebraic framework using dual spaces to establish general existence and uniqueness theorems for various interpolation problems, deriving explicit dual basis formulas that enable efficient construction of interpolants for applications ranging from Lagrange polynomials and Hermite splines to trigonometric interpolation and heterogeneous sensor data.

Original authors: Andronick Arutyunov, Nikolay Korgin

Published 2026-06-23✓ Author reviewed ⓘ

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andronick Arutyunov, Nikolay Korgin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to draw a smooth line connecting a series of dots on a piece of paper. Sometimes you just know the location of the dots. Other times, you might know the location of a dot and the direction the line is going when it hits that dot (like a car's speed), or even how fast that direction is changing (like a car accelerating).

This paper is about creating a single, universal "rulebook" for drawing these lines, no matter what kind of information you have or what kind of line you want to draw.

Here is the breakdown of their approach using simple analogies:

1. The Problem: Too Many Different Tools

Usually, if you want to connect dots, you use one tool (like a ruler for straight lines). If you want to connect dots with curves that also match a specific speed, you use a different, more complex tool (like a spline). If you are dealing with waves (like sound or tides), you use a third tool (trigonometry).

The authors say this is messy. It's like having a different key for every single door in a house. They want one "Master Key" that opens every door.

2. The Solution: The "Dual Space" (The Question and Answer Game)

The authors use a concept from linear algebra called Dual Spaces. Think of it as a game of "Question and Answer":

The Function Space (The Answers): Imagine a box full of different types of lines (polynomials, waves, etc.). These are the potential answers.
The Dual Space (The Questions): Your data points are actually "questions" you ask the lines.
- "What is your height at point A?" (Value)
- "What is your slope at point B?" (Velocity/First Derivative)
- "How fast is your slope changing at point C?" (Acceleration/Second Derivative)

The paper argues that if you have exactly as many "Questions" as you have "Degrees of Freedom" in your box of lines, there is exactly one perfect line that answers all your questions correctly.

3. The Magic Trick: The "Dual Basis"

Once you have your set of questions, the authors show you how to build a special set of "Answer Keys" called a Dual Basis.

Think of these keys as "Master Switches."

Switch 1 is designed so that if you turn it on, it answers "Yes" to Question 1 and "No" to everything else.
Switch 2 answers "Yes" to Question 2 and "No" to the rest.
And so on.

To draw your final line, you don't need to solve a giant, complicated math puzzle every time. You just take your specific data values (the answers you want) and mix these "Master Switches" together.

Formula: Final Line = (Value at A × Switch 1) + (Speed at B × Switch 2) + ...

This makes the process fast and predictable. You calculate the "Switches" once, and then you can plug in any new data instantly.

4. What They Actually Did

The paper applies this "Master Key" idea to three specific scenarios:

Standard Polynomials (The Smooth Curves): They showed how to build these "Switches" for standard curves (like Cubic Hermite splines) where you know the position and speed at two points. They even gave the exact formulas for 5th-degree curves (which handle position, speed, and acceleration).
Trigonometric Polynomials (The Waves): They applied the same logic to wave-like functions (sines and cosines). This is useful for things that repeat, like seasons or rotating machinery. They found the "Switches" for these waves and noted a catch: if your points are too far apart (specifically, a full circle apart), the math breaks down, just like a wave that repeats itself perfectly might confuse the system.
Mixed Sensors (The "Heterogeneous" Scenario): This is a practical example they tackled. Imagine a robot car where one sensor tells you the position at time 1, a different sensor tells you the speed at time 2, and a third tells you the acceleration at time 3. These sensors might not be perfectly synchronized. The authors created a "Switch" system that can handle this messy, mixed-up data without needing to force the sensors to agree on the exact same time.

5. The Bottom Line

The paper doesn't claim to have invented a new type of car or a new medical device. Instead, it claims to have provided a unified mathematical framework.

It says: "Stop treating every interpolation problem as a unique, isolated puzzle. Instead, view them all as a system of questions and answers. If you set up your 'Dual Basis' (your Master Switches) correctly, you can solve for any combination of values, speeds, or accelerations, whether you are using straight lines, curves, or waves, with a single, consistent method."

They demonstrated this works for:

Planning robot paths (kinematics).
Handling data from different, unsynchronized sensors.
Fitting waves to periodic data.

The core takeaway is efficiency and universality: One method, one set of rules, applied to many different types of data.

Technical Summary: Interpolations: a linear algebra approach

Problem Statement
The paper addresses the fragmentation often found in applied interpolation problems, where methods such as Lagrange polynomials, Hermite splines, and trigonometric interpolation are treated as disconnected collections of formulas. In practical scenarios, such as reconstructing trajectories for moving objects or fusing data from heterogeneous sensors (e.g., measuring position, velocity, and acceleration at distinct, unsynchronized points), practitioners must frequently switch between schemes or preprocess data in ad-hoc ways. This paper proposes a unified framework to resolve these issues by viewing interpolation not as a collection of specific formulas, but as a systematic linear algebra problem involving dual spaces.

Methodology
The core methodology relies on the theory of finite-dimensional vector spaces and their duals. The authors formalize the concept of a "pointwise interpolation scheme," defined by a set of points $\{t_i\}$ and a corresponding set of linear functionals (evaluations of the function or its derivatives) acting on a chosen function space $F$ (e.g., polynomials $\mathbb{R}[t]$ or trigonometric polynomials $\mathbb{T}[t]$ ).

Dual Space Formulation: Interpolation conditions are treated as linear functionals in the dual space $F^*$ . The existence and uniqueness of an interpolant are established by ensuring that the set of functionals forms a basis for $F^*$ .
Dual Basis Construction: By fixing a basis in the dual space corresponding to the interpolation conditions, the authors derive a corresponding dual basis in the primal space $F$ .
Explicit Construction: The interpolating function is constructed as a linear combination of the dual basis functions, where the coefficients are exactly the prescribed data values (function values, derivatives, etc.).
Matrix Representation: The relationship between a chosen basis of $F$ (e.g., monomials) and the dual basis is encoded in a matrix $A_{\varepsilon, v}$ (often a confluent Vandermonde matrix). The coefficients of the dual basis polynomials are obtained by inverting this matrix.

Key Contributions and Results

Unified Framework: The paper demonstrates that Lagrange interpolation, Hermite splines, and trigonometric interpolation are manifestations of the same underlying linear algebraic structure. The choice of function space (polynomial vs. trigonometric) and the specific scheme (values only vs. values and derivatives) are decoupled from the analysis of existence and uniqueness.
General Existence and Uniqueness Theorem: Theorem 2.10 establishes that if the dimension of the function space equals the number of interpolation constraints and the associated linear functionals form a basis of the dual space, a unique solution exists.
Explicit Dual Basis Formulas:
- Polynomial Case: The authors derive explicit formulas for dual basis polynomials for Hermite splines of orders $n=0$ (linear), $n=1$ (cubic), and $n=2$ (quintic). They show that for a scheme involving $n$ derivatives at two points, the minimal degree polynomial required is $2n+1$ .
- Trigonometric Case: The framework is extended to trigonometric polynomials. The paper provides closed-form expressions for dual basis functions in the space $T_2[t]$ (spanned by $\sin t, \sin 2t, \cos t, \cos 2t$ ) for a scheme involving values and first derivatives at two points. The authors analyze degeneracy conditions, noting that the determinant of the system vanishes when the grid step is an integer multiple of $2\pi$ .
- General Formula: Theorem 5.1 provides a general matrix-based formula for computing the dual basis for any interpolation scheme, applicable to both polynomial and trigonometric spaces.
Heterogeneous Sensor Interpolation: The paper proposes a specific interpolation scheme for scenarios where different sensors provide different types of data (e.g., position at $t_1, t_2$ , velocity at $t_3$ , and acceleration at $t_4$ ) without requiring synchronization. A unique cubic polynomial solution is derived for this mixed-constraint scheme, with explicit conditions for non-degeneracy.
Affine Structure of Solutions: For cases where the polynomial degree exceeds the minimum required, the paper characterizes the set of all solutions as an affine space, with the minimal-degree solution serving as the particular solution.

Significance and Claims
The authors claim that this approach offers a "systematic, dimension-aware" method that decouples the analysis of interpolation conditions from the choice of basis functions. The primary significance lies in:

Efficiency: Once the dual basis is computed for a specific scheme, constructing the interpolant for any new set of data values is reduced to a simple linear combination, avoiding the need to re-solve the system of equations.
Versatility: The framework naturally handles complex, non-standard interpolation problems (such as those with heterogeneous sensor data) that do not fit traditional textbook definitions.
Physical Interpretation: The paper suggests that this method is particularly useful for trajectory planning and kinematics, where one can analyze the limiting values of derivatives (velocity, acceleration) to ensure they adhere to physical constraints of the modeled object.

The paper does not claim to introduce new interpolation algorithms in the traditional sense but rather provides a rigorous, unifying theoretical lens that simplifies the derivation and analysis of existing and novel interpolation schemes. The results are demonstrated through examples in kinematics and periodic function interpolation, suggesting applicability in trajectory planning, signal processing, and sensor fusion.

1. The Problem: Too Many Different Tools

2. The Solution: The "Dual Space" (The Question and Answer Game)

3. The Magic Trick: The "Dual Basis"

4. What They Actually Did

5. The Bottom Line

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