Approximation theory studies how to replace a mathematical object with a simpler one while controlling what is lost. For functions, that usually means choosing a manageable family of curves and measuring how closely one of them represents a target over a declared domain. The gold standard is an approximation whose error is meaningful for the intended use, whose assumptions are visible and whose claimed accuracy survives an independent check.
This guide connects interpolation, least squares, minimax approximation, polynomial bases, rational functions and local models. Its worked cases show why two correct methods can produce different answers, why a small residual need not mean a small error, and why accurate function values do not automatically give accurate derivatives or decisions. The examples are constructed teaching problems. Use the Gold Standard master for the wider collection and the Education parent for the goal of learning that transfers into unfamiliar situations.
- Specify the target, domain, permitted approximation family and error measure before choosing an algorithm.
- Distinguish matching observations from controlling error across an interval or predicting new observations.
- Compare methods using the same information, computational budget and intended output.
- Separate representation error, data uncertainty and numerical calculation error.
- Test the operation that actually matters, including derivatives, integrals, thresholds or extrapolation when relevant.
Choose your approximation theory route
Chapters 1–3: Define the problem and measure error
Chapters 4–6: Compare objectives and representations
Chapters 7–9: Understand convergence and stability
Chapters 10–12: Test evidence and practical consequences
Chapter index
Define the problem and measure error · Chapters 1–3
1. What is being approximated?
Compare objectives and representations · Chapters 4–6
4. Interpolation is a constraint, not an error certificate
Understand convergence and stability · Chapters 7–9
7. Local approximation and the benefit of subdivision
Test evidence and practical consequences · Chapters 10–12
10. Sampling, noise and unsupported detail
Transfer and continue learning · Chapters 13–14
CHAPTER 1 OF 14 · Define the problem and measure error
1. What is being approximated?
For estimation, rounding, significant figures and introductory numerical methods, use How Mathematical Approximation Works. This guide concentrates on the mathematical theory of approximating functions: choosing a function space, defining an error norm, finding a best member and understanding convergence. Approximation algorithms for discrete optimization form another related field, with different performance guarantees.
An approximation is a substitute chosen for a purpose. A map approximates geography, a polynomial can approximate a function, and a finite set of basis functions can approximate a spatial field. The mathematical question is not whether the substitute is identical to the original. It is whether the difference is controlled in the way the task requires.
For a function problem, begin with f, the target, and a set V of permitted approximants. If V is the set of polynomials of degree at most two, every candidate has the form a+bx+cx². If V consists of piecewise linear functions on a fixed grid, the candidates have different freedom: separate slopes on adjoining intervals, usually joined continuously.
Four decisions define the problem
The domain tells you where accuracy is requested. The family tells you what forms are allowed. The error measure tells you which discrepancies matter. The information available tells you what you can actually construct or verify. Changing any of these decisions can change the answer even when the target formula stays the same.
Approximating a known formula is different from fitting measurements of an unknown process. In the first case, you may be able to evaluate f at additional points to test a candidate. In the second, measurement noise, sampling limitations and model assumptions affect what “the target” means. A good fit to observed data may still be a poor description of the underlying process.
Also distinguish an exact mathematical best approximation from the result of a finite numerical algorithm. The best member of V may be defined by an optimization problem. A computed result might only approximately solve that problem because of rounding, stopping tolerances or imperfect searches for error extrema.
Write a concrete task statement before selecting a method: “Replace this function on [0,1] by a quadratic with maximum absolute error below 0.01.” That statement is much more informative than “fit a good curve.” It gives a reader something to verify and gives you a reason to reject a visually attractive answer that fails the actual requirement.
CHAPTER 2 OF 14 · Define the problem and measure error
2. Error measures change what best means
Let e(x) = f(x)−p(x). A maximum-error measure considers the largest magnitude of e over the domain. A squared-error measure averages or integrates e². An absolute-error measure averages or integrates |e|. These measures reward different compromises, so their minimizers need not agree.
Consider three observed values: zero, zero and nine. Approximate all three by a single constant c. Minimizing the sum of squared discrepancies gives c = 3, their mean. The squared error is then 9+9+36 = 54. Choosing c = 0 instead gives 81, so the mean wins under this objective.
For the sum of absolute discrepancies, c = 0 gives total error nine, whereas c = 3 gives twelve. A median minimizes this objective. For the maximum absolute discrepancy, balance the extremes zero and nine: c = 4.5 gives maximum error 4.5. No constant can do better because it must remain within that distance of both extremes.
None of these answers is a contradiction
The mean, median and midpoint of the extreme values answer different questions. If a rare large discrepancy is especially costly, the maximum-error objective may fit the purpose. If the task genuinely values average squared discrepancy, the least-squares answer may be appropriate. If robustness to isolated extreme observations matters, an absolute-error objective can be useful, though the reason for those extremes still deserves investigation.
Units also matter. Squared error has squared output units; its square root returns to the original units. Relative error divides by a target magnitude and can become unstable or undefined near zero. A mixed criterion, combining an absolute allowance with a relative allowance, often expresses practical tolerances more clearly than either alone.
Never report only “the error” without stating its definition. A root-mean-square error of 0.01 does not imply every point is within 0.01. Conversely, a maximum-error bound may be conservative for most of the domain. The usefulness of a number depends on the question it answers, not only on whether it is small.
CHAPTER 3 OF 14 · Define the problem and measure error
3. A continuous example with two correct answers
Approximate f(x) = x² on [−1,1] by a constant c. First minimize the integrated squared error J(c) = ∫(x²−c)² dx over that interval. Expanding and integrating gives J(c) = 2/5−4c/3+2c². Differentiating with respect to c gives J′(c) = −4/3+4c, so the minimum occurs at c = 1/3.
The second derivative is four, confirming a strict minimum. Substitution gives J(1/3) = 8/45. Dividing by the interval length two gives mean squared error 4/45. Its square root is approximately 0.29814. The maximum absolute error of this same constant is 2/3, reached at the endpoints.
Now minimize the worst discrepancy
Over [−1,1], x² ranges from zero to one. To minimize the maximum distance from a constant to every value in that range, choose c = 1/2. Its error is −1/2 at zero and +1/2 at both endpoints. The maximum absolute error is therefore 1/2, smaller than the 2/3 produced by the least-squares constant.
But J(1/2) = 7/30, larger than 8/45. The minimax constant improves the worst error while sacrificing integrated squared accuracy. You cannot compare the two constants fairly by silently switching objectives after seeing the answers.
This example is also a warning about sampling. If you approximate the integral with only a few sample points, their placement and weights define a discrete problem. Sampling only the endpoints would suggest c = 1, which is a poor representative of the whole interval. More samples do not automatically repair a biased distribution of samples.
The Remez exchange and minimax article develops the worst-error viewpoint. The present example gives a direct reason for needing that viewpoint: the approximation that best balances typical squared discrepancies can leave a larger worst-case discrepancy than the application permits.
CHAPTER 4 OF 14 · Compare objectives and representations
4. Interpolation is a constraint, not an error certificate
Interpolation requires a candidate to reproduce specified values exactly, at least in ideal arithmetic. For distinct sample inputs, a polynomial of sufficiently high permitted degree can satisfy those conditions. What happens between the inputs is a separate question. Zero error at the nodes does not imply zero error everywhere.
Take f(x) = x² on [0,1] and interpolate at the endpoints. The straight line through (0,0) and (1,1) is p(x) = x. At x = 1/2 the error f−p is −1/4. The data conditions are satisfied perfectly, yet the largest absolute error over the interval is 1/4.
Now use the same linear family but minimize maximum absolute error. The candidate q(x) = x−1/8 has error x²−x+1/8. It takes values +1/8 at zero and one, and −1/8 at one half. Its maximum error is 1/8. Allowing small endpoint errors has halved the worst error across the interval.
Why the second line cannot be uniformly improved
Suppose another line improved strictly on 1/8 at all three listed points. Relative to q, it would need to move upward at both endpoints, because f−q is positive there, and downward at the midpoint, where f−q is negative. But a line’s midpoint value is the average of its endpoint values. That required pattern is impossible. This elementary argument establishes the optimality of q for this example.
Interpolation is valuable when exact reproduction of trusted samples is the appropriate contract. It may be inappropriate when samples contain noise or when the real goal is uniform accuracy. Neither interpolation nor approximation should be chosen merely because its plot looks smoother.
For noisy data, ask whether discrepancies reflect measurement error, unresolved dynamics or incorrect assumptions. Exact reproduction can reproduce those errors. For an exact function sampled deliberately, ask whether the nodes cover difficult regions and whether independent points confirm the behaviour between them. The construction conditions and the validation conditions should not be confused.
CHAPTER 5 OF 14 · Compare objectives and representations
5. Least squares as a geometric projection
A least-squares approximation in a linear space chooses coefficients so that the residual is orthogonal to the allowed directions, using the inner product that defines the problem. In a continuous problem that inner product may be an integral. In a data-fitting problem it may be a weighted sum over observations.
Approximate x² on [0,1] by a+bx in the integrated squared sense. Orthogonality to one gives ∫(x²−a−bx) dx = 0, or 1/3−a−b/2 = 0. Orthogonality to x gives 1/4−a/2−b/3 = 0. Solving yields b = 1 and a = −1/6.
Thus the least-squares line is p(x) = x−1/6. Its endpoint errors are +1/6; its midpoint error is −1/12. These differ from the minimax line’s balanced ±1/8. Direct integration gives squared error 1/180 for the least-squares line, compared with 7/960 for the minimax line. Each wins its own objective.
The discrete matrix version
For observations yᵢ at inputs xᵢ, construct a matrix with columns equal to the chosen basis functions evaluated at those inputs. With a linear model, the columns might be one and x. The coefficient vector minimizes the norm of A c−y. The residual is orthogonal to the column space for an unconstrained ordinary least-squares solution.
The NumPy least-squares reference documents a numerical interface for this problem, including rank and singular-value information. Those outputs matter when different coefficient combinations produce nearly indistinguishable sampled values. A solver returning numbers does not prove that the data determine those numbers robustly.
Weights change the geometry. Giving an observation a larger weight makes its residual more costly, but weights should express justified priorities or uncertainty assumptions. They should not be tuned merely to make an inconvenient point disappear. If you change weights, you have changed the approximation problem and should reevaluate the result under the intended application criterion.
CHAPTER 6 OF 14 · Compare objectives and representations
6. Choosing a representation that fits the structure
Global polynomials, rational functions, trigonometric functions and piecewise polynomials each describe different kinds of structure efficiently. The choice is not a competition for the most sophisticated name. It is a question of what family can represent the relevant behaviour economically and reliably.
A polynomial can represent x² exactly with three coefficient slots. A rational function can represent 1/(1+x) exactly with a constant numerator and linear denominator. Forcing a polynomial to mimic that reciprocal function over a wide interval may require more terms. Conversely, a rational denominator adds opportunities for poles that a polynomial does not have.
The Rational Interpolation root explains how denominator behaviour must be checked. A fitted pole is not automatically a physical singularity, and a good sample fit does not establish that the quotient remains well behaved throughout the operating interval.
Basis and family are different choices
Writing a degree-n polynomial in a Chebyshev basis does not change the set of degree-n polynomials available. It changes the coordinates used to describe them. That can improve scaling and numerical handling, but it does not add information to the data or remove the need to choose an error objective.
The Chebyshev Approximation root connects basis choice to cosine-spaced sampling and recurrence-based evaluation. Use it for that specific construction. Here the broader principle is that representation and algorithm should support the mathematical objective rather than obscure it.
Periodic functions may suit trigonometric representations. Localised changes may suit piecewise descriptions. Known symmetry can reduce the number of needed terms. Known positivity or monotonicity may require constrained approximation, because a generic fit need not preserve those properties between samples.
Use structure only when it is justified. Imposing even symmetry on an asymmetric process can produce a neat, compact and systematically wrong model. A representation is useful because it encodes the right restrictions, not because restrictions are inherently beneficial.
CHAPTER 7 OF 14 · Understand convergence and stability
7. Local approximation and the benefit of subdivision
A global polynomial uses one expression across the entire domain. A piecewise model divides the domain and assigns a simpler expression to each part. Subdivision can concentrate effort where the function changes rapidly and avoid making every region pay for one difficult feature.
For f(x) = x² on a small interval [a,a+h], let p be the straight line joining the endpoint values. Direct algebra gives p(x)−x² = (x−a)(a+h−x). The error is nonnegative and reaches h²/4 at the midpoint. The formula is exact for this target.
On [0,1], one segment has maximum error 1/4. Two equal segments have h = 1/2 and maximum error 1/16. Four equal segments have maximum error 1/64. Halving segment width reduces the maximum error by a factor of four in this example. This is a concrete convergence pattern, not merely a smoother-looking picture.
Continuity requirements affect the model
The piecewise linear interpolant is continuous at the shared nodes, but its slopes generally jump. If the application needs smooth derivatives, a different construction is required. Cubic splines use piecewise cubics with continuity conditions and additional endpoint conditions to determine the pieces.
The official SciPy CubicSpline documentation specifies a twice continuously differentiable interpolant and explains endpoint choices such as natural, clamped and periodic conditions. These conditions are part of the mathematical model. They should not be treated as cosmetic switches with no effect on the result.
For a general sufficiently differentiable function, curvature affects the error of linear interpolation. Regions of larger curvature may need smaller intervals. However, an adaptive method must estimate that difficulty from reliable information; noisy finite differences can suggest false curvature.
The finite-element analysis article extends local approximation to fields over meshes. Its lesson applies beyond engineering: increasing local resolution is meaningful only when you also check which quantities converge and whether the governing model remains appropriate.
CHAPTER 8 OF 14 · Understand convergence and stability
8. Convergence is a statement with conditions
To say that approximations converge is to specify what approaches what, in which sense, as some resource increases. The resource might be degree, number of segments, sample count or basis size. The sense might be pointwise convergence, integrated squared convergence or uniform convergence.
Pointwise convergence means the error tends to zero at each fixed input. Uniform convergence means one error bound eventually works across the whole domain. The second statement is stronger. A moving narrow error peak can shrink in width while remaining large in height, making some aggregate errors small without reducing the maximum error comparably.
A concrete narrow-peak calculation
On [0,1], define eₕ(x) = max(1−x/h,0) for 0 < h ≤ 1. Its maximum is one for every h. Its integrated squared size is ∫₀ʰ(1−x/h)² dx = h/3. Thus its L² norm is √(h/3), which tends to zero as h shrinks, even though its maximum never decreases.
This example shows why integrated accuracy cannot automatically stand in for worst-case accuracy. It also shows why a coarse validation grid can miss a narrow feature. Whether that feature matters depends on the application, but it must not disappear from the reasoning simply because an average looks good.
Regularity affects convergence rates. Smooth targets often admit more efficient approximation than targets with corners or jumps. A continuous polynomial cannot converge uniformly on a closed interval to a function with a jump, because a uniform limit of continuous functions is continuous. Increasing degree cannot defeat that logical restriction.
When observing convergence numerically, compare errors under controlled changes. Keep the target, domain, units and error measure fixed. If you change the test grid along with the model, make sure the comparison remains meaningful. Report an observed rate as an observation over the tested range unless a theorem with verified assumptions supports a stronger claim. A few decreasing numbers do not establish the eventual behaviour at arbitrarily high resolution.
CHAPTER 9 OF 14 · Understand convergence and stability
9. Best approximation and stable computation
The best approximation in a family and the output of an algorithm are different objects. Let v* be a best approximant to f in a chosen norm, and let v̂ be the computed result. The triangle inequality gives ||f−v̂|| ≤ ||f−v*|| + ||v*−v̂||. The first term concerns what the family can achieve; the second concerns how closely the computation reaches it.
This separation helps diagnosis. If the family cannot represent a sharp feature at the chosen resolution, a more accurate linear solver will not remove that limitation. If the coefficient system is poorly conditioned, increasing the family size may make the calculation less reliable even while the theoretical best error decreases.
A stable representation matters
Suppose two basis functions are almost identical on every sampled input. Large changes in their coefficients can cancel, leaving nearly the same sampled curve. The observations then determine certain combinations of coefficients much more clearly than the individual coefficients. A small residual does not prove stable coefficient recovery.
Scale the input domain, inspect rank information and use appropriate numerical solvers. Orthogonal or well-scaled bases can help. Avoid forming an explicit inverse merely because the algebra can be written that way. A numerical method should preserve useful structure and expose uncertainty rather than bury it inside large cancelling coefficients.
Input uncertainty also matters. Even exact arithmetic cannot make a highly sensitive problem insensitive to uncertain observations. Repeating the calculation at higher precision can diagnose rounding problems, but it cannot create information absent from the input data.
For a practical test, perturb observations within justified uncertainty bounds and examine the changes in predictions over the intended domain. Large coefficient changes with small prediction changes may be acceptable for some uses; large prediction changes are a more direct concern. If the coefficients themselves have physical interpretations, their stability becomes part of the required output. Always distinguish stability of the fitted curve from stability of a particular parameterization.
CHAPTER 10 OF 14 · Test evidence and practical consequences
10. Sampling, noise and unsupported detail
Finitely many samples cannot identify an unrestricted function. Given any interpolant through sample inputs x₁,…,xₙ, add K times the product (x−x₁)…(x−xₙ). The modified function agrees at every sample input for any constant K, yet can differ greatly elsewhere. Information between samples requires assumptions or additional observations.
That elementary construction is a useful antidote to overconfidence. A dense dataset may provide strong evidence under reasonable smoothness assumptions, but exact agreement with a finite table alone does not prove uniform accuracy. The function family and sampling design carry part of the claim.
More freedom can fit the noise
Suppose measurements are yᵢ = f(xᵢ)+ηᵢ, with ηᵢ representing measurement error. A highly flexible interpolant can reproduce every ηᵢ as if it were a real feature of f. Reducing the training residual may therefore worsen predictions at new inputs.
A lower-degree fit or a smoothing penalty can trade exact reproduction for resistance to noise. That trade needs a stated objective. If a method penalizes roughness, explain what form of roughness is penalized and how the penalty strength is selected. The resulting curve is not simply “the data made smooth”; it reflects an additional preference.
Validation data should represent the intended use. Randomly withholding points from a strongly correlated time series may give an easier prediction task than forecasting a future interval. Likewise, a test set concentrated in a flat region says little about a sharp transition elsewhere.
Keep data used for construction separate from data used to select settings and data reserved for final assessment. If the same checks are repeatedly used to tune the model, acknowledge that they have become part of model selection. The honest question is not whether the procedure ever saw a point labelled “test,” but whether the final evidence was independent of the choices being judged.
When additional data are expensive, choose checks where competing approximations disagree or where consequences are greatest. Those checks can be more informative than adding many observations where every candidate already behaves similarly.
CHAPTER 11 OF 14 · Test evidence and practical consequences
11. An error budget that follows the actual task
A practical approximation can accumulate several errors: a simplified physical model, uncertain data, a restricted approximation family, incomplete numerical optimization and finite-precision evaluation. These errors should not be automatically added as if they were independent random variables. Their relationships determine the appropriate combination rule.
If valid absolute bounds apply to successive differences, the triangle inequality gives a conservative sum. For example, suppose a target-to-model bound is 0.02, the model-to-approximant bound is 0.005 and the evaluation bound is 0.001, all in the same output units and throughout the same domain. A total bound of 0.026 follows. A root-sum-square combination would require a different probabilistic justification.
Check the quantity being consumed
Uniform value error E over an interval of length L implies integral error at most LE for ordinary integrable functions. But value accuracy alone does not control derivatives. Consider e(x) = ε sin(kx): its magnitude never exceeds ε, while its derivative has magnitude up to εk. Taking k large makes derivative error large without increasing the value error.
Root locations are also sensitive to slope. Near a simple root, a local estimate often divides value error by the magnitude of the target slope. Near a flat or multiple root, that reasoning can break down or predict much larger location uncertainty. A threshold decision made near the tolerance boundary should reflect that uncertainty.
If the application cares about a maximum, a derivative, a crossing time or a ranking, validate that output directly when possible. A small average error in the original curve is evidence about a different quantity. It may support the decision, but it does not automatically certify it.
Record whether each contribution is a theorem-backed bound, a measured discrepancy, a statistical uncertainty estimate or an engineering allowance. These categories can coexist in one project, but presenting them all as interchangeable “error bars” conceals their different meanings. A clear budget explains what remains uncertain and which additional test would reduce that uncertainty most effectively.
CHAPTER 12 OF 14 · Test evidence and practical consequences
12. A repair sequence for a disappointing approximation
Begin with the contract. Confirm the target, units, domain and desired output. Many apparent approximation failures are actually input-coordinate mistakes or mismatched objectives. A method optimized for average squared error should not be called defective merely because another method achieves a smaller maximum error.
Then verify simple cases. Constants, linear functions and one known basis function can reveal coefficient-order errors, normalization errors and faulty interval maps. Check construction residuals, but do not stop there. Compare at independent points, including boundaries and regions of rapid variation.
Let the error pattern guide the next change
If the discrepancy is spread smoothly across the domain, increasing degree or changing the family may help. If it is localized near a corner, subdivision may be more efficient. If errors grow outside the sampled range, the problem may be unsupported extrapolation. If predictions change greatly under small data perturbations, investigate conditioning or excessive flexibility.
If increasing resolution improves one quantity but worsens another, examine the downstream operation. Derivatives may expose noise that function values conceal. Peak values may be sensitive to local sampling. A fitted curve can become visually smoother while a scientifically relevant feature becomes less accurate.
Do not repair a failure only by removing the evidence that revealed it. Deleting an inconvenient observation, clipping an oscillation or shortening the displayed domain may sometimes be justified, but each changes the problem and requires an explanation grounded in the data or intended use.
After a change, repeat both the failed test and the previously successful checks. Preserve the earlier result long enough to make the comparison reproducible. Use a fixed independent validation set where appropriate, while recognizing that repeated tuning against it gradually reduces its independence.
The How X Works hub provides routes into applications where these choices become concrete. Whether approximating a curve, a field or a computational response, the repair should address the observed mechanism of failure. More computation is useful when it supplies the missing resolution or accuracy; otherwise it can make an unjustified answer look more precise.
CHAPTER 13 OF 14 · Transfer and continue learning
13. Changed-context transfer: one curve, three jobs
Consider the constructed response f(x) = x² on [0,2], and suppose only a straight-line surrogate is permitted. First, the job is to match the endpoint values. The interpolating line is p(x) = 2x. At x = 1 it predicts two instead of one, and its maximum absolute error is one.
Second, the job is to minimize maximum absolute error across the interval. Scaling the example from chapter four gives q(x) = 2x−1/2. Its errors are +1/2 at zero and two, and −1/2 at one. The maximum is therefore one half. It deliberately misses both endpoints to improve the worst discrepancy.
Third, the job is to minimize integrated squared error. Scaling the least-squares calculation gives r(x) = 2x−2/3. Its maximum absolute error is 2/3. It is neither the endpoint interpolant nor the minimax line. The task, not the target formula alone, determines the appropriate coefficients.
Change the output requirement
The exact integral of x² from zero to two is 8/3. The interpolating line integrates to four. The minimax line integrates to three. The least-squares line integrates to 8/3, exactly matching this integral. That equality follows here from residual orthogonality to constants; it does not mean the line matches the function pointwise.
Now require the surrogate to remain nonnegative on the entire interval. Both q and r violate that constraint near zero. Their unconstrained optimality statements no longer answer the revised problem. You must solve a constrained problem or choose a suitable family rather than clipping the output and retaining the old optimality claim.
Finally, suppose the values came from uncertain measurements instead of the exact formula. The preceding exact error calculations would no longer be available without additional assumptions about the true response. You could still compare fitted candidates on independent measurements, but you would need to report what those measurements establish.
This exercise tests transfer because every change is small in wording and substantial in meaning. A learner who notices the changed objective, output or constraint is using approximation theory. A learner who reuses the same coefficients merely because the curve still looks quadratic has missed the central decision.
CHAPTER 14 OF 14 · Transfer and continue learning
14. Questions and a route to dependable practice
Is an approximation simply an inaccurate answer?
It is a deliberately simplified answer with a stated purpose and controlled limitations. An exact expression may be expensive to evaluate or less useful than a compact representation with a proved tolerance. The weakness is not approximation itself; it is an unstated or untested claim about its accuracy.
Does the best approximation always exist and remain unique?
That depends on the family, domain and norm. Classical polynomial minimax approximation for a continuous function on a closed interval has strong existence and uniqueness results, summarized in the NIST approximation reference. Those conclusions should not be transferred automatically to every nonlinear family, constraint set or data-fitting problem.
Should I always increase the degree?
Increase it when evidence indicates that representation capacity is the limiting factor and the computation remains stable. More terms can be wasteful for a poorly chosen family, harmful for noisy data or ineffective near a discontinuity. Compare resolution changes with independent errors and with the cost of evaluating the resulting model.
What is the most useful beginner project?
Reproduce the constant and linear approximations in this guide. For each one, write the objective before solving, then compute both integrated squared error and maximum absolute error. Plotting can help, but the algebra should explain the difference. Next compare global polynomials with piecewise linear approximations of the same target under a fixed evaluation budget.
Keep a short record containing the function or dataset, domain, units, family, objective, construction procedure and independent checks. Add one failure case: noise, a corner, a narrow peak or a changed downstream requirement. Explain what failed and why the repair addresses that failure.
The subject is learned when you can defend the approximation in the language of its use. You should be able to say what is guaranteed, what is estimated and what remains unknown. The Education parent connects that capability to a wider standard: understanding becomes durable when it guides a sound decision after the familiar example has changed.