Working definition
DLMF 18.2.1 defines orthogonality on an interval by an integral: the product of two distinct members against a weight function integrates to zero. The weight is constrained at 18.2.1_5, which requires it to be non-negative, to have positive total mass, and to have all moments finite. Discrete analogues replace the integral by a sum over an infinite set at 18.2.2 or a finite one at 18.2.3, and 18.2.4_5 states the general form against a Lebesgue-Stieltjes measure. Two three-term recurrences follow, at 18.2.8 and 18.2.10, each carrying a strict positivity condition under which Favard theorem gives the converse, so that a family satisfying such a recurrence is orthogonal for some measure.
Notation
p sub n of x for the n-th member of the familyw of x for the weight and mu for the general measureA sub n, B sub n and C sub n for the recurrence coefficientsAssumptions
- Orthogonality is relative to a declared weight or measure; the same polynomials are not orthogonal against a different one.
- The weight conditions at 18.2.1_5 require non-negativity, positive total mass and finite moments, so a weight failing any of these defines no such family.
- The converse direction depends on the strict positivity conditions at 18.2.9_5 and 18.2.11_2, not on the recurrence shape alone.
Invariants
- Continuous orthogonality against a weight, DLMF 18.2.1
- The weight must be non-negative with positive mass and finite moments, DLMF 18.2.1_5
- The three-term recurrences, DLMF 18.2.8 and 18.2.10
Reproducible procedure
- Name the weight or measure before calling a family orthogonal, because the term is otherwise incomplete.
- Check the weight against the conditions at 18.2.1_5 rather than assuming a positive-looking function qualifies.
- Where a family is generated by recurrence, confirm the positivity conditions if orthogonality is being inferred from the recurrence rather than the other way round.
Error and boundary controls
- The chapter states the recurrences without claiming they are numerically stable. Forward recurrence can amplify rounding, and the reference does not settle the direction to use.
- A finite discrete family at 18.2.3 is orthogonal only up to degree N; beyond that the relation does not hold and is not claimed to.
- Orthogonality against a weight says nothing about conditioning of the resulting expansion coefficients for a particular function.
What this does not establish
The chapter defines orthogonality and its recurrences. It does not establish the numerical stability of evaluating them, and orthogonality holds against a declared weight rather than as an intrinsic property.
Explicit applications
0 cross-domain bridges
This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 18: Orthogonal Polynomials · National Institute of Standards and Technology
Establishes: Orthogonality conditions in the continuous, discrete and measure-theoretic settings with the positivity and moment conditions attached, and the two standard three-term recurrence forms together with the positivity that makes the converse hold.
Boundary: The chapter states the conditions and the recurrences. It does not establish that evaluating a recurrence in finite precision is stable, and orthogonality is defined against a declared weight rather than being a property a polynomial family has on its own.
Direct answer
- DLMF 18.2.1 defines orthogonality on an interval by an integral: the product of two distinct members against a weight function integrates to zero. The weight is constrained at 18.2.1_5, which requires it to be non-negative, to have positive total mass, and to have all moments finite. Discrete analogues replace the integral by a sum over an infinite set at 18.2.2 or a finite one at 18.2.3, and 18.2.4_5 states the general form against a Lebesgue-Stieltjes measure. Two three-term recurrences follow, at 18.2.8 and 18.2.10, each carrying a strict positivity condition under which Favard theorem gives the converse, so that a family satisfying such a recurrence is orthogonal for some measure.
Mechanism and method
- Name the weight or measure before calling a family orthogonal, because the term is otherwise incomplete.
- Check the weight against the conditions at 18.2.1_5 rather than assuming a positive-looking function qualifies.
- Where a family is generated by recurrence, confirm the positivity conditions if orthogonality is being inferred from the recurrence rather than the other way round.
What is measured
- Continuous orthogonality against a weight, DLMF 18.2.1
- The weight must be non-negative with positive mass and finite moments, DLMF 18.2.1_5
- The three-term recurrences, DLMF 18.2.8 and 18.2.10
Limitations
- The chapter states the recurrences without claiming they are numerically stable. Forward recurrence can amplify rounding, and the reference does not settle the direction to use.
- A finite discrete family at 18.2.3 is orthogonal only up to degree N; beyond that the relation does not hold and is not claimed to.
- Orthogonality against a weight says nothing about conditioning of the resulting expansion coefficients for a particular function.
- Orthogonality is relative to a declared weight or measure; the same polynomials are not orthogonal against a different one.
- The weight conditions at 18.2.1_5 require non-negativity, positive total mass and finite moments, so a weight failing any of these defines no such family.
- The converse direction depends on the strict positivity conditions at 18.2.9_5 and 18.2.11_2, not on the recurrence shape alone.
What this does not establish
- The chapter defines orthogonality and its recurrences. It does not establish the numerical stability of evaluating them, and orthogonality holds against a declared weight rather than as an intrinsic property.