Secondary Measure

An auxiliary positive measure derived from an initial moment measure so that its secondary-polynomial sequence becomes orthogonal, with the pair linked by a reciprocal Stieltjes-transform relation.

Version

v3 · 2026-09-06 · History

Domain-specific #

2733

Origin domain

mathematics

Subdomain

orthogonal polynomials

Aliases

Secondary orthogonality measure, Associated secondary measure

Core Idea

Let a positive measure \(\rho\) with sufficiently many finite moments generate orthogonal polynomials \(P_n\). Their associated or secondary polynomials \(Q_n\) can be defined by a divided-difference transform such as

\[ Q_n(x)=\int \frac{P_n(t)-P_n(x)}{t-x}\,d\rho(t), \]

with convention-dependent signs and normalizations. A secondary measure \(\mu\) associated with \(\rho\) is a positive measure for which the \(Q_n\) form an orthogonal family.

For a normalized initial measure and a declared Stieltjes-transform convention, the construction is encoded by a reciprocal relation of the form.

Scope of Application

Secondary measures belong to the theory of orthogonal polynomials, Jacobi recurrences, moment problems, and Stieltjes transforms. They support operator identities between weighted \(L^2\) spaces, explicit calculations for classical polynomial families, and iterative sequences of normalized measures. Examples associated with shifted Legendre, Laguerre, Hermite, and Chebyshev systems illustrate how the transform relation can produce explicit densities and Fourier coefficients.

Clarity

State the sign convention for \(S_\rho\), whether measures are normalized to probability mass, the normalization of \(P_n\) and \(Q_n\), the interval, and all moment or density hypotheses. Without these declarations, formulas that differ only conventionally may look contradictory.

Manages Complexity

The construction replaces a direct search for an orthogonalizing weight for \(Q_n\) with a transform-level update. Recurrence coefficients, moments, and analytic boundary values can then be used systematically instead of recomputing every pairwise integral.

Abstract Reasoning

1. Fix \(\rho\), its support, moments, and Stieltjes-transform convention.
2. Construct and normalize the orthogonal polynomials \(P_n\).
3. Apply the divided-difference operator to obtain \(Q_n\).
4. Derive the reciprocal Stieltjes expression proposed for \(\mu\).
5. Verify that the expression is the transform of a positive measure.
6. Recover its density or moments when possible.
7. Check orthogonality and norm relations for the \(Q_n\).
8. Normalize and iterate only after recording the mass introduced at each step.

Knowledge Transfer

The portable pattern is to construct a new carrier that makes a transformed basis orthogonal, using an analytic transform to move between carriers. The proposed immediate parent is Measure.

Relationships to Other Abstractions

Current abstraction Secondary Measure Domain-specific

Parents (1) — more general patterns this builds on

• Secondary Measure is a kind of Measure Prime

  Measure is the proposed immediate parent.

Hierarchy paths (2) — routes to 2 parentless roots

• Secondary Measure → Measure → Aggregation → Micro Macro Linkage

Show alternative path (1)

Neighborhood in Abstraction Space

Secondary Measure sits in a sparse region of the domain-specific corpus (89^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Moment Problems & Discrete Approximation (7 abstractions)

Nearest neighbors

• Legendre Polynomials — 0.80
• Euclidean Space — 0.80
• Hamburger moment problem — 0.79
• Moment matrix — 0.79
• Finite Difference Coefficient — 0.78

Computed from structural-signature embeddings · 2026-09-08