TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exact high-rate limit for a positive compact one-dimensional compander

Statement

Let $a < b$ and let $f, g : [a, b] \to (0, \infty)$ be $C^{1}$ probability densities. For the compander of Positive C1 compact one-dimensional compander, the distortions satisfy

$\lim_{N \to \infty} N^{2} D_{N} (f, g) = \frac{1}{12} \int_{a}^{b} \frac{f (x)}{g (x)^{2}} d x .$

This is an exact limit for the specified compact construction. It assumes neither an unbounded source nor negligible overload, and it makes no claim about unrestricted optimal quantizers or dimensions greater than one.

Facts & Assumptions

Given: A fixed interval $[a, b]$ with $a < b$ and positive normalized $C^{1}$ densities $f, g$ .

[F1]

Positive C1 compact one-dimensional compander: with $ϕ = G_{g}^{- 1}$ , the $N$ cells have cumulative-coordinate endpoints $j / N$ , reproduction points are $ϕ ((j - \frac{1}{2}) / N)$ , and $D_{N} (f, g)$ is the integral of squared error against density $f$ on $[a, b]$ .

[F2]

A positive C1 point density has a C2 cumulative inverse: $ϕ$ is an increasing $C^{2}$ bijection, $ϕ^{'} = 1 / (g \circ ϕ)$ and $ϕ^{''} = - g^{'} \circ ϕ / (g \circ ϕ)^{3}$ , including continuous endpoint extensions.

[F3]

The compact compander is a measurable finite quantizer: the cells are Borel and $D_{N} (f, g)$ is finite for each $N$ .

[F4]

Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ : a continuous real function on an interval has the primitive $A (x) = \int_{x_{0}}^{x} h$ , with $A^{'} = h$ , and the integral of a continuous derivative equals the endpoint difference.

[F5]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : the derivative of a composition of differentiable Euclidean maps is the composition of their derivatives.

[F6]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : differentiable real functions obey the product and quotient rules where the denominator is nonzero.

[F7]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : for a function continuous on a closed interval and differentiable in its interior, its endpoint increment equals an interior derivative times the interval length.

[F8]

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: every continuous real function on a nonempty compact metric space is bounded and attains its maximum and minimum.

[F9]

Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous: a continuous map on a compact metric space is uniformly continuous.

Proof

1.1F1F2F3F4F5

Put $v_{j} = (j - \frac{1}{2}) / N$ , $I_{j} = [(j - 1) / N, j / N]$ , and $F (u) = f (ϕ (u)) / g (ϕ (u))$ . For $h_{j} (x) = f (x) (x - ϕ (v_{j}))^{2}$ , its primitive $A_{j} (x) = \int_{a}^{x} h_{j} (s) d s$ satisfies $A_{j}^{'} = h_{j}$ by [F4]. The chain rule [F5] applied to $A_{j} \circ ϕ$ on $I_{j}$ , followed by endpoint evaluation using [F4], gives $D_{N} (f, g) = \sum_{j = 1}^{N} \int_{I_{j}} F (u) (ϕ (u) - ϕ (v_{j}))^{2} d u$ , because [F2] gives $ϕ^{'} = 1 / (g \circ ϕ)$ and [F1] gives the reproduction point $ϕ (v_{j})$ . The distortion is well-defined by [F3].

1.2F2F5F6F7F8algebra

By the chain rule [F5] and quotient rule [F6], $F$ is $C^{1}$ ; the functions $ϕ^{'}$ , $ϕ^{''}$ , $F$ and $F^{'}$ are continuous on compact $[0, 1]$ , so [F8] gives finite bounds $M_{1} = ∥ ϕ^{'} ∥_{\infty}$ , $M_{2} = ∥ ϕ^{''} ∥_{\infty}$ , $M_{F} = ∥ F ∥_{\infty}$ , and $M_{F^{'}} = ∥ F^{'} ∥_{\infty}$ , independent of $N, j$ . If $δ = u - v_{j}$ and $e = ϕ (u) - ϕ (v_{j}) - ϕ^{'} (v_{j}) δ$ , [F7] applied to $ϕ^{'}$ and $F$ , together with [F4], gives $∣ e ∣ \leq M_{2} ∣ δ ∣^{2} / 2$ and $∣ F (u) - F (v_{j}) ∣ \leq M_{F^{'}} ∣ δ ∣$ . Expanding the square therefore bounds the absolute difference under consideration by $M_{F^{'}} M_{1}^{2} ∣ δ ∣^{3} + M_{F} (M_{1} M_{2} ∣ δ ∣^{3} + M_{2}^{2} ∣ δ ∣^{4} / 4)$ . Since $∣ δ ∣ \leq 1 / (2 N) \leq 1 / 2$ , this is at most $C ∣ δ ∣^{3}$ for $C = M_{F^{'}} M_{1}^{2} + M_{F} M_{1} M_{2} + M_{F} M_{2}^{2} / 8$ , which is independent of $N, j$ .

2.1F4step 1.1step 1.2algebra

On $I_{j}$ , direct integration gives $\int_{I_{j}} δ^{2} d u = 1 / (12 N^{3})$ and $\int_{I_{j}} ∣ δ ∣^{3} d u = 1 / (32 N^{4})$ . Thus step 1.2 gives $D_{N} (f, g) = \frac{1}{12 N^{3}} \sum_{j = 1}^{N} H (v_{j}) + E_{N}$ with $∣ E_{N} ∣ \leq C / (32 N^{3})$ , where $H (u) : = F (u) ϕ^{'} (u)^{2} = f (ϕ (u)) / g (ϕ (u))^{3}$ .

3.1step 2.1F4F7F9algebra

Multiplying by $N^{2}$ gives $N^{2} D_{N} (f, g) = \frac{1}{12 N} \sum_{j = 1}^{N} H (v_{j}) + N^{2} E_{N}$ , with $∣ N^{2} E_{N} ∣ \leq C / (32 N) \to 0$ . By [F9], $H$ is uniformly continuous on $[0, 1]$ . Given $ε > 0$ , choose $δ > 0$ such that $∣ H (u) - H (v) ∣ < ε$ whenever $∣ u - v ∣ < δ$ . Let $P (u) : = \int_{0}^{u} H (s) d s$ ; [F4] gives $P^{'} = H$ and $P (1) - P (0) = \int_{0}^{1} H$ . For $N$ with $1 / (2 N) < δ$ , the mean value theorem [F7] on each $I_{j}$ gives $ξ_{j} \in ((j - 1) / N, j / N)$ with $P (j / N) - P ((j - 1) / N) = H (ξ_{j}) / N$ . Since $∣ ξ_{j} - v_{j} ∣ < 1 / (2 N) < δ$ , telescoping gives $∣ \frac{1}{N} \sum_{j = 1}^{N} H (v_{j}) - \int_{0}^{1} H (u) d u ∣ \leq \frac{1}{N} \sum_{j = 1}^{N} ∣ H (v_{j}) - H (ξ_{j}) ∣ < ε$ . Thus the midpoint sums converge to $\int_{0}^{1} H (u) d u$ .

3.2F2F4F5step 2.1

Let $P (x) = \int_{a}^{x} f (s) / g (s)^{2} d s$ . The integrand is continuous because $g$ is positive on the compact interval, so [F4] gives $P^{'} = f / g^{2}$ . By [F5] and [F2], $(P \circ ϕ)^{'} (u) = f (ϕ (u)) / g (ϕ (u))^{3} = H (u)$ . Evaluating this primitive at $0, 1$ by [F4], using $ϕ (0) = a$ and $ϕ (1) = b$ , gives $\int_{0}^{1} H (u) d u = \int_{a}^{b} f (x) / g (x)^{2} d x$ .

4.1step 3.1step 3.2∎

Steps 3.1 and 3.2 identify the limit of $N^{2} D_{N} (f, g)$ with the displayed integral, proving the theorem.

Depends on

Used by

Dependency tree · two levels

61 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)
Robert Gallager, 6.450 Principles of Digital Communications I, Chapter 3: Quantization (standard reference, not scraped)
Robert M. Gray and David L. Neuhoff, Quantization, IEEE Transactions on Information Theory 44(6) (1998) (standard reference, not scraped)