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Generating-coefficient estimate for the quadratic CQ datum

Statement

Let $0 < α < 1$ , put $β = 2 - α$ , and define $c_{m} : = \frac{Γ (m + β + 1)}{Γ (β + 1) Γ (m + 1)} (m \geq 0),$ $s_{1} : = 1, s_{n} : = c_{n - 1} + c_{n - 2} (n \geq 2) .$ There is a constant $C_{α} < \infty$ such that for every $n \geq 1$ , $∣ s_{n} - \frac{2 n^{β}}{Γ (β + 1)} ∣ \leq C_{α} n^{β - 1} .$ For $n \geq 1$ , $s_{n}$ is the coefficient of $z^{n}$ in the real Taylor series about $0$ of $z (1 + z) (1 - z)^{α - 3}$ on $(- 1, 1)$ .

Facts & Assumptions

Given: $0 < α < 1$ , $β = 2 - α$ , and $n \geq 1$ .

[F1]

Backward-Euler convolution quadrature for a Caputo equation: for $h > 0$ and $v^{n} = n^{2}$ , ${\overset{ˉ}{\partial}}_{h}^{α} v^{n} = h^{- α} \sum_{j = 0}^{n} ω_{j} (n - j)^{2}$ . Thus the unscaled weight convolution is $h^{α} {\overset{ˉ}{\partial}}_{h}^{α} v^{n} = \sum_{j = 0}^{n} ω_{j} (n - j)^{2}$ .

[F2]

Backward-Euler fractional weights and positive partial sums and On positive reals, the principal branch power agrees with the published real power: for real $z \in (- 1, 1)$ , $\sum_{j \geq 0} ω_{j} z^{j} = (1 - z)^{α}$ in the real-power convention.

[F3]

The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ : $Γ (x + 1) = x Γ (x)$ for $x > 0$ .

[F4]

The real Gamma function by Euler's integral: $Γ (x) > 0$ for $x > 0$ and $Γ (1) = 1$ .

[F5]

Gautschi's inequality for the real Gamma function: for $x > 0$ and $0 \leq q \leq 1$ , $x^{1 - q} \leq Γ (x + 1) / Γ (x + q) \leq (x + 1)^{1 - q}$ .

[F6]

Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence: if the absolute-term ratios tend to a value below $1$ , the series converges absolutely.

[F7]

A real power series about a centre, its interval of convergence, and its radius in $[0, + \infty]$ and Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius: power series may be differentiated term by term inside their radius.

[F8]

Continuity and derivatives of positive-base real powers, The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , and 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 $g (c) \neq 0$ : positive-base powers obey exponent laws and the chain/product rules, with $(x^{q})^{'} = q x^{q - 1}$ for $x > 0$ .

[F9]

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 $f (b) - f (a) = f^{'} (c) (b - a)$ : a differentiable function with zero derivative on an interval is constant there.

[F10]

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges: for $∣ z ∣ < 1$ , $\sum_{m \geq 0} z^{m} = (1 - z)^{- 1}$ .

[F11]

Inside the common radius the product of two power-series sums is represented by the Cauchy product of their coefficients: inside the common radius, the product of two power-series sums has the Cauchy-product coefficients.

[F12]

A power-series sum is infinitely differentiable inside its radius and satisfies $a_{n} = f^{(n)} (c) / ι (n!)$ at its centre: coefficients of equal power-series sums near their centre agree.

Proof

1.1givenF3F4F6F7F8F9F12algebra

By [F3] and $Γ (1) = 1$ in [F4], $c_{0} = 1$ and $c_{m + 1} / c_{m} = (m + β + 1) / (m + 1)$ . These terms are positive by [F4], and for every fixed real $z$ with $0 < ∣ z ∣ < 1$ the ratio of successive absolute terms of $\sum c_{m} z^{m}$ tends to $∣ z ∣$ . Hence [F6] gives absolute convergence there; for $z = 0$ only the constant term remains. The coefficient recurrence yields $(m + 1) c_{m + 1} = (m + β + 1) c_{m}$ ; by [F7], coefficient comparison gives $(1 - z) C^{'} (z) = (β + 1) C (z)$ for $C (z) : = \sum_{m \geq 0} c_{m} z^{m}$ . The derivative of $C (z) (1 - z)^{β + 1}$ is therefore zero by [F8]. It equals $1$ at $z = 0$ , so [F9] and [F8] give $C (z) = (1 - z)^{- (β + 1)}$ for real $z \in (- 1, 1)$ .

2.1F1F2F7F8F10F11F12step 1.1algebra

Starting from [F10], termwise differentiation by [F7] gives $\sum_{m \geq 0} m z^{m} = \frac{z}{(1 - z)^{2}}, \sum_{m \geq 0} m^{2} z^{m} = z \frac{d}{d z} (\frac{z}{(1 - z)^{2}}) = \frac{z (1 + z)}{(1 - z)^{3}}$ for real $z \in (- 1, 1)$ . Multiplying by the real symbol in [F2] and using [F8] gives $z (1 + z) (1 - z)^{α - 3}$ . By [F11], its $z^{n}$ coefficient is the unscaled convolution $\sum_{j = 0}^{n} ω_{j} (n - j)^{2} = h^{α} {\overset{ˉ}{\partial}}_{h}^{α} v^{n}$ for $v^{n} = n^{2}$ ; by step 1.1 it is $1$ for $n = 1$ and $c_{n - 1} + c_{n - 2}$ for $n \geq 2$ . This proves the coefficient assertion.

2.2step 1.1F3F5F8F9algebra

Put $r = 1 - α = β - 1 \in (0, 1)$ . For $n \geq 2$ , apply [F5] with $x = n - α$ and $q = α$ . It gives $(n - α)^{r} \leq Γ (n + 1 - α) / Γ (n) \leq (n + 1 - α)^{r}$ . By [F3], $c_{n - 1} = \frac{(n + 1 - α) Γ (n + 1 - α)}{Γ (β + 1) Γ (n)} .$ Consequently $c_{n - 1}$ lies between $(n + 1 - α) (n - α)^{r} / Γ (β + 1)$ and $(n + 1 - α)^{r + 1} / Γ (β + 1)$ . For the upper endpoint, the mean value theorem gives $(n + 1 - α)^{r + 1} - n^{r + 1} \leq (r + 1) (n + 1)^{r} \leq C n^{r}$ . For the lower endpoint, write $∣ (n + 1 - α) (n - α)^{r} - n^{r + 1} ∣ \leq (n + 1 - α) ∣ (n - α)^{r} - n^{r} ∣ + (1 - α) n^{r} .$ Because $n \geq 2$ , $n - α \geq n / 2$ ; the mean value theorem bounds $∣ n^{r} - (n - α)^{r} ∣ \leq r α (n / 2)^{r - 1}$ , and $n + 1 - α \leq 3 n / 2$ . Thus this lower endpoint also differs from $n^{r + 1}$ by at most a constant depending on $α$ times $n^{r}$ . Dividing by the fixed positive $Γ (β + 1)$ proves Thus $∣ c_{n - 1} - n^{β} / Γ (β + 1) ∣ \leq C_{α} n^{β - 1}$ after enlarging the constant.

3.1step 2.2F8F9algebra∎

For $n \geq 3$ , apply the same estimate with $n - 1$ in place of $n$ to $c_{n - 2}$ . Also $∣ n^{β} - (n - 1)^{β} ∣ \leq β n^{β - 1}$ by the mean value theorem. Hence $∣ c_{n - 2} - n^{β} / Γ (β + 1) ∣ \leq C_{α} n^{β - 1}$ after another enlargement. The cases $n = 1, 2$ have finite coefficients and are covered by enlarging $C_{α}$ once more. Adding the two estimates proves the claim.

Depends on

Used by

First-order CQ error for a compatible quadratic datum Theorem

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78 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

Ralf Hiptmair, Advanced Numerical Methods for Computational Science and Engineering, lecture notes (2023), §3.3.2 (standard reference, not scraped)
C. Lubich, Convolution Quadrature and Discretized Operational Calculus I, Numerische Mathematik 52 (1988), §1 (standard reference, not scraped)