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Backward-Euler fractional weights and positive partial sums

Statement

Let $0 < α < 1$ and let $(ω_{j})$ be the weights from Backward-Euler convolution quadrature for a Caputo equation. Then $ω_{0} = 1$ , $ω_{j} < 0$ for every $j \geq 1$ , and for every complex $z$ with $∣ z ∣ < 1$ , $\sum_{j = 0}^{\infty} ω_{j} z^{j} = (1 - z)_{Log}^{} .$ For $n \geq 0$ define $p_{n} : = \sum_{j = 0}^{n} ω_{j} .$ Then $p_{n} = \prod_{k = 1}^{n} (1 - \frac{α}{k}) = \frac{Γ (n + 1 - α)}{Γ (1 - α) Γ (n + 1)} > 0,$ where the empty product for $n = 0$ is $1$ . For every $n \geq 1$ , $p_{n} \geq \frac{(n + 1)^{- α}}{Γ (1 - α)} .$

Facts & Assumptions

Given: $0 < α < 1$ and the coefficient recurrence in Backward-Euler convolution quadrature for a Caputo equation.

[F1]

Backward-Euler convolution quadrature for a Caputo equation: $ω_{0} = 1$ and $ω_{j} = ((j - 1 - α) / j) ω_{j - 1}$ for $j \geq 1$ .

[F2]

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

[F3]

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

[F4]

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

[F5]

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges and If $0 \leq a_{k} \leq b_{k}$ eventually, convergence of $\sum b_{k}$ gives convergence of $\sum a_{k}$ , and divergence of $\sum a_{k}$ gives divergence of $\sum b_{k}$ : $\sum_{j \geq 0} r^{j}$ converges for $∣ r ∣ < 1$ , and a series dominated termwise in absolute value by it converges absolutely.

[F6]

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: a power series may be differentiated term by term inside its radius of convergence.

[F7]

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 $x > 0$ .

[F8]

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

[F9]

A power-series sum is infinitely differentiable inside its radius and satisfies $a_{n} = f^{(n)} (c) / ι (n!)$ at its centre: the coefficients of a power series equal to another power series near its centre agree, since they are the normalized derivatives at that centre.

[F10]

Principal-branch generating function for backward-Euler convolution quadrature: for $∣ z ∣ < 1$ , the coefficient series is the principal-branch power $(1 - z)_{Log}^{}$ .

[F11]

On positive reals, the principal branch power agrees with the published real power: when $z$ is real and $∣ z ∣ < 1$ , $(1 - z)_{Log}^{} = (1 - z)^{α}$ with the real-power convention.

Proof

1.1givenF1F5algebra

The recurrence [F1] gives $ω_{1} = - α < 0$ and, for $j \geq 2$ , $ω_{j} = ((j - 1 - α) / j) ω_{j - 1} < 0$ , since $0 < (j - 1 - α) / j < 1$ . It also gives $∣ ω_{j} ∣ \leq 1$ for every $j$ . Hence $W (z) : = \sum_{j \geq 0} ω_{j} z^{j}$ converges absolutely for $∣ z ∣ < 1$ by [F5], and its radius is at least $1$ .

2.1step 1.1F10F11

By [F10], the complex generating function equals the principal-branch power for $∣ z ∣ < 1$ . On the real interval $(- 1, 1)$ , [F11] identifies it with $(1 - z)^{α}$ in the real-power convention.

3.1step 2.1F5F6F7F8F9algebra

Since $p_{n} = \sum_{j = 0}^{n} ω_{j}$ , [F8] and the geometric series [F5] give $P (z) : = \sum_{n \geq 0} p_{n} z^{n} = W (z) / (1 - z) = (1 - z)^{α - 1}$ for real $z \in (- 1, 1)$ . Differentiate this real series by [F6] and the right side by [F7]. Comparing coefficients in $(1 - z) P^{'} (z) = (1 - α) P (z)$ yields $(n + 1) p_{n + 1} = (n + 1 - α) p_{n}$ for $n \geq 0$ .

4.1step 3.1F2F3algebra

Iterating the recurrence from step 3.1 gives $p_{n} = \prod_{k = 1}^{n} (k - α) / k$ . By [F3], $\prod_{k = 1}^{n} (k - α) = \frac{Γ (n + 1 - α)}{Γ (1 - α)}, \prod_{k = 1}^{n} k = Γ (n + 1) .$ Thus the displayed Gamma formula follows; all factors are positive, and [F2] confirms the Gamma denominator is positive. This also covers $n = 0$ with the empty product.

5.1step 4.1F2F4algebra∎

For $n \geq 1$ , apply [F4] with $x = n$ and $s = 1 - α$ . It gives $n^{α} \leq Γ (n + 1) / Γ (n + 1 - α) \leq (n + 1)^{α}$ . Taking reciprocals and using step 4.1 yields $p_{n} \geq (n + 1)^{- α} / Γ (1 - α)$ .

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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)