# Fungrim entry: 005478

$\left(\begin{cases} \sum_{k=0}^{N} {a}_{k} < 2 \pi, & N = 0\\\sum_{k=0}^{N} {a}_{k} \le 2 \pi, & N \ge 1\\ \end{cases}\right) \;\implies\; \left(\sum_{n=1}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} n\right) = -\frac{1}{2} + \int_{0}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} x\right) \, dx\right)$
Assumptions:$N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; {a}_{k} \in \left(0, \infty\right)$
References:
• https://www.carma.newcastle.edu.au/resources/jon/sinc-sums.pdf
TeX:
\left(\begin{cases} \sum_{k=0}^{N} {a}_{k} < 2 \pi, & N = 0\\\sum_{k=0}^{N} {a}_{k} \le 2 \pi, & N \ge 1\\ \end{cases}\right) \;\implies\; \left(\sum_{n=1}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} n\right) = -\frac{1}{2} + \int_{0}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} x\right) \, dx\right)

N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; {a}_{k} \in \left(0, \infty\right)
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
Pi$\pi$ The constant pi (3.14...)
Product$\prod_{n} f(n)$ Product
Sinc$\operatorname{sinc}(z)$ Sinc function
Infinity$\infty$ Positive infinity
Integral$\int_{a}^{b} f(x) \, dx$ Integral
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
OpenInterval$\left(a, b\right)$ Open interval
Source code for this entry:
Entry(ID("005478"),
Formula(Implies(Cases(Tuple(Less(Sum(Subscript(a, k), For(k, 0, N)), Mul(2, Pi)), Equal(N, 0)), Tuple(LessEqual(Sum(Subscript(a, k), For(k, 0, N)), Mul(2, Pi)), GreaterEqual(N, 1))), Equal(Sum(Product(Sinc(Mul(Subscript(a, k), n)), For(k, 0, N)), For(n, 1, Infinity)), Add(Neg(Div(1, 2)), Integral(Product(Sinc(Mul(Subscript(a, k), x)), For(k, 0, N)), For(x, 0, Infinity)))))),
Variables(N, a),
Assumptions(And(Element(N, ZZGreaterEqual(0)), Element(Subscript(a, k), OpenInterval(0, Infinity)))),
References("https://www.carma.newcastle.edu.au/resources/jon/sinc-sums.pdf"))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC