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Fungrim entry: 2a8ec9

({k=0Nak<2π,N=0k=0Nak2π,N1)    (n=k=0Nsinc ⁣(akn)=k=0Nsinc ⁣(akx)dx)\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=-\infty}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} n\right) = \int_{-\infty}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} x\right) \, dx\right)
Assumptions:NZ0  and  ak(0,)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=-\infty}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} n\right) = \int_{-\infty}^{\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
Sumnf(n)\sum_{n} f(n) Sum
Piπ\pi The constant pi (3.14...)
Productnf(n)\prod_{n} f(n) Product
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Infinity\infty Positive infinity
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
OpenInterval(a,b)\left(a, b\right) Open interval
Source code for this entry:
Entry(ID("2a8ec9"),
    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, Neg(Infinity), Infinity)), Integral(Product(Sinc(Mul(Subscript(a, k), x)), For(k, 0, N)), For(x, Neg(Infinity), 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"))

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

2020-08-27 09:56:25.682319 UTC