Multi-interval approximation for EvalChebyshevFunction

Library Questions
1

Hello,

Can I use a multi-interval domain for EvalChebyshevFunction?
I want to approximate (\frac{\sin(\pi x)}{\pi x}) with polynomials, and the values of (x) would be near integers.
Specifically, all (x) would be in the intervals ((i-\epsilon, i+\epsilon)) for (i = -N, \ldots, 0, \ldots, N).

In this case, if I use a long interval like [-N, N], it causes a lot of noise.
Can you give me some advice on how to deal with this issue?
I want to know if I can use multi-interval approximation as in the bootstrapping procedure.

Thank you.

2

The sine function is a special case because it is periodic. I don’t know how this would be done for a general function.

For the usage of this function in the CKKS bootstrapping, it is instructive to look at the papers https://eprint.iacr.org/2018/153.pdf, https://eprint.iacr.org/2018/1043.pdf and https://eprint.iacr.org/2020/1203.pdf, and look at the code for CKKS bootstrapping in OpenFHE in openfhe-development/src/pke/lib/scheme/ckksrns/ckksrns-fhe.cpp at main · openfheorg/openfhe-development · GitHub.

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