Is it ok to send part of digital signature if we have bandwidth constraints? When creating digital signatures is it ok to send part of it when we have bandwidth constraints?
No, it is generally not OK to send only part of the signature, because then it can no longer be checked; that's unless we remove only very little of the signature; or unless the term signature is used for what really is a symmetric Message Authentication Code (in which case shortening only reduces the security, perhaps acceptably).



For any signature scheme, we can remove a few bits of the signature, say $k$, and define a verification scheme that accepts the signature if any of the possible $2^k$ original signatures matching the truncated signature is acceptable. The risk of forgery is increased by at most $2^k$ but might remain acceptable. The work to verify the signature is increased by at most $2^k$ (including not at all, if we count a deterministic secret-key MAC as a signature scheme, for it is just as easy to verify a truncated such MAC as it is to verify the original MAC).

When there's a single bit to trim from an RSA signature (as is often the case when a signature is enciphered using a different modulus having the same number of bits as the modulus used in the signature), there's a standard trick avoiding a doubling of the verifier's work: if the RSA signature is $S$, send $\widetilde S=\min(S,N-S)$, which saves 1 bits, and let the verifier compute $V={\widetilde S}^e\bmod N$; either $V$ or $N-V$ is the normal padded message representative, the right one being determinable by some characteristic of the padded message representative, like being even or having its few high bits clear.

When there are bandwidth constraints, more radical options are:


Giving up public verification and using a deterministic secret-key MAC, which is king in term of compactness.
Using a signature scheme allowing message recovery, such that (at least some of) the message is embedded in the signature, saving bandwidth. The standard such RSA schemes are defined by ISO/IEC 9796-2 free partial preview, which allows for example to convey any $b$-byte message as a cryptogram of $\min(256, b+22)$ bytes, when using 2048-bit RSA keys and SHA-1 (but beware that scheme 1, common in the Smart Card industry, could be vulnerable to attack in a chosen-message setup, and has no proven security reduction in other setups).
Using schemes designed for short signatures with appendix, such a BLS; unfortunately, this is relatively new, and no international standard has emerged, or even is in the work AFAIK.