Incorrect, lightweight wallets most certainly have enough data to verify signatures via merkle root. How well they do it is another story, but it's is not unable. – arshbot – 2018-08-17T19:50:22.587

2Merkle roots are completely unrelated to signatures. The merkle root is about the block, not about signatures. – Andrew Chow – 2018-08-17T19:52:10.340

1Interesting, I assumed because the merkle root was reliant on the transaction hash it thus had some data of the signature in it - but according to the whitepaper: "He can't check the transaction for himself, but by linking it to a place in the chain, he can see that a network node has accepted it, and blocks added after it further confirm the network has accepted it." Which means it's really just relying on the network, not doing proper validation itself. I stand corrected – arshbot – 2018-08-17T19:55:19.550

Lightweight wallets know each input of a transaction, thus they should be able to fetch every previous transaction, up to the coinbase transaction. I've also learned that Satoshi once said: "The recipient just needs to verify it back to a depth that is sufficiently far back in the block chain, which will often only require a depth of 2 transactions. All transactions before that can be discarded." – Chris Chen – 2018-12-12T22:38:50.147

No, lightweight wallets can't fetch previous transactions. A full node will not relay transactions that are already in the blockchain. So a lightweight wallet would be unable to fetch previous transactions once they confirm. A lightweight wallet could fetch blocks, and from there the previous transactions. But they don't know which blocks, so they would have to fetch all of them.... and now we have described a full node. – Andrew Chow – 2018-12-12T23:24:10.177

I see Electrum, which should be a typical lightweight wallet, supports loading transaction from blockchain, by txid. – Chris Chen – 2019-01-14T05:33:28.537

Furthermore, I see Electrum can tell whether a 0-conf transaction has a confirmed parent or not. – Chris Chen – 2019-01-14T05:40:58.460

Beyond the viability of validating signatures for lightweight wallets, I think whether such validation actually makes sense might be more important. In some other word, will validating signatures improve security for them? – Chris Chen – 2019-01-14T06:02:35.593

1Yes, validating signatures does improve security. Being able to validate signatures means that you can partially check if a transaction is valid. This means that you are less likely to accept invalid transactions. However, signature validation is not the only thing for checking if a transaction is valid. You also need to check that the transaction spends coins that exist, and you can't really do that without validating the entire blockchain. – Andrew Chow – 2019-01-14T15:00:49.603