Albert Barnes Commentary Acts 10:43

"To him bear all the prophets witness, that through his name every one that believeth on him shall receive remission of sins." — Acts 10:43 (ASV)

To him give, etc. See Barnes on Luke 24:27, 44.

That through his name, etc. This was implied in what the prophets said. See Romans 10:11. It was not, indeed, expressly affirmed that those who believed in him would be pardoned; but this was implied in what they said. They promised a Messiah; and their religion consisted mainly in believing in a Messiah to come. See the reasoning of the apostle Paul in Romans 4.