Under the Articles of the Constitution, can the president of the United States be impeached on the basis of felonies he or she committed prior to being sworn in as president?
The answer depends on what you mean by "can."
Impeachment is essentially a political process largely unreviewable by the judiciary (other than perhaps for gross procedural deficiencies like convicting on the basis of a coin toss1). If the House does impeach, it can impeach. I take your question then to also be asking whether it is conceivable that the House would impeach for pre-presidential acts. That is unknown, but I don't think it has been ruled out by history.
The Constitution Annotated explains:
The type of behavior that qualifies as impeachable conduct, and the circumstances in which impeachment is an appropriate remedy for such actions, are thus determined by, among other things, competing political interests, changing institutional relationships among the three branches of government, and legislators’ interaction with and accountability to the public. The weight of historical practice, rather than judicial precedent, is thus central to understanding the nature of impeachment in the United States.
the absence of impeachment proceedings directed against particular conduct in the past does not mean that such conduct would not be deemed impeachable in different circumstances. For example, certain conduct giving rise to impeachment might not have occurred or attracted notice at an earlier time. Understandings of impeachable behavior might also change over time to recognize impeachment as available for a wider range of behavior than has been previously recognized.
Until now, however, impeachment of an official has always been based on acts during the official's term. This might indicate that the present understanding of what constitutes an impeachable offence does not include conduct prior to a person's appointment as president. Or, it may simply be because such situation has not yet arisen.
For example, if Congress were to learn that a president, prior to their election, had illegally obtained national secrets and illegally used those secrets to come to agreements with foreign adversaries to help them get elected, that could conceivably be a basis on which the House would impeach. I have only included the caveat of illegality because your question asks about felonies, but even this is not a requirement for impeachment. It is widely understood that impeachment is not limited to codified crimes. The history of impeachment includes many "offences" which are of a purely political character.
1. See Justice Souter's concurrence in Nixon v. United States, 506 U.S. 224 (1993), in which he envisioned that there might be scenarios that call for more "searching review of impeachment proceedings," like if an officer was convicted "upon a coin toss, or upon a summary determination that an officer of the United States was simply 'a bad guy'."