By Howard Schneider

WASHINGTON (Reuters) -Federal Reserve officials, in their first public comments since the U.S. central bank cut interest rates by half a percentage point, laid out on Friday the depth of the debate over the move, with one governor saying inflation was now so weak the large reduction was needed and another arguing price pressures remain so strong a smaller cut would have been better.

Fed Governors Christopher Waller and Michelle Bowman have been close in spirit through much of the central bank's battle against inflation as advocates of faster and more robust rate increases to keep it contained.

But Waller said in an interview with CNBC that recent data convinced him the Fed needed to cut rates faster because it is at risk of undershooting its inflation target, while Bowman in a separate statement worried the half-percentage-point cut sent the wrong signal with inflation still above the central bank's 2% goal.

Waller said data released in the days before the policy-setting Federal Open Market Committee's two-day meeting this week led him to believe that the central bank's preferred index of inflation, the personal consumption expenditures price index, was "softening much faster than I thought it was going to. And that is what put me over the edge to say, look, I think 50 (basis points) is the right thing to do."

With the same information in hand, however, Bowman said that while she agreed it was time to cut rates given how much inflation has slowed, prices were still increasing at a roughly 2.5% rate on a year-over-year basis, and "the Committee's larger policy action could be interpreted as a premature declaration of victory."

"Moving at a measured pace toward a more neutral policy stance will ensure further progress in bringing inflation down," she said in explaining her dissent in favor of a quarter-percentage-point cut.

It was the first dissent by a member of the Fed's Board of Governors in 19 years, and highlighted the still unresolved issue of how fully Fed Chair Jerome Powell had the backing of the FOMC's 12 voting members and seven non-voting participants in beginning a new cycle of rate-cutting with the 50-basis-point reduction.

Only voting members of the committee, including the seven Fed governors and five of the 12 reserve bank presidents at any given meeting, can dissent.

Economic projections issued by other Fed policymakers alongside the policy statement on Wednesday showed many seemed inclined towards a quarter-percentage-point cut, though the "dot plot" reflecting officials' rate outlook does not indicate how many of them were non-voters with no option to register an objection.