A day after trimming his Socialist government to an inner core of 15 ministers, President Francois Mitterrand tonight warned French citizens of a tough struggle as they seek to overcome France's growing economic problems.

Speaking in a rare nationwide television broadcast, Mitterrand emerged from a lengthy period of self-imposed silence to defend the economic policies of his 22-month-old administration of Socialists and Communists. He appealed to citizens to help the new government win an economic battle on three fronts--inflation, unemployment and foreign trade--by saving more and buying French goods rather than foreign products.

The 14-minute broadcast from the Elysee Palace evidently was designed to reassert Mitterrand's leadership following a setback in municipal elections earlier this month and a string of economic failures culminating in the devaluation of the franc earlier this week. The president seemed eager to dispel the sense of political vacuum that was created here during the past two weeks while he was pondering the government shake-up and France's future economic course.

"It is high time to stop this infernal machine inflation . . . . That is why I shall struggle, and the government with me, against this evil, and I shall mobilize the country to this end," Mitterrand said.

Details of a new austerity package to accompany the franc devaluation are to be revealed Friday at a meeting of the new government headed by Prime Minister Pierre Mauroy, who led the last government and was reappointed by Mitterrand yesterday. The package is expected to include measures to cut imports and reduce French families' purchasing power.

The tone of Mitterrand's speech tonight, together with the composition of the new government, reflected the difficult political and economic compromises he is making. As the first Socialist leader of France in more than two decades, he is torn between political pressures from his left-wing electorate to protect working-class living standards and the demands of bankers and industrialists for more rigorous economic policies.

Political commentators here said that, while Mitterrand is aware of the scale of France's economic problems, he also is convinced that nothing can be achieved without support of the trade unions and left-wing activists.

Mitterrand tonight insisted that greater social progress had been achieved under his administration than in the previous half century. But he also acknowledged that progress in reducing inflation, which is running at about 9 percent, had been unsatisfactory.

"Yes, this policy is right . . . in spite of the difficulties we have made way on this arduous road and carried out more reforms in a few months . . . than France has known for half a century," he said.

As if to underline the urgency of the new government's task, figures were issued today showing that France ran up a trade deficit of more than $1 billion in February. Although this was an improvement over January, it was still far short of the government's objective of reducing the deficit to $6 billion this year and eliminating it by the end of 1984.

Mitterrand said that France had no intention of isolating itself from the rest of Europe or forgetting its obligations to the North Atlantic Treaty Organization. This passage appeared to be designed to dispel fears among other Europeans, particularly West Germany, that France might be tempted to follow more protectionist policies as a way out of its economic difficulties.

Describing the government's economic program as "a plan of rigor adapted to the times," Mitterrand said it would include measures to reduce the trade imbalance, cut inflation, keep the social security system in balance, develop savings and maintain a hold on government spending.