Hope stirs throughout the great Republic that the economic glooms afflicting it since early 1980 may finally be dispersing. Yet recovery will be slow. The federal government has through taxes and regulations enfeebled our productive capacity. It has also impoverished our citizenry through years of inflation and high taxes.

Older cities, particularly in the industrial Midwest, are expected to recover most slowly of all. Since the early 1960s, cities had increasingly looked to Washington to bail them out of their financial problems, but since 1978 the federal money spigot has been closing.

Cities that have been heavily reliant on federal largess -- for instance, New York -- are the models of the past. This summer, the public-private mix established in Indianapolis has been boomed as the model for the 1980s. Begun in the 1970s under Mayor Richard Lugar (who is now a U.S. senator) and perpetuated under Mayor William Hudnut, the Indianapolis plan fused the energies and capacities of business, foundations and private citizens to carry on government's affairs more economically and robustly. Indianapolis is not the only city to bring in the energies of its corporations, foundations and private citizens. Battle Creek, Trenton, Rochester and Pittsburgh are also using this public-private model successfully. But Indianapolis is being watched. The activities that its planners have long been anticipating are suddenly coming to life.

This week it opens its first annual International Violin Competition. The event might more likely be expected in Paris or Vienna, but it is in Indianapolis because it fits in so neatly with the regional development that Indianapolis has been following for the past decade, namely: building on established strengths with private, foundation and public funds. The Indiana University School of Music is one of the world's preeminent incubators of musical talent. From this base, Indianapolis is launching its contest. Using the enormous expertise and influence available to it, the organizers of the International Violin Competition of Indianapolis will be able to ensure the winner an initiation into concertizing that the young Paganini would never have imagined -- 30 North American engagements and 20 in Europe. Moreover, for two weeks music lovers will be drawn to Indianapolis to listen to serious music. Such civilized tourists may not spend as extravagantly as the gamblers in Atlantic City, but then they are not as given to leaping from hotel windows either.

The city will be ready for its whistling guests, for through a judicious mix of private and public money plus such devices as tax concessions, it has rebuilt its inner city with luxury hotels, shopping malls, restaurants and amusement parks. These have been built to serve another tourist attraction created on the basis of an established strength: medicine.

Thanks to the Eli Lilly pharmaceutical corporation and major medical research facilities at Indiana University's Medical Center, medical research has thrived for years. Noting the swarm of joggers and other health freaks multiplying in the land, Indianapolis' planners began investing in a sports research center -- jogging can bring pain. They built exquisite athletic facilities for a National Sports Festival held last month, and now claim that the city is "the amateur sports capital of the United States." On a far wider scale than the violin competition, this Sports Festival attracted attention and tourists' dollars to the city. And now an $80 million domed stadium is being built to lure professional athletic teams.

Some of the benefits derived from this growth have merely to do with the city's morale or sense of itself. Other benefits, however, are more tangible, involving jobs and investment. What is essential to cities like Indianapolis is that new companies begin and prosper. David L. Birch of MIT's Program on Neighborhood and Regional Change writes that the major creation of new jobs in the 1980s will be from new business, generally employing 100 or fewer employees.

Aware of this and of the huge numbers of new entrepreneurs who fail not because their products are duds but because their knowledge of business is shaky, Indianapolis' Lilly Endowment is funding a national center in the city to train entrepreneurs. The center was established in July with cheers from such crusading capitalists as former Treasury secretary William Simon and Nobel Prize-winning economist Milton Friedman.

Is the Indianapolis plan the blueprint for urban revival? The Reaganites obviously will think so, but it is too early to say. What can be said is that the effort shows a strategic intelligence and an awareness of economic realities. Washington cannot bankroll American cities forever. Somewhere along the line, Americans must create more wealth lest the economic glooms hang on.