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As director of special projects for Francis Ford Coppola’s company American Zoetrope, he produced movies like Paul Schrader’s “Mishima: A Life in Four Chapters” (1985), a complicated film about Yukio Mishima, the eccentric Japanese author who killed himself publicly in 1970 — a passion project that Mr. Schrader has described as “the definition of an unfinanceable project.” Mr. Luddy was its tireless booster and supporter, funding it early on with his American Express card.