From Wikipedia
Jeff Alexander (July 2, 1910 – December 23, 1989) was an American conductor, arranger, and composer of film, radio and television scores.
Born Myer Goodhue Alexander in Seattle, Washington, Alexander began performing in his teens as a singer and dancer in vaudeville productions. He then began playing piano and composing big band music.
In 1947, he moved to Los Angeles and began writing film and, later, television scores. His first film project was the score for Shall We Dance. He ultimately composed the scores to 35 films, including The Tender Trap (1955), Jailhouse Rock (1957), Kid Galahad (1962), and Support Your Local Sheriff! (1969).