There are no copyright markings on either side of the postcard, as can be seen on front and back views.
Ayres and his fellow musicians started the band known as "Fashions in Music" in 1937, continuing with it into the 1940s. This postcard was created by the band as a promotional item for it.
Mitchell Ayres no longer referred to the band he was leading by 1948; at the time he was signed for The Chesterfield Supper Club, his group was known as Mitchell Ayres and his Orchestra, or the Mitchell Ayres Orchestra. The name "Fashions in Music" was no longer used.
Mitchell Ayres died in 1969, so this item is well within the pre-1978 guidelines for unmarked material.
United States Copyright Office page 2 "Visually Perceptible Copies The notice for visually perceptible copies should contain all three elements described below. They should appear together or in close proximity on the copies.
2 The year of first publication. If the work is a derivative work or a compilation incorporating previously published material, the year date of first publication of the derivative work or compilation is sufficient. Examples of derivative works are translations or dramatizations; an example of a compilation is an anthology. The year may be omitted when a pictorial, graphic, or sculptural work, with accompanying textual matter, if any, is reproduced in or on greeting cards, postcards, stationery, jewelry, dolls, toys, or useful articles.
Note that it may still be copyrighted in jurisdictions that do not apply the rule of the shorter term for US works (depending on the date of the author's death), such as Canada (50 p.m.a.), Mainland China (50 p.m.a., not Hong Kong or Macao), Germany (70 p.m.a.), Mexico (100 p.m.a.), Switzerland (70 p.m.a.), and other countries with individual treaties.