Would any of these songs from the big band era be considered real jazz?

I'm making a compialation of the greatest jazz recordings of all time. I got to the big band era and I'm having a hard time distinguishing between real jazz and pop songs. For instance, would the early ballads by Nat King Cole and Frank Sinatra be considered jazz? I know most of the Glenn Miller classics... show more I'm making a compialation of the greatest jazz recordings of all time. I got to the big band era and I'm having a hard time distinguishing between real jazz and pop songs. For instance, would the early ballads by Nat King Cole and Frank Sinatra be considered jazz? I know most of the Glenn Miller classics would be considered jazz but what about the Dorseys? Here are a few I'm having trouble on.

Mack The Knife - Louis Armstrong (1955)
T'Ain't Whatcha Do - Jimmie Lunceford Band (1939)
Blues In The Night - Woody Herman w/his Orchestra (v) (1941)
Tenderly - Randy Brooks & His Orchestra (1946)
Sometimes I'm Happy - Lester Young (I) (1943)
Don't Be That Way - Benny Goodman Orchestra (1938)
The Blues My Naughty Sweetie Gives To Me - Sidney Bechet & His Hot Six (1951)
Why Don't You Do Right? - Benny Goodman Band w/Peggy Lee (v) (1942)
The Man I Love - Benny Goodman Quartet (I) (1937)
It's Only A Paper Moon - Nat "King" Cole Trio (1944)
Deep Purple - Larry Clinton Orchestra w/Bea Wain, (v) (1938))
A Tisket, A Tasket - Ella Fitzgerald w/Chick Webb Band (1937)
Oh! Look At Me Now - Tommy Dorsey Band w/Frank Sinatra & The Pied Pipers (1941)
Tangerine - Jimmy Dorsey Band w/Bob Eberly & Helen O'Connell (v) (1941)
Perfidia - Jimmy Dorsey Orchestra (1941)
Straighten Up & Fly Right - Nat "King" Cole Trio (1942)
Let's Get Away From It All - Tommy Dorsey Band w/Frank Sinatra, & The Pied Pipers (1941)
And The Angels Sing - Benny Goodman Orchestra w/Ziggy Elman (I) & Martha Tilton (v) (1939)
Taking A Chance On Love - Benny Goodman w/Helen Forrest (v) (1940-s, 1943-r)
Yes Indeed! - Tommy Dorsey Band w/Jo Stafford (v) (1941)
Night And Day - Billie Holiday (1939)
Song of India - Tommy Dorsey Orchestra (1937)
Moonglow - Artie Shaw & His Orchestra (1940)
The Man I Love - Billie Holiday (live, 1946) / (studio, 1939)
On The Sunny Side Of The Street - Tommy Dorsey Band w/The Clark Sisters (v) (1944)
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