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to speak or think favorably of; pronounce or consider agreeable or good; judge favorably:
to approve the policies of the administration.
to consent or agree to:
Father approved our plan to visit Chicago.
to confirm or sanction formally; ratify:
The Senate promptly approved the bill.
Obsolete.

to demonstrate; show.
to make good; attest.
to prove by trial.
to convict.

to speak or consider favorably (sometimes followed by of):
Mother didn’t approve of him. The boss wouldn’t approve of the plan. He said that he approved.
Contemporary Examples

Catsoulis aptly—and, I think, approvingly—calls this “a method that spits on context.”
Tears of Gaza: Why Context Matters Sigal Samuel September 24, 2012

Peter approvingly cites the refusal of A.B. Yehoshua and David Grossman to go speak in Ariel.
Both Sides Now Yehudah Mirsky March 19, 2012

Historical Examples

In which attitude, with Mrs. Vint patting him approvingly on the back, they were surprised by Paul Carrick.
The Atlantic Monthly, Volume 18, No. 105, July 1866 Various

“That’s the way to talk, darlint,” said his mother, approvingly.
Brave and Bold Horatio Alger

“Ah; that proves you have a Kind Heart,” remarked the Scarecrow, approvingly.
The Tin Woodman of Oz L. Frank Baum

“Good for you, Max; I can see you’ve got an idea,” cried out Jim, approvingly.
With Trapper Jim in the North Woods Lawrence J. Leslie

“Them’s my sentiments,” nodded Mr. Dibbs, approvingly, helping himself to more wine and pine-apple.
Grif B. L. (Benjamin Leopold) Farjeon

“My, but you are giving me a whole lot,” he said, watching her approvingly and encouragingly.
David Dunne Belle Kanaris Maniates

“Yes; that is all very nicely told,” the old gentleman said approvingly.
Little Folks (November 1884) Various

“You do manage to have a good time, all right,” he said approvingly.
Four Little Blossoms at Brookside Farm Mabel C. Hawley

verb
when intr, often foll by of. to consider fair, good, or right; commend (a person or thing)
(transitive) to authorize or sanction
(transitive) (obsolete) to demonstrate or prove by trial
verb
(transitive) (law) to improve or increase the value of (waste or common land), as by enclosure
v.

c.1300, “to demonstrate, prove;” mid-14c., “to attest (something) with authority,” from Old French aprover (Modern French approuver) “approve, agree to,” from Latin approbare “to assent to as good, regard as good,” from ad- “to” (see ad-) + probare “to try, test something (to find if it is good),” from probus “honest, genuine” (see prove).

The meaning extended late 14c. to “to sanction, endorse, confirm formally” then to “assent to (something) as good” (early 15c.), especially in reference to the actions of authorities, parliaments, etc. Related: Approved; approving.