India has always influenced West in a lot of ways. India has been a perennial theme in Hollywood films from times of yore- what with epic movies like The Discovery of India, Indiana Jones and the Temple of Doom, Octopussy etc. But these were movies that heavily revolved around Indian settings and characters, and I wouldn't call them an influence in the actual films.
They were Indian because they had to be.
But, there has been a noticeable trend in the mainstream films- like Dirty Harry, Scarface, Matrix etc where despite of their clearly Western themes, they are heavily dosed with Indian motifs.
Here's a list:
1. Scarface:
This movie needs little introduction- Al Pacino's bete noir film that set the trend for the many movies that followed with the bad-man-with-a-heart theme.
As much as the movie was famous- it's flagship song "Push it to the limit"
http://hardshipalarmnot.blogspot.com/2007/10/push-it-to-limit-ultimate-song-to.htmlwas phenomenally popular (in fact the whole soundtrack is a classic in 80's music). In what is primarily a Rock song, you find a short Veena interlude in the song to slow down the tempo of the song once it reaches a peak. Veena as we all know is a variant of a traditional Indian instrument.
2.Dirty Harry (Magnum Force):
In what is considered to be a quintessential American cop film, it is surprising to find that the background score is a medley of Western instruments with a dominating tabla score. The music is used to portray suspense and from the success of the film you can be assured that it was a good combination.
3. A Little Princess (Ramayan: Morning Raga):
This is a highly emotional film involving an orphaned little girl as the central character, and it's score derives itself from many Indian themes, one being the Ramayan. This musical piece- a flute rendition depicting sunrise is a lilting and soothing number, one which can be found only from Indian songs.
4. Matrix Revolutions (Navras):
The most famous of all Indian themes in Western films- the Navras song from Matrix Revolutions- a strong and powerful completely Indian song in a western movie. It was a revelation that shook the world as much as it pleasantly surprised India. The song was more talked about than the whole movie.