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icedagecs 2022-07-03 Answered
Let u have the cosine series representation
u = k 1 = 0 k 2 = 0 a k _ cos ( 2 π k 1 x L 1 ) cos ( 2 π k 2 y L 2 )
What are the coefficients A k _ in
u 2 = k 1 = 0 k 2 = 0 A k _ cos ( 2 π k 1 x L 1 ) cos ( 2 π k 2 y L 2 )
in terms of the a k _ ?
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Answers (2)

trantegisis
Answered 2022-07-04 Author has 20 answers
In 1D,
( a k cos ( k t ) ) 2 = k = 0 j = 0 a k a j cos ( k t ) cos ( j t ) = 1 2 k = 0 j = 0 a k a j ( cos ( ( k j ) t ) + cos ( ( k + j ) t ) ) = 1 2 k = 0 j = k a k a k j cos ( k t ) + 1 2 k = 0 j = k a k a j k cos ( k t )
so that
A k = j = k a k a k j + j = k a k a j k = 1 2 a k 2 + j = 1 a k a j .

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letumsnemesislh
Answered 2022-07-05 Author has 6 answers
cos m x cos n x = cos ( m + n ) x + cos ( m n ) x 2
4 A i j = m n = i p q = j a m n a p q + m + n = i p q = j a m n a p q + m n = i p + q = j a m n a p q + m + n = i p + q = j a m n a p q

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