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 _ ?

Question

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      _      ?

trantegisis · Accepted Answer

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        .

letumsnemesislh · Accepted Answer

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

Let u have the cosine series representation u = <munderover> ∑

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