The purpose of this problem is that I want to prove that for any λ integrable function f on a bounded closed interval [a,b] holds lim n → ∞ ∫ [ a , b ] f ( x ) sin ⁡ ( n x ) d λ = 0.I have submitted a proof below.

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The purpose of this problem is that I want to prove that for any   λ integrable function f on a bounded closed interval [a,b] holds      lim          n      →      ∞            ∫          [      a      ,      b      ]        f  (  x  )  sin  ⁡  (  n  x  )  d  λ  =  0.I have submitted a proof below.

treccinair · Accepted Answer

The smooth functions with compact suppport             C        0          ∞        (  [  a  ,  b  ]  ) functions are dense over the Lebesgue integral functions       L    1    [  a  ,  b  ]. Then for every f there is a sequence   (      f          k            )    k    ⊂            C        0          ∞        (  [  a  ,  b  ]  ) such that   ‖  f  −      f    k        ‖    1        →          k      →      ∞            0  . We observe that                              |                      ∫            a            b                                f            k                    (          x          )          sin          ⁡          (          n          x          )          d          x          |                =                  |          −                      1            n                                ∫            a            b                                f            k            ′                    (          x          )          cos          ⁡          (          n          x          )          d          x          |                =                  1                      n            2                                    |                      ∫            a            b                                f            k            ″                    (          x          )          sin          ⁡          (          n          x          )          d          x          |                ≤                              M            k                                n            2                                    →                      n            →            ∞                                    0                    Where       M    k    =      sup          x      ∈      [      a      ,      b      ]            |        f    k    ″    (  x  )      |    (  b  −  a  )  &amp;lt;  ∞. Now                              |                      ∫            a            b                    f          (          x          )          sin          ⁡          (          n          x          )          d          x          |                                    ≤                  |                      ∫            a            b                                f            k                    (          x          )          sin          ⁡          (          n          x          )          d          x          |                +                  |                      ∫            a            b                    (          f          (          x          )          −                      f            k                    (          x          )          )          sin          ⁡          (          n          x          )          d          x          |                                                  ≤                              M            k                                n            2                          +        ‖                  f          k                −        f                  ‖          1                          →                      n            →            ∞                          ‖                  f          k                −        f                  ‖          1                          →                      k            →            ∞                          0.

The purpose of this problem is that I want to prove that for any λ integrable

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