Oscillatory Cancellation of Integral

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Under which conditions for a real function $f(x)$ is
$$
left| int_1^x left fleft(fracxwright) - fleft(leftlfloor fracxw rightrfloorright) right cosleft(wright) , rm dw right|
$$
finite, as $xrightarrow infty$.



The point here is that the $cos $ function is present for cancellation of most parts. I was thinking to expand
$$
fleft(fracxwright) = fleft(leftlfloorfracxwrightrfloorright) + f'left(leftlfloorfracxwrightrfloorright) underbraceleft(fracxw - leftlfloorfracxwrightrfloor right)_cal O(1) + cal Oleft( left(fracxw - leftlfloorfracxwrightrfloor right)^2 right)
$$
so that $|f'(x)|$ must at least be bounded as $xrightarrow infty$.



But is it true that for any $cal O(1)$ function $g(x)$
$$
left| int_1^x g(w) cosleft(wright) , rm dw right|
$$
is finite as $xrightarrow infty$? I mean it is obviously not true if I choose $g(x)=cos(x)$, so there must be some restrictions as in $cos(x)$ and $g(x)$ are not correlated.



I hope it is clear what I want. I was trying to make it somewhat rigorous.










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    Under which conditions for a real function $f(x)$ is
    $$
    left| int_1^x left fleft(fracxwright) - fleft(leftlfloor fracxw rightrfloorright) right cosleft(wright) , rm dw right|
    $$
    finite, as $xrightarrow infty$.



    The point here is that the $cos $ function is present for cancellation of most parts. I was thinking to expand
    $$
    fleft(fracxwright) = fleft(leftlfloorfracxwrightrfloorright) + f'left(leftlfloorfracxwrightrfloorright) underbraceleft(fracxw - leftlfloorfracxwrightrfloor right)_cal O(1) + cal Oleft( left(fracxw - leftlfloorfracxwrightrfloor right)^2 right)
    $$
    so that $|f'(x)|$ must at least be bounded as $xrightarrow infty$.



    But is it true that for any $cal O(1)$ function $g(x)$
    $$
    left| int_1^x g(w) cosleft(wright) , rm dw right|
    $$
    is finite as $xrightarrow infty$? I mean it is obviously not true if I choose $g(x)=cos(x)$, so there must be some restrictions as in $cos(x)$ and $g(x)$ are not correlated.



    I hope it is clear what I want. I was trying to make it somewhat rigorous.










    share|cite|improve this question

























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      Under which conditions for a real function $f(x)$ is
      $$
      left| int_1^x left fleft(fracxwright) - fleft(leftlfloor fracxw rightrfloorright) right cosleft(wright) , rm dw right|
      $$
      finite, as $xrightarrow infty$.



      The point here is that the $cos $ function is present for cancellation of most parts. I was thinking to expand
      $$
      fleft(fracxwright) = fleft(leftlfloorfracxwrightrfloorright) + f'left(leftlfloorfracxwrightrfloorright) underbraceleft(fracxw - leftlfloorfracxwrightrfloor right)_cal O(1) + cal Oleft( left(fracxw - leftlfloorfracxwrightrfloor right)^2 right)
      $$
      so that $|f'(x)|$ must at least be bounded as $xrightarrow infty$.



      But is it true that for any $cal O(1)$ function $g(x)$
      $$
      left| int_1^x g(w) cosleft(wright) , rm dw right|
      $$
      is finite as $xrightarrow infty$? I mean it is obviously not true if I choose $g(x)=cos(x)$, so there must be some restrictions as in $cos(x)$ and $g(x)$ are not correlated.



      I hope it is clear what I want. I was trying to make it somewhat rigorous.










      share|cite|improve this question















      Under which conditions for a real function $f(x)$ is
      $$
      left| int_1^x left fleft(fracxwright) - fleft(leftlfloor fracxw rightrfloorright) right cosleft(wright) , rm dw right|
      $$
      finite, as $xrightarrow infty$.



      The point here is that the $cos $ function is present for cancellation of most parts. I was thinking to expand
      $$
      fleft(fracxwright) = fleft(leftlfloorfracxwrightrfloorright) + f'left(leftlfloorfracxwrightrfloorright) underbraceleft(fracxw - leftlfloorfracxwrightrfloor right)_cal O(1) + cal Oleft( left(fracxw - leftlfloorfracxwrightrfloor right)^2 right)
      $$
      so that $|f'(x)|$ must at least be bounded as $xrightarrow infty$.



      But is it true that for any $cal O(1)$ function $g(x)$
      $$
      left| int_1^x g(w) cosleft(wright) , rm dw right|
      $$
      is finite as $xrightarrow infty$? I mean it is obviously not true if I choose $g(x)=cos(x)$, so there must be some restrictions as in $cos(x)$ and $g(x)$ are not correlated.



      I hope it is clear what I want. I was trying to make it somewhat rigorous.







      real-analysis oscillatory-integral






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      edited Sep 10 at 21:20

























      asked Sep 10 at 21:08









      Diger

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