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Let <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«/math»</span> be the <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mi»§#960;«/mi»«/math»</span>- periodical function <br /><br /> <div style="text-align: center;"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»a«/mi»«mo»-«/mo»«mi»sin«/mi»«mfenced»«mfrac»«mi»x«/mi»«mn»2«/mn»«/mfrac»«/mfenced»«mo»,«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mfenced close=¨]¨ open=¨[¨»«mrow»«mn»0«/mn»«mo»,«/mo»«mi»§#960;«/mi»«/mrow»«/mfenced»«/math»</span><br /><br /> <div style="text-align: justify;">which meets the condition <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»f«/mi»«mo»(«/mo»«mo»-«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span> if <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§nbsp;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mfenced close=¨]¨ open=¨[¨»«mrow»«mo»-«/mo»«mi»§#960;«/mi»«mo»,«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/math»</span><br /><br />a) The value of a that makes the constant term of <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»SFT«/mi»«mo»(«/mo»«mi»f«/mi»«mo»)«/mo»«/math»</span> vanish: {1:SA:=#a}<br /><br />b) The series of Fourier of <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«/math»</span> is: <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mrow»«mo»§#8721;«/mo»«mo largeop=¨true¨»(«/mo»«/mrow»«mrow»«mi»n«/mi»«mo»§#10878;«/mo»«mn»1«/mn»«/mrow»«/munder»«/math»</span> {1:SA:=#an} <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»(«/mo»«mi»nx«/mi»«mo»)«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»+«/mo»«/math»</span> {1:SA:=#bn} <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»sin«/mi»«mo»(«/mo»«mi»nx«/mi»«mo»)«/mo»«/math»</span> <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo largeop=¨true¨»)«/mo»«/math»</span><br />c) The <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»SFT«/mi»«mo»(«/mo»«mi»f«/mi»«mo»)«/mo»«/math»</span> is pointwise convergent to <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«/math»</span> according to the Dirichlet criteria (T/F): {1:SA:=T}<br />d) The <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»SFT«/mi»«mo»(«/mo»«mi»f«/mi»«mo»)«/mo»«/math»</span> cannot be uniformly convergent to <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«/math»</span> since <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«/math»</span> is not continuous<span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8704;«/mo»«mo»§nbsp;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi»§#8477;«/mi»«/math»</span> (T/F): {1:SA:=F}<br />e) Using <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»SFT«/mi»«mo»(«/mo»«mi»f«/mi»«mo»)«/mo»«/math»</span>, the result of <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mo»§#8721;«/mo»«mrow»«mi»n«/mi»«mo»§#10878;«/mo»«mn»1«/mn»«/mrow»«/munder»«mfrac»«mn»1«/mn»«mrow»«mn»4«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»+«/mo»«munder»«mo»§#8721;«/mo»«mrow»«mi»n«/mi»«mo»§#10878;«/mo»«mn»1«/mn»«/mrow»«/munder»«mfrac»«msup»«mfenced»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/msup»«mrow»«mn»4«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/math»</span> is: {1:SA:=#sol}<br />f) Using <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»SFT«/mi»«mo»(«/mo»«mi»f«/mi»«mo»)«/mo»«/math»</span>, it can be proved that <span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mo»§#8721;«/mo»«mrow»«mi»n«/mi»«mo»§#10878;«/mo»«mn»1«/mn»«/mrow»«/munder»«mfrac»«mrow»«mi»n«/mi»«msup»«mfenced»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/mrow»«mrow»«mn»4«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«msqrt»«mn»2«/mn»«/msqrt»«mi»§#960;«/mi»«/mrow»«mn»8«/mn»«/mfrac»«/math»</span> (T/F): {1:SA:=F} </div></div>
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