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All the functions that one comes across be it continuous or discontinuous is often represented by Fourier series.
The%20summing%20up%20of%206%20sign%20functions%20that%20have%20discrete%20amplitudes%20and%20harmonised%20frequencies%20is%20called%20fourier%20series.%20With%20different%20combinations%20of%20sines%20and%20cosines%20allows%20individuals%20to%20observe%20arbitrary%20periodic%20signals.%20A%20signal%20that%20basically%20gets%20repeated%20after%20a%20period%20of%20time.%20Harmonic%20analysis%20is%20called%20the%20study%20of%20fourier%20series%20and%20helps%20in%20breaking%20down%20the%20arbitrary%20periodic%20function%28f%29%26nbsp%3B%20that%20further%20helps%20in%20studying%20the%20individual%20terms%20and%20the%20later%20on%20combined%20again%20to%20solve%20the%20initial%20problem.%20If%20a%20function%20is%20not%20periodic%2C%20then%20instead%20of%20using%20fourier%20series%2C%20fourier%20transform%20is%20used.%3Cbr%3E%3Cbr%3EPeter%20Gustav%20Lejeune%20Dirichlet%20gave%20a%20theorem%20for%20a%20discontinuous%20function%20which%20states%20that%3A%20If%20f%28x%29%20is%20a%20periodic%20of%20period%202%u03C0%2C%20and%20if%20between%20%u2212%u03C0%20and%20%u03C0%20it%20is%20single-valued%2C%20has%20a%20finite%20number%20of%20maximum%20and%20minimum%20values%2C%20and%20a%20finite%20number%20of%20discontinuities%2C%20and%20if%20R%20%u03C0%20%u2212%u03C0%20%7Cf%28x%29%7Cdx%20is%20finite%2C%20then%20the%20Fourier%20series%20converges%20to%20f%28x%29%20at%20all%20the%20points%20where%20f%28x%29%20is%20continuous%3B%20at%20jumps%2C%20the%20Fourier%20series%20converges%20to%20the%20midpoint%20of%20the%20jump%20%28This%20includes%20jumps%20that%20occur%20at%20%B1%u03C0%20for%20the%20periodic%20function%29.%26nbsp%3B%3Cbr%3E%3Cbr%3EThe%20fourier%20series%20is%20only%20used%20for%20analysing%20the%20components%20of%20frequency%20that%20have%20a%20set%20of%20given%20frequencies%20of%20a%20function.%20An%20example%20of%20discrete%20fourier%20transform-%26nbsp%3B%20H%28%u03C9%29%20%3D%20Z%20%u221E%20%u2212%u221E%20h%28t%29e%20i%u03C9tdt%20%283%29%20h%28t%29%20%3D%201%202%u03C0%20Z%20%u221E%20%u2212%u221E%20H%28%u03C9%29e%20%u2212i%u03C9td%u03C9.%20Many%20applications%20use%20the%20fourier%20series%20for%20experimental%20analysis%20or%20accurate%20mathematical%20techniques%20for%20a%20rigorous%20outcome.%3Cbr%3E%3Ca%20class%3D%22mention%22%20data-mention%3D%22%23maths%22%20data-user-id%3D%22X6s3JnXvspv2%22%20href%3D%22/topic/maths%22%3E%23maths%3C/a%3E%20%3Ca%20class%3D%22mention%22%20data-mention%3D%22%23Science%22%20data-user-id%3D%226RhBHV5lGAXq%22%20href%3D%22/topic/science%22%3E%23Science%3C/a%3E%20%3Ca%20class%3D%22mention%22%20data-mention%3D%22%23Application%22%20data-user-id%3D%22BORWj0H7Bjmk%22%20href%3D%22/topic/application%22%3E%23Application%3C/a%3E%26nbsp%3B
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