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Let C[-pi,pi] be the set of continuous function from [-pi,pi] to C. Endow this with usual inner product (<f,g>= integral from -pi to pi of f multiplied by g conjugate, and let ||.|| be the corresponding norm).

Let h(n) be fourier coefficent of f

Now, |h(n)|<_ 1/2pi( ||f||.||e^int||) by schwarz inequaity

=1/pi . 1/ 2^(0.5) ||f|| since ||e^int|| =(pi +pi)^0.5

Do you agree?

Let h(n) be fourier coefficent of f

Now, |h(n)|<_ 1/2pi( ||f||.||e^int||) by schwarz inequaity

=1/pi . 1/ 2^(0.5) ||f|| since ||e^int|| =(pi +pi)^0.5

Do you agree?

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