Many sine functions

Adding many sine functions yields a so-called Fourier series (see page 1074). The pictures below show Sum[Sin[n x]/n, {n, k}]

Sum[Sin[n x]/n, {n, k}] for various numbers of terms k. Apart from a glitch that gets narrower with increasing k (the so-called Gibbs phenomenon), the result has a simple triangular form. Other so-called Fourier series in which the coefficient of Sin[m x]

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Sin[m x] is a smooth function of m for all integer m yield similarly simple results.

The pictures below show Sum[Sin[n² x]/n², {n, k}]

Sum[Sin[\!\(\*SuperscriptBox[\(n\),\(2\)]\) x]/\!\(\*SuperscriptBox[\(n\),\(2\)]\),{n,k}, where in effect all coefficients of Sin[m x]

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Sin[m x] other than those where m is a perfect square are set to zero. The result is a much more complicated curve. Note that for x of the form p π/q

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p π/q, the k = ∞

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k = ∞ sum is just

(π/q/(2q))² Sum[Sin[n² p π/q]/Sin[n π/(2q)]², {n, q - 1}]

\!\(\*SuperscriptBox[\((\[Pi]/(2q))\),\(2\)]\) \!\(\*SuperscriptBox[\(Sum[Sin[\!\(\*SuperscriptBox[\(n\),\(2\)]\) p \[Pi]/q]/Sin[n \[Pi]/(2q)]\),\(2\)]\),{n,q-1}]

The pictures below show Sum[Cos[2ⁿ x], {n, k}]

Sum[Cos[\!\(\*SuperscriptBox[\(2\),\(n\)]\) x],{n,k}] (as studied by Karl Weierstrass in 1872). The curves obtained in this case show a definite nested structure, in which the value at a point x is essentially determined directly from the base 2 digit sequence of x. (See also page 1080.)

The curves below are approximations to Sum[Cos[2ⁿ x]/2^{a n}, {n, ∞}]

Sum[Cos[\!\(\*SuperscriptBox[\(2\),\(n\)]\) x]/\!\(\*SuperscriptBox[\(2\),\(a n\)]\),{n,\[Infinity]}]. They can be thought of as having dimensions 2 - a

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2 - a and smoothed power spectra ω^{-(1 + 2a)}

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\!\(\*SuperscriptBox[\(\[Omega]\),\(-(1+2a)\)]\).