Nobody Ever Told You What the Fourier Series Actually Does

Comments · 3

• CR

  @CraigNull · 3 weeks ago

  &quot;The universe hums in circles&quot;<br><br>I&apos;d rephrase this as, the universe *can be expressed with circles*. But it can also be expressed in many other ways too, in an exactly analogous matter.<br><br>This idea, that an arbitrary function or wave can be decomposed into a sum of a given sequence of fundamental functions, is not unique to sines and cosines. There are also orthogonal sequences of polynomials, wherein any function on an interval can be expressed as a sum of those specific polynomials. Examples are Hermite polynomials, Chebyshev polynomials (in the appropriate context in which orthogonality applies, anyway. Chebyshev polynomials are for functions restricted to [-1, 1], for instance).<br><br>You can make orthonormal sequences of triangular waves, then express any function as a sum of those. Off the top of my head I don&apos;t recall the convergence or smoothness criteria, because it sounds dicey to express a smooth function as a sum of jagged ones. But you&apos;ll still get pointwise convergence, at least. Does that mean the universe hums in triangles? Maybe so, if you find triangle functions innately appealing.<br><br>In fact, you can start with ANY sequence of linearly independent functions you like, f1, f2, f3, ... then, express an arbitrary function F as a linear combination of them, a1f1 + a2f2 + ... in the following manner: first, apply the Gram-Schmidt process to f1, f2, f3, .... to create another sequence of functions, g1, g2, g3, ... that are mutually orthogonal, where g1 = f1, g2 is a linear combination of f1 and f2, and in general gk is a linear combination of f1, f2, ..., fk. Then you can use a Fourier-style integral formula to find the coefficients with which arbitrary function F can be expressed as a linear combination of the g1, g2, g3, ... . Then back-substitute the gk functions for the fk you started with. Thereby, you&apos;ve expressed an arbitrary function F as a linear combination of any infinite sequence of linearly independent functions you wish. See also the topic of Wavelet Analysis for furtherance of the topic of express arbitrary signals as a sum of arbitrary component pieces.<br><br>The sinusoid of specifically trigonometric functions is not unique in this regard. You might say, they&apos;re special because they&apos;re prettier, or their origin from circles makes them more natural. It&apos;s often useful, specifically because due to the smoothness of these functions, regularity, boundedness over the whole real line, and that they satisfy the simple differential equation y&apos;&apos; + y = 0 (or harmonics thereof), which makes expressing functions as sums of sines and cosines uniquely useful in applications in physics and elsewhere. But within the question of: when can you express an arbitrary function as a sum of a given sequence of functions, and how might you go about it in a simple way, sines and cosines are not distinguished

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• MA

  @Marian-r3k · 2 weeks ago

  la IA hace los guiones mas aburridos ...