In-class Demonstration: Fourier Approximations

By Russell Blyth and Mike May, S.J.

restart:with(plots):

Warning, the name changecoords has been redefined

We start by using Maple to perform some tedious integrals for us.

Verify that the functions used as basis vectors in the approximating subspace are orthogonal.

Int(sin(i*Pi*x)*sin(j*Pi*x),x=-1..1) = int(sin(i*Pi*x)*sin(j*Pi*x),x=-1..1); Int(sin(i*Pi*x)*sin(i*Pi*x),x=-1..1) = int(sin(i*Pi*x)*sin(i*Pi*x),x=-1..1); Int(sin(i*Pi*x)*cos(j*Pi*x),x=-1..1) = int(sin(i*Pi*x)*cos(j*Pi*x),x=-1..1); Int(cos(i*Pi*x)*cos(j*Pi*x),x=-1..1) = int(cos(i*Pi*x)*cos(j*Pi*x),x=-1..1); Int(cos(i*Pi*x)*cos(i*Pi*x),x=-1..1) = int(cos(i*Pi*x)*cos(i*Pi*x),x=-1..1);

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That left us with some work still to do. Try again. This time we first tell Maple that i and j are integers (Maple assumes i and j are nonzero):

assume(i,integer): assume(j,integer): interface(showassumed=0):

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

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtRiQ2Jy1JKG1zdWJzdXBHRiU2Jy1JI21vR0YlNjRRJiZpbnQ7RigvJSVmb3JtR1EhRigvJSZmZW5jZUdRJmZhbHNlRigvJSpzZXBhcmF0b3JHRjovJSdsc3BhY2VHUSQwZW1GKC8lJ3JzcGFjZUdGPy8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGKC8lJXNpemVHUSMxMkYoLyUrZm9yZWdyb3VuZEdRLlsxNDQsMTQ0LDE0NF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvSSttc2VtYW50aWNzR0YlUSZpbmVydEYoLUkjbW5HRiU2OVEpJm1pbnVzOzFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdGVy8lJWJvbGRHRjovJSdpdGFsaWNHRjovJSp1bmRlcmxpbmVHRjovJSpzdWJzY3JpcHRHRjovJSxzdXBlcnNjcmlwdEdGOi8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GKC8lK2JhY2tncm91bmRHRmduLyUnb3BhcXVlR0Y6LyUrZXhlY3V0YWJsZUdGOi8lKXJlYWRvbmx5R1EldHJ1ZUYoLyUpY29tcG9zZWRHRjovJSpjb252ZXJ0ZWRHRjovJStpbXNlbGVjdGVkR0Y6LyUscGxhY2Vob2xkZXJHRjovJTBmb250X3N0eWxlX25hbWVHRlQvJSptYXRoY29sb3JHRmBwLyUvbWF0aGJhY2tncm91bmRHRmduLyUrZm9udGZhbWlseUdGYW8vJSxtYXRodmFyaWFudEdRJ25vcm1hbEYoLyUpbWF0aHNpemVHRlctRlxvNjlGS0Zfb0Zib0Zkb0Zmb0Zob0Zqb0ZccEZecEZhcEZjcEZlcEZncEZqcEZccUZecUZgcUZicUZkcUZmcUZocUZqcUZdci8lMXN1cGVyc2NyaXB0c2hpZnRHUSIyRigvJS9zdWJzY3JpcHRzaGlmdEdRIjBGKC1GJDYlLUYkNiUtSSNtaUdGJTY5USRzaW5GKEZfb0Zib0Zkb0Zmb0Zob0Zqb0ZccEZecEZhcEZjcEZlcEZncEZqcEZccUZecUZgcUZicUZkcUZmcUZocUZqcUZdci1GMjYzUTAmQXBwbHlGdW5jdGlvbjtGKC9GNlEmaW5maXhGKEY4RjtGPUZARkJGREZGRklGTEZORlBGUkZVL0ZZRmBwRmVuLUYkNiUtRjI2M1EiKEYoL0Y2USdwcmVmaXhGKC9GOUZpcEY7L0Y+US50aGlubWF0aHNwYWNlRigvRkFGXnQvRkNGaXBGREZGRklGTEZORlBGUkZVRmRzRmVuLUYkNiMtRiQ2Jy1GXHM2OVEiaUYoRl9vRmJvRmRvL0Znb0ZpcEZob0Zqb0ZccEZecEZhcEZjcEZlcEZncEZqcEZccUZecUZgcUZicUZkcUZmcUZocS9GW3JRJ2l0YWxpY0YoRl1yLUYyNjNRMSZJbnZpc2libGVUaW1lcztGKEZic0Y4RjtGPUZARkJGREZGRklGTEZORlBGUkZVRmRzRmVuLUZcczY5USNQaUYoRl9vRmJvRmRvRmZvRmhvRmpvRlxwRl5wRmFwRmNwRmVwRmdwRmpwRlxxRl5xRmBxRmJxRmRxRmZxRmhxRmpxRl1yRlt1LUZcczY5USJ4RihGX29GYm9GZG9GaHRGaG9Gam9GXHBGXnBGYXBGY3BGZXBGZ3BGanBGXHFGXnFGYHFGYnFGZHFGZnFGaHFGaXRGXXItRjI2M1EiKUYoL0Y2UShwb3N0Zml4RihGXHRGO0ZddC9GQVEydmVyeXRoaW5tYXRoc3BhY2VGKEZgdEZERkZGSUZMRk5GUEZSRlVGZHNGZW5GW3UtRiQ2JS1GXHM2OVEkY29zRihGX29GYm9GZG9GZm9GaG9Gam9GXHBGXnBGYXBGY3BGZXBGZ3BGanBGXHFGXnFGYHFGYnFGZHFGZnFGaHFGanFGXXJGX3MtRiQ2JUZncy1GJDYjLUYkNictRlxzNjlRImpGKEZfb0Zib0Zkb0ZodEZob0Zqb0ZccEZecEZhcEZjcEZlcEZncEZqcEZccUZecUZgcUZicUZkcUZmcUZocUZpdEZdckZbdUZedUZbdUZhdUZkdS1JJ21zcGFjZUdGJTYmLyUnaGVpZ2h0R1EnMC4wfmV4RigvJSZ3aWR0aEdRJzAuM35lbUYoLyUmZGVwdGhHRl53LyUqbGluZWJyZWFrR1ElYXV0b0YoLUYyNjRRMCZEaWZmZXJlbnRpYWxEO0YoRmpzRjhGO0Y9RkBGQkZERkZGSUZMRk5GUEZSRlVGWEZlbkZobkZhdS1GMjYzUSI9RihGYnNGOEY7L0Y+US90aGlja21hdGhzcGFjZUYoL0ZBRl54RkJGREZGRklGTEZORlBGUkZVRmRzRmVuLUZcbzY5USIwRihGX29GYm9GZG9GZm9GaG9Gam9GXHBGXnBGYXBGY3BGZXBGZ3BGanBGXHFGXnFGYHFGYnFGZHFGZnFGaHFGanFGXXI3Iy8tSSRJbnRHNiQlKnByb3RlY3RlZEdGKjYkKiYtSSRzaW5HRmd4NiMqKEkjaXxpckdGKCIiIkkjUGlHRmh4RmB5SSJ4R0YoRmB5RmB5LUkkY29zR0ZneDYjKihJI2p8aXJHRihGYHlGYXlGYHlGYnlGYHlGYHkvRmJ5OyEiIkZgeSIiIQ==

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

We need to also integrate against 1:

Int(sin(i*Pi*x)*1,x=-1..1) = int(sin(i*Pi*x)*1,x=-1..1); Int(cos(i*Pi*x)*1,x=-1..1) = int(cos(i*Pi*x)*1,x=-1..1); Int(1*1,x=-1..1) = int(1*1,x=-1..1);

To compensate for the value of this final integral, we must remember to half the coefficient of this constant function later.

Graph several of these integrands as a reality check - note that in each case it appears that the signed area from x=-1 to x=1 has the appropriate value (0 or 1):

plot(sin(2*Pi*x)*sin(3*Pi*x),x=-1..1);

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

plot(sin(3*Pi*x)*cos(4*Pi*x),x=-1..1);

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plot(cos(5*Pi*x)*cos(3*Pi*x),x=-1..1);

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plot(sin(3*Pi*x)*sin(3*Pi*x),x=-1..1);

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Example

Find Fourier approximations for f(x) = x.

We compute the Fourier coefficients for f(x) = x

Int(x*sin(j*Pi*x),x=-1..1) = int(x*sin(j*Pi*x),x=-1..1); Int(x*cos(j*Pi*x),x=-1..1) = int(x*cos(j*Pi*x),x=-1..1); Int(x,x=-1..1) = int(x,x=-1..1);

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Note there are only sine terms (not surprisingly, since f is an odd function).

Next use Maple to write a general expression for the order n approximation

Fapprox := (x,n) -> sum(-2*(-1)^j/(j*Pi)*sin(j*Pi*x),j=1..n);

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We can now plot (say) a 10 term approximation against f(x).

eval(Fapprox(x,10)); plot([Fapprox(x,10),x],x=-2..2,y = -2..2, title = " 10th Order Fourier approximation to y =x", titlefont = [HELVETICA,12], color = [green,blue]);

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