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Functional Approximation through Finite Fourier Series INSTITUTO DE ESTUDIOS SUPERIORES DE TAMAULIPAS MEXICO, MMVI Prepared by David Macias Ferrer E-mail: david.macias@iest.edu.mx Madero City, Mexico URL: http://www.geocities.com/dmacias_iest/MyPage.html
<Text-field style="Heading 1" layout="Heading 1">Goals</Text-field> To aproximate a Piecewise Continuous Function through Trigonometric Polynomials commonly called Fourier Partial Sums or Finite Fourier Series. To show the convergence of these aproximation via Bessel's Inequality and using Maple spreadsheets. To verify the Weiersstras's Theorem applied to case. To show the powerful Maple 8 graphics tools to visualize the application of Weierstrass's Theorem.
<Text-field style="Heading 1" layout="Heading 1">Fourier Partial Sums</Text-field> The theory of approximation of functions is one of the central branches in mathematical analysis and has been developed over a number of decades. In 1822 the French mathematician Jean Baptiste Joseph Fourier (1768-1830) had completed his important essay "Th\303\251orie Analytique de la Chaleur". Although in his work, the current "Fourier series" don't appear, Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable. In 1900 the Hungarian mathematician Lip\303\263t Fej\303\251r (1880-1959) published a fundamental summation theorem for Fourier series; he worked on power series and on potential theory. Much of his work is on Fourier series and their singularities but he also contributed to Approximation Theory. Another important contribution to Approximation Theory was the work of the russian mathematician Pafnuty Lvovich Chebyshev (1821-1894) "Th\303\251orie des M\303\251canismes Connus sous le nom de Parall\303\251logrammes" published in 1854. It was in this work that his famous Chebyshev Orthogonals Polynomials appeared for the first time. In this work the notion NiMyLSUkYWJzRzYjLCYtJSJmRzYjJSJ4RyIiIi0lIlBHRiohIiIlKGVwc2lsb25H is found. We will use the trigonometric polynomials approximation of the form: NiMvLSUiZ0c2IyUieEcsJiomJiUiYUc2IyIiISIiIiIiIyEiIkYuLSUkc3VtRzYkLCYqJiZGKzYjJSJuR0YuLSUkY29zRzYjKiZGOEYuRidGLkYuRi4qJiYlImJHRjdGLi0lJHNpbkdGO0YuRi4vRjg7Ri4lIk5HRi4= to particular case of a periodic piecewise continuous function NiMtJSJmRzYjJSJ4Rw== defined on closed interval NiM3JCwkJSNQaUchIiJGJQ==. restart: The piecewise continuous function NiMtJSJmRzYjJSJ4Rw== is given by: f :=x->piecewise(-Pi<=x and x<0,-2*x,0<x and x<=Pi,3*x); Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmMzEsJEkjUGlHRishIiI5JDJGMiIiISwkRjIhIiMzMkY0RjIxRjJGMCwkRjIiIiRGJUYlRiU= the graphic is: plot(f(x),x=-Pi..Pi,color=blue); 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 We will calculate the Fourier Coefficients: cos(n*Pi):=(-1)^n; KSEiIkkibkc2Ig== sin(n*Pi):=0; IiIh a[0]:=1/Pi*int(f(x),x=-Pi..Pi); LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiYiIiM= a[n]:=1/Pi*int((f(x)*cos(n*x)),x=-Pi..Pi); LCQqKEkjUGlHJSpwcm90ZWN0ZWRHISIiLCYpRiZJIm5HNiIiIiJGJkYrRitGKSEiIyIiJg== simplify(%,trig); LCQqKEkjUGlHJSpwcm90ZWN0ZWRHISIiLCYpRiZJIm5HNiIiIiJGJkYrRitGKSEiIyIiJg== which may rewritten as: a[n]:=(5/(n^2*Pi))*((-1)^n-1); LCQqKEkjUGlHJSpwcm90ZWN0ZWRHISIiLCYpRiZJIm5HNiIiIiJGJkYrRitGKSEiIyIiJg== b[n]:=1/Pi*int((f(x)*sin(n*x)),x=-Pi..Pi); KiZJI1BpRyUqcHJvdGVjdGVkRyEiIiwmKigpRiVJIm5HNiIiIiJGKUYlRiNGKyIiIyooKUYlLCZGK0YrRilGK0YrRilGJUYjRisiIiRGKw== b[n]:=(-1)^(n+1)/n; KiYpISIiLCYiIiJGJkkibkc2IkYmRiZGJ0Yk The following theorem is verified Theorem.- Let NiMtJSJmRzYjJSJ4Rw== be a piecewise continuous function on NiM3JCwkJSNQaUchIiJGJQ== the Fourier coefficients associated satisfy the following relationships: NiMvLSUmbGltaXRHNiQmJSJhRzYjJSJuRy9GKiUpaW5maW5pdHlHIiIh and NiMvLSUmbGltaXRHNiQmJSJiRzYjJSJuRy9GKiUpaW5maW5pdHlHIiIh with the Maple commands the previous theorem is verified: limit('a[n]','n'=infinity)=limit(a[n],n=infinity); Ly1JJmxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQmSSJhR0YoNiNJIm5HRigvRi1JKWluZmluaXR5R0YmIiIh limit('b[n]','n'=infinity)=limit(b[n],n=infinity); Ly1JJmxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQmSSJiR0YoNiNJIm5HRigvRi1JKWluZmluaXR5R0YmIiIh The Finite Fourier Series associated to function is: NiMvLSUiZ0c2IyUieEcsJiooIiImIiIiIiIlISIiJSNQaUdGK0YrLSUkU3VtRzYkLCYqKkYqRisqJiklIm5HIiIjRitGLkYrRi0sJiksJEYrRi1GNkYrRitGLUYrLSUkY29zRzYjKiZGNkYrRidGK0YrRisqKClGOiwmRjZGK0YrRitGK0Y2Ri0tJSRzaW5HRj1GK0YrL0Y2O0YrJSlpbmZpbml0eUdGKw== NiMtJSN8aXJmRzYjJSJ4Rw== Exist several types of convergence criteria for functional approximations. One of the most useful for Fourier Approximations is NiMqJCklIkxHIiIjIiIi. This clearly implies the following result: Theorem. Bessel's Inequality. Let NiMtJSJmRzYjJSJ4Rw== be a function defined on NiM3JCwkJSNQaUchIiJGJQ== such that NiMqJCktJSJmRzYjJSJ4RyIiIyIiIg== has a finite integral on NiM3JCwkJSNQaUchIiJGJQ==. If NiMmJSJhRzYjJSJuRw== and NiMmJSJiRzYjJSJuRw== are the Fourier coefficient of the function NiMtJSJmRzYjJSJ4Rw==, then we have: NiMxLCYqJikmJSJhRzYjIiIhIiIjIiIiRishIiJGLC0lJHN1bUc2JCwmKiQpJkYoNiMlIm5HRitGLEYsKiQpJiUiYkdGNUYrRixGLC9GNjtGLCUpaW5maW5pdHlHRiwqKEYsRiwlI1BpR0YtLSUkSW50RzYkKiQpLSUiZkc2IyUieEdGK0YsL0ZIOywkRj9GLUY/Riw= In particular the series NiMtJSRzdW1HNiQsJiokKSYlImFHNiMlIm5HIiIjIiIiRi4qJCkmJSJiR0YrRi1GLkYuL0YsO0YuJSlpbmZpbml0eUc= is convergent. We will use this criterion but using the inequality: NiMxLCYqJikmJSJhRzYjIiIhIiIjIiIiRishIiJGLC0lJHN1bUc2JCwmKiQpJkYoNiMlIm5HRitGLEYsKiQpJiUiYkdGNUYrRixGLC9GNjtGLCUiTkdGLCooRixGLCUjUGlHRi0tJSRJbnRHNiQqJCktJSJmRzYjJSJ4R0YrRiwvRkg7LCRGP0YtRj9GLA== this is, the Fourier partial series for NiMvJSJuRyUiTkc=. Using the Maple command we have: 1/Pi*Int(('f(x)^2'),x=-Pi..Pi)>='a[0]^2'/2+sum('a[n]^2+b[n]^2','n'=1..N); MSwmKiQmSSJhRzYiNiMiIiEiIiMjIiIiRiotSSRzdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2JCwmKiQmRiY2I0kibkdGJ0YqRiwqJCZJImJHRidGNkYqRiwvRjc7RixJIk5HRidGLComSSNQaUdGMCEiIi1JJEludEdGLzYkKiQtSSJmR0YnNiNJInhHRidGKi9GSDssJEY/RkBGP0Ys where: 1/Pi*Int((f(x)^2),x=-Pi..Pi)=1/Pi*int((f(x)^2),x=-Pi..Pi); LyomSSNQaUclKnByb3RlY3RlZEchIiItSSRJbnRHNiRGJUkoX3N5c2xpYkc2IjYkKiQtSSpwaWVjZXdpc2VHRiU2JjMxLCRGJEYmSSJ4R0YrMkY0IiIhLCRGNCEiIzMyRjZGNDFGNEYkLCRGNCIiJCIiIy9GNDtGM0YkIiIiLCQqJEYkRj4jIiM4Rj0= I[B]=evalf(1/Pi*int((f(x)^2),x=-Pi..Pi)); LyZeIyIiIjYjSSJCRzYiJCIrdiZHb0YlISIp with Fourier coefficients we will use a spreadsheet to show the convergence of Fourier partial series NiMlIm5H NiMmJSJhRzYjJSJuRw== NiMqJCkmJSJhRzYjJSJuRyIiIyIiIg== NiMmJSJiRzYjJSJuRw== NiMqJCkmJSJiRzYjJSJuRyIiIyIiIg== NiMsJiomIyIiIiIiI0YmKiQpJiUiYUc2IyIiIUYnRiZGJkYmLSUkc3VtRzYkLCYqJCkmRis2IyUibkdGJ0YmRiYqJCkmJSJiR0Y1RidGJkYmL0Y2O0YmJSJOR0Ym NiMlJmRlbHRhRw== NiMiIiE= NiMkIitOOylSJnkhIio= NiMkIitfRl1vaCEiKQ== NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIit3OEQlMyQhIik= NiMkIisqPnhEPiIhIik= NiMiIiI= NiMkIStoKSk0JD0kISIq NiMkIitPPUA4NSEiKQ== NiMkIiIiIiIh NiMkIiIiIiIh NiMkIis3S1koPiUhIik= NiMkIilqYE96ISIp NiMiIiM= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKysrXSEjNQ== NiMkIisrKysrRCEjNQ== NiMkIis3S1lBVSEiKQ== NiMkIilqYE9hISIp NiMiIiQ= NiMkISs3bHhPTiEjNQ== NiMkIis1KXkzRCIhIzU= NiMkIitMTExMTCEjNQ== NiMkIis2NjY2NiEjNQ== NiMkIis2SjNZVSEiKQ== NiMkIilrYXVJISIp NiMiIiU= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKysrRCEjNQ== NiMkIisrKytdaSEjNg== NiMkIis2SkxfVSEiKQ== NiMkIilrYVxDISIp NiMiIiY= NiMkIStXJlJLRiIhIzU= NiMkIitPKlE2aSIhIzY= NiMkIisrKysrPyEjNQ== NiMkIisrKysrUyEjNg== NiMkIitdVSZ6RCUhIik= NiMkIilEVigpPSEiKQ== NiMiIic= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStubW1tOyEjNQ== NiMkIit6eHh4RiEjNg== NiMkIitHP3RnVSEiKQ== NiMkIilabDQ7ISIp NiMiIig= NiMkIStFLDcnXCchIzY= NiMkIitwdyYqPlUhIzc= NiMkIitIOWRHOSEjNQ== NiMkIitHaiIzLyMhIzY= NiMkIitTWz5qVSEiKQ== NiMkIilOUGo4ISIp NiMiIik= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKytdNyEjNQ== NiMkIisrK11pOiEjNg== NiMkIitTdHZrVSEiKQ== NiMkIilONzI3ISIp NiMiIio= NiMkISshb14oSFIhIzY= NiMkIitGW0hXOiEjNw== NiMkIis2NjY2NiEjNQ== NiMkIissemNNNyEjNg== NiMkIitQajltVSEiKQ== NiMkIilRQW81ISIp NiMiIzU= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKysrNSEjNQ== NiMkIisrKysrNSEjNg== NiMkIitQajluVSEiKQ== NiMkIihRQW8qISIp NiMiIzY= NiMkISs7LW1JRSEjNg== NiMkIitzSlA/cCEjOA== NiMkIisiNDQ0NCohIzY= NiMkIis1R1lrIykhIzc= NiMkIisoKT4vb1UhIik= NiMkIigpZSd5KSEiKQ== NiMiIzc= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStMTExMJCkhIzY= NiMkIitXV1dXcCEjNw== NiMkIitKa3RvVSEiKQ== NiMkIihXQDQpISIp NiMiIzg= NiMkIStdL1wkKT0hIzY= NiMkIit2aWBaTiEjOA== NiMkIisjcDJCcCghIzY= NiMkIitqKGZyImYhIzc= NiMkIitBT09wVSEiKQ== NiMkIihgXFkoISIp NiMiIzk= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStWciZHOSghIzY= NiMkIis8My8tXiEjNw== NiMkIitFUSgpcFUhIik= NiMkIihcWiZwISIp NiMiIzo= NiMkIStdL1wkKT0hIzY= NiMkIit2aWBaTiEjOA== NiMkIitubW1tbSEjNg== NiMkIitYV1dXVyEjNw== NiMkIitYUE5xVSEiKQ== NiMkIihJW1onISIp NiMiIzs= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKytdaSEjNg== NiMkIisrK0QxUiEjNw== NiMkIitxVnVxVSEiKQ== NiMkIigwVTMnISIp NiMiIzw= NiMkISswMXI5OSEjNg== NiMkIisnNDE5KyMhIzg= NiMkIitUSE4jKWUhIzY= NiMkIis3dz9nTSEjNw== NiMkIiswLzZyVSEiKQ== NiMkIihxIj1kISIp NiMiIz0= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStjYmJiYiEjNg== NiMkIithKD5rMyQhIzc= NiMkIitaIT45RiUhIik= NiMkIihHJjRhISIp NiMiIz4= NiMkISsiSD05NSIhIzY= NiMkIitfQTc4NyEjOA== NiMkIismKnk6al8hIzY= NiMkIisuSjNxRiEjNw== NiMkIisnPTM8RiUhIik= NiMkIigqUT9eISIp NiMiIz8= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKysrXSEjNg== NiMkIisrKysrRCEjNw== NiMkIisnPWU+RiUhIik= NiMkIigqUXFbISIp NiMiI0A= NiMkIStuJFt1IikpISM3 NiMkIitxJlJaeCghIzk= NiMkIitpWiE+dyUhIzY= NiMkIisncHR2RSMhIzc= NiMkIis9Rj5zVSEiKQ== NiMkIihkZWolISIp NiMiI0E= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStYWFhYWCEjNg== NiMkIistZDZtPyEjNw== NiMkIitJJCpSc1UhIik= NiMkIihYI0hXISIp NiMiI0I= NiMkIStcNyJ6QCghIzc= NiMkIishR0MpNF8hIzk= NiMkIisoM0V5TSUhIzY= NiMkIitvImYuKj0hIzc= NiMkIit3TmZzVSEiKQ== NiMkIigqKlxCJSEiKQ== NiMiI0M= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStubW1tVCEjNg== NiMkIis2NjZPPCEjNw== NiMkIisoPW5GRiUhIik= NiMkIigpUWhTISIp NiMiI0Q= NiMkISs6LD88ZyEjNw== NiMkIitBKHAxaSQhIzk= NiMkIisrKysrUyEjNg== NiMkIisrKysrOyEjNw== NiMkIiszMyRIRiUhIik= NiMkIihueCpRISIp NiMiI0U= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStZUTpZUSEjNg== NiMkIitUKipHejkhIzc= NiMkIitQKHlJRiUhIik= NiMkIihRKVxQISIp NiMiI0Y= NiMkISt5ImVINCYhIzc= NiMkIisrQiNRZiMhIzk= NiMkIisvUHEuUCEjNg== NiMkIis4QHVyOCEjNw== NiMkIiswJj1LRiUhIik= NiMkIihxK2gkISIp NiMiI0c= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStyJkc5ZCQhIzY= NiMkIisvLV52NyEjNw== NiMkIitjZ010VSEiKQ== NiMkIig+RFskISIp NiMiI0g= NiMkISthMlJtViEjNw== NiMkIitBb2AxPiEjOQ== NiMkIitpZUZbTSEjNg== NiMkIitVMTEqPSIhIzc= NiMkIitwb1l0VSEiKQ== NiMkIigxPE8kISIp NiMiI0k= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStMTExMTCEjNg== NiMkIis2NjY2NiEjNw== NiMkIishKXpkdFUhIik= NiMkIigmZl1LISIp NiMiI0o= NiMkISt2dipbeSQhIzc= NiMkIitsXGFLOSEjOQ== NiMkIitfayFlQSQhIzY= NiMkIitGRmVTNSEjNw== NiMkIityTW90VSEiKQ== NiMkIigvXjkkISIp NiMiI0s= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKytESiEjNg== NiMkIisrXWlsKCohIzg= NiMkIitGNnl0VSEiKQ== NiMkIihbdS8kISIp NiMiI0w= NiMkISshcHhBSiQhIzc= NiMkIitdJD1yNCIhIzk= NiMkIitJSUlJSSEjNg== NiMkIitha3QjPSohIzg= NiMkIiteUyhRRiUhIik= NiMkIihDWCZIISIp NiMiI00= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStyazxUSCEjNg== NiMkIitNIT4wbCkhIzg= NiMkIitjMCdSRiUhIik= NiMkIig+IW9HISIp NiMiI04= NiMkISsmemJII0ghIzc= NiMkIishZXFPYSkhIzo= NiMkIitkRzlkRyEjNg== NiMkIiswYEVqIikhIzg= NiMkIitWSS91VSEiKQ== NiMkIihLYnkjISIp NiMiI08= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISt5eHh4RiEjNg== NiMkIislUVxncighIzg= NiMkIisuLTd1VSEiKQ== NiMkIihzJDNGISIp NiMiI1A= NiMkIStdIVslKWYjISM3 NiMkIitwQSQ+dichIzo= NiMkIisuRnEtRiEjNg== NiMkIissPmcvdCEjOA== NiMkIitDUj51VSEiKQ== NiMkIiheWWojISIp NiMiI1E= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStaKnk6aiMhIzY= NiMkIitheD9EcCEjOA== NiMkIit3SkV1VSEiKQ== NiMkIigqUmxEISIp NiMiI1I= NiMkISspKnA3REIhIzc= NiMkIitkYkAxYSEjOg== NiMkIitrRDVrRCEjNg== NiMkIitmPml1bCEjOA== NiMkIitqJUhWRiUhIik= NiMkIig3IipcIyEiKQ== NiMiI1M= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKysrRCEjNg== NiMkIisrKytdaSEjOA== NiMkIitqPlJ1VSEiKQ== NiMkIig3bVYjISIp NiMiI1Q= NiMkIStucncjNCMhIzc= NiMkIis6V256ViEjOg== NiMkIishUkMhUkMhIzY= NiMkIit2KlIpW2YhIzg= NiMkIisqKT1YdVUhIik= NiMkIignb3dCISIp NiMiI1U= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsiUV80USMhIzY= NiMkIitUVSQqb2MhIzg= NiMkIit5JjNYRiUhIik= NiMkIigoKio+QiEiKQ== NiMiI1Y= NiMkISt2WGQkKj0hIzc= NiMkIityWWkmZSQhIzo= NiMkIismUiJlREIhIzY= NiMkIitEKUckM2EhIzg= NiMkIis/SWN1VSEiKQ== NiMkIihiYkUjISIp NiMiI1c= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISt0c3NzQSEjNg== NiMkIitkIypHbF4hIzg= NiMkIit0WWh1VSEiKQ== NiMkIigtUkAjISIp NiMiI1g= NiMkIStMWF9APCEjNw== NiMkIis9bmtqSCEjOg== NiMkIitBQUFBQSEjNg== NiMkIisvO0ZRXCEjOA== NiMkIitfVm11VSEiKQ== NiMkIihCVTsjISIp NiMiI1k= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStWSSJSPCMhIzY= NiMkIis+eiplcyUhIzg= NiMkIis2O3J1VSEiKQ== NiMkIihrcDYjISIp NiMiI1o= NiMkIStzMSE+ZCIhIzc= NiMkIitCPCgzWiMhIzo= NiMkIit1JmZ3NyMhIzY= NiMkIitqXyRwXyUhIzg= NiMkIitGcnZ1VSEiKQ== NiMkIihbOTIjISIp NiMiI1s= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStMTEwkMyMhIzY= NiMkIit3eEZTViEjOA== NiMkIitJMCFbRiUhIik= NiMkIihYIUc/ISIp NiMiI1w= NiMkIStcI280VyIhIzc= NiMkIismXCpRdz8hIzo= NiMkIitGaiIzLyMhIzY= NiMkIisiR0pcOyUhIzg= NiMkIisoUVVbRiUhIik= NiMkIigpPScpPiEiKQ== NiMiI10= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKysrPyEjNg== NiMkIisrKysrUyEjOA== NiMkIisoUSMpW0YlISIp NiMkIigpPVk+ISIp NiMiI14= NiMkISssKVFkSyIhIzc= NiMkIitvTGVkPCEjOg== NiMkIis5VnlnPiEjNg== NiMkIitFXm5XUSEjOA== NiMkIis1NSNcRiUhIik= NiMkIihsdiE+ISIp NiMiI18= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStCcDJCPiEjNg== NiMkIitfW0EpcCQhIzg= NiMkIisjKnomXEYlISIp NiMkIigkZXE9ISIp NiMiI2A= NiMkISssIil6QjchIzc= NiMkIisjeiJvKFwiISM6 NiMkIitgQ3onKT0hIzY= NiMkIitoZCkqZk4hIzg= NiMkIitVUCpcRiUhIik= NiMkIihMWyQ9ISIp NiMiI2E= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStfPSY9Jj0hIzY= NiMkIisjR2IkSE0hIzg= NiMkIitPIUddRiUhIik= NiMkIihSMCE9ISIp NiMiI2I= NiMkIStpJ3lKOCIhIzc= NiMkIishKVE0JUciISM6 NiMkIis9PT09PSEjNg== NiMkIitCXnkwTCEjOA== NiMkIitBNzF2VSEiKQ== NiMkIihgdHciISIp NiMiI2M= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsnRzlkeSIhIzY= NiMkIis2YngpPSQhIzg= NiMkIis1SjR2VSEiKQ== NiMkIihsYXQiISIp NiMiI2Q= NiMkISsnM2tBMCIhIzc= NiMkIisyKGZzNSIhIzo= NiMkIitsZlFhPCEjNg== NiMkIis5LCh5MiQhIzg= NiMkIisrUzd2VSEiKQ== NiMkIih2WHEiISIp NiMiI2U= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStKejhDPCEjNg== NiMkIiswO2xzSCEjOA== NiMkIitGUDp2VSEiKQ== NiMkIihbW24iISIp NiMiI2Y= NiMkISthWzsoeiohIzg= NiMkIis9UlcpZiohIzs= NiMkIithXyJccCIhIzY= NiMkIis9eHRzRyEjOA== NiMkIitdRD12VSEiKQ== NiMkIihEZ2siISIp NiMiI2c= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStubW1tOyEjNg== NiMkIit6eHh4RiEjOA== NiMkIitHLkB2VSEiKQ== NiMkIihaIz07ISIp NiMiI2g= NiMkIStoIjNVOSohIzg= NiMkIisqR2E7TykhIzs= NiMkIitpVU1SOyEjNg== NiMkIis0J1x1byMhIzg= NiMkIisnR1BfRiUhIik= NiMkIigqRyJmIiEiKQ== NiMiI2k= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStFSyFIaCIhIzY= NiMkIis7b1gsRSEjOA== NiMkIissTEV2VSEiKQ== NiMkIih1X2MiISIp NiMiI2o= NiMkISskejxXYikhIzg= NiMkIit5aiF5SighIzs= NiMkIisoZSx0ZSIhIzY= NiMkIitHal8+RCEjOA== NiMkIitwJilHdlUhIik= NiMkIigxK2EiISIp NiMiI2s= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrK11pOiEjNg== NiMkIitdaVNUQyEjOA== NiMkIiskKUhKdlUhIik= NiMkIigjZjo6ISIp NiMiI2w= NiMkISspUSwqPiEpISM4 NiMkIitGPSk9VichIzs= NiMkIitROllROiEjNg== NiMkIisvUidvTyMhIzg= NiMkIis7bkx2VSEiKQ== NiMkIihmPVwiISIp NiMiI20= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISs6Ojo6OiEjNg== NiMkIis4VG8mSCMhIzg= NiMkIit0J2ZgRiUhIik= NiMkIigtKm85ISIp NiMiI24= NiMkISsrPSdSYCghIzg= NiMkIitTIWVnbiYhIzs= NiMkIis4dGAjXCIhIzY= NiMkIitKd21GQSEjOA== NiMkIisyP1F2VSEiKQ== NiMkIihvbFciISIp NiMiI28= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStOIyllcTkhIzY= NiMkIitkKEhFOyMhIzg= NiMkIitMT1N2VSEiKQ== NiMkIihVXFUiISIp NiMiI3A= NiMkISstaikzNCghIzg= NiMkIitibzFHXSEjOw== NiMkIitpYEZcOSEjNg== NiMkIit2ISpSK0AhIzg= NiMkIisob0NhRiUhIik= NiMkIigpKVFTIiEiKQ== NiMiI3E= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStIOWRHOSEjNg== NiMkIitHaiIzLyMhIzg= NiMkIismNFhhRiUhIik= NiMkIighWyRRIiEiKQ== NiMiI3I= NiMkISsxengmbychIzg= NiMkIitARScqcFchIzs= NiMkIisvMlgzOSEjNg== NiMkIisnUUxQKT4hIzg= NiMkIit4XFl2VSEiKQ== NiMkIigpZmo4ISIp NiMiI3M= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsqKSkpKSkpUSIhIzY= NiMkIitZQixIPiEjOA== NiMkIituVVt2VSEiKQ== NiMkIigzVk0iISIp NiMiI3Q= NiMkISsjWz5XSichIzg= NiMkIitSJCo9KClSISM7 NiMkIis5SScpcDghIzY= NiMkIit4WV93PSEjOA== NiMkIitzSV12VSEiKQ== NiMkIiguYksiISIp NiMiI3U= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISteOE5eOCEjNg== NiMkIit1LzpFPSEjOA== NiMkIitNOF92VSEiKQ== NiMkIihUc0kiISIp NiMiI3Y= NiMkIStmTjt0ZiEjOA== NiMkIishSG95YyQhIzs= NiMkIitMTExMOCEjNg== NiMkIit4eHh4PCEjOA== NiMkIitbIlJiRiUhIik= NiMkIihGJSpHIiEiKQ== NiMiI3c= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISt1JSp5OjghIzY= NiMkIitTPklKPCEjOA== NiMkIitoa2J2VSEiKQ== NiMkIig5QEYiISIp NiMiI3g= NiMkISs/QyUpZWMhIzg= NiMkIitgKFxBPyQhIzs= NiMkIisqSCwoKUgiISM2 NiMkIitrXWknbyIhIzg= NiMkIitmTGR2VSEiKQ== NiMkIig7X0QiISIp NiMiI3k= NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsjR14/RyIhIzY= NiMkIishXGJPayIhIzg= NiMkIisneipldlUhIik= NiMkIih6KFE3ISIp NiMiI3o= NiMkISs8VnBvYCEjOA== NiMkIitueUcjKUchIzs= NiMkIismeUFlRSIhIzY= NiMkIitCdEktOyEjOA== NiMkIitbZWd2VSEiKQ== NiMkIihGRkEiISIp NiMiIyEp NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsrKytdNyEjNg== NiMkIisrK11pOiEjOA== NiMkIit0OWl2VSEiKQ== NiMkIigtcj8iISIp NiMiIyIp NiMkISs6RUkrXiEjOA== NiMkIit3JzM4ZyMhIzs= NiMkIissemNNNyEjNg== NiMkIistejpDOiEjOA== NiMkIitUbmp2VSEiKQ== NiMkIihNPT4iISIp NiMiIyMp NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsmPjcmPjchIzY= NiMkIislKio0c1siISM4 NiMkIis4O2x2VSEiKQ== NiMkIihpcDwiISIp NiMiIyQp NiMkISsmR1g6JlshIzg= NiMkIitsIlxQTiMhIzs= NiMkIit4Iz5bPyIhIzY= NiMkIishXCplXjkhIzg= NiMkIitgaG12VSEiKQ== NiMkIihBQzsiISIp NiMiIyUp NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISshPncvPiIhIzY= NiMkIitmTkI8OSEjOA== NiMkIitELm92VSEiKQ== NiMkIihdI1s2ISIp NiMiIyYp NiMkIStfR2I/WSEjOA== NiMkIittMyZcOCMhIzs= NiMkIispZXFrPCIhIzY= NiMkIitXSTMlUSIhIzg= NiMkIisoPSVwdlUhIik= NiMkIigpUU02ISIp NiMiIycp NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISspcCF6aTYhIzY= NiMkIisyQTNfOCEjOA== NiMkIiszeHF2VSEiKQ== NiMkIihuMzciISIp NiMiIygp NiMkIStrSm4wVyEjOA== NiMkIisuYyo0JT4hIzs= NiMkIisoR0QlXDYhIzY= NiMkIishXHk2SyIhIzg= NiMkIitSNHN2VSEiKQ== NiMkIihPdzUiISIp NiMiIykp NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStPT09PNiEjNg== NiMkIis4QksiSCIhIzg= NiMkIitfUXR2VSEiKQ== NiMkIihCWjQiISIp NiMiIyop NiMkIStdPFcwVSEjOA== NiMkIitKU2RvPCEjOw== NiMkIisxYmZCNiEjNg== NiMkIitob1lpNyEjOA== NiMkIismXFlkRiUhIik= NiMkIighMyMzIiEiKQ== NiMiIyEq NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISs2NjY2NiEjNg== NiMkIissemNNNyEjOA== NiMkIitUKWVkRiUhIik= NiMkIihNKHA1ISIp NiMiIyIq NiMkIStDb2I9UyEjOA== NiMkIismKil6W2giISM7 NiMkIisqNCwqKTQiISM2 NiMkIitET2UyNyEjOA== NiMkIitMNHh2VSEiKQ== NiMkIihVdzAiISIp NiMiIyMq NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStBbCZwMyIhIzY= NiMkIisiW3U5PSIhIzg= NiMkIitbRnl2VSEiKQ== NiMkIihGZS8iISIp NiMiIyQq NiMkIStoIWVRJVEhIzg= NiMkIit6V194OSEjOw== NiMkIis8KW9fMiIhIzY= NiMkIitIST9jNiEjOA== NiMkIitEVnp2VSEiKQ== NiMkIihdVS4iISIp NiMiIyUq NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkISsoeUhRMSIhIzY= NiMkIis7UXRKNiEjOA== NiMkIitVYyFlRiUhIik= NiMkIihMSC0iISIp NiMiIyYq NiMkIStYJjMubyQhIzg= NiMkIisqNG5XTiIhIzs= NiMkIit6OmpfNSEjNg== NiMkIitUSy4zNiEjOA== NiMkIitPbiJlRiUhIik= NiMkIihSPSwiISIp NiMiIycq NiMkIiIhRiQ= NiMkIiIhRiQ= NiMkIStubW1UNSEjNg== NiMkIitYJXBdMyIhIzg= NiMkIisoZUZlRiUhIik= NiMkIigpKTQrIiEiKQ== NiMiIygq NiMkIStaJHpwXyQhIzg= NiMkIitKJGVSQyIhIzs= NiMkIitOeSM0LiIhIzY= NiMkIissQSJHMSIhIzg= NiMkIitGI1FlRiUhIik= NiMkIidbLioqISIp In this spreadsheet for NiMvJSJuRyIjKCo=, NiMvJSZkZWx0YUctJSRhYnNHNiMsJjcjLCYqJikmJSJhRzYjIiIhIiIjIiIiRjEhIiJGMi0lJHN1bUc2JCwmKiQpJkYuNiMlIm5HRjFGMkYyKiQpJiUiYkdGO0YxRjJGMi9GPDtGMiUiTkdGMkYyLCQqJiomRjJGMiUjUGlHRjNGMi0lJEludEc2JCokKS0lImZHNiMlInhHRjFGMi9GUDssJEZHRjNGR0YyRjJGMw== and NiMvLSUmTGltaXRHNiQlJmRlbHRhRy8lIm5HJSlpbmZpbml0eUciIiE=. Although the convergence is slow NiMvLSUmTGltaXRHNiQ3IywmKiYpJiUiYUc2IyIiISIiIyIiIkYvISIiRjAtJSRzdW1HNiQsJiokKSZGLDYjJSJuR0YvRjBGMCokKSYlImJHRjlGL0YwRjAvRjo7RjAlIk5HRjAvRjolKWluZmluaXR5Ry0lJkZsb2F0RzYkIit2JkdvRiUhIik= For n = 1000, we have: abs((a[0]^2/2+Sum('a[n]^2+b[n]^2','n' = 1 .. 1000))-(1/Pi)*Int(f(x)^2,x = -Pi .. Pi)); LUkkYWJzRyUqcHJvdGVjdGVkRzYjLCgqJEkjUGlHRiQiIiMjIiNEIiIpLUkkU3VtRzYkRiRJKF9zeXNsaWJHNiI2JCwmKiQmSSJhR0YxNiNJIm5HRjFGKSIiIiokJkkiYkdGMUY3RilGOS9GODtGOSIlKzVGOSomRighIiItSSRJbnRHRi82JCokLUkqcGllY2V3aXNlR0YkNiYzMSwkRihGQUkieEdGMTJGTCIiISwkRkwhIiMzMkZORkwxRkxGKCwkRkwiIiRGKS9GTDtGS0YoRjlGQQ== evalf(%); JCImXCoqKiEiKQ==
<Text-field style="Heading 1" layout="Heading 1">Weierstrass Approximation Theorem</Text-field> Theorem. (Weierstrass).- If NiMtJSJmRzYjJSJ4Rw== is a continuous real-valued function on NiM3JCUiYUclImJH and if any NiMyIiIhJShlcHNpbG9uRw== given, then there exist a polynomial NiMtJSJQRzYjJSJ4Rw== on NiM3JCUiYUclImJH such that: NiMyLSUkYWJzRzYjLCYtJSJmRzYjJSJ4RyIiIi0lIlBHRiohIiIlKGVwc2lsb25H for all NiMlInhH on NiM3JCUiYUclImJH. In our problem NiMtJSJQRzYjJSJ4Rw== represent a trigonometric polynomial and NiMvLSUiUEc2IyUieEctJSJnR0Ym. f :=x->piecewise(-Pi<=x and x<0,-2*x,0<=x and x<=Pi,3*x); Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmMzEsJEkjUGlHRishIiI5JDJGMiIiISwkRjIhIiMzMUY0RjIxRjJGMCwkRjIiIiRGJUYlRiU= g1:=x->5/4*Pi+sum(5/(n^2*Pi)*((-1)^n-1)*cos(n*x)+(-1)^(n+1)/n*sin(n*x),n = 1 .. 3); Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZJI1BpRyUqcHJvdGVjdGVkRyMiIiYiIiUtSSRzdW1HNiRGK0koX3N5c2xpYkdGJTYkLCYqKkkibkdGJSEiI0YqISIiLCYpRjhGNiIiIkY4RjtGOy1JJGNvc0dGMTYjKiZGNkY7OSRGO0Y7Ri0qKClGOCwmRjtGO0Y2RjtGO0Y2RjgtSSRzaW5HRjFGPkY7RjsvRjY7RjsiIiRGO0YlRiVGJQ== g2:=x->5/4*Pi+sum(5/(n^2*Pi)*((-1)^n-1)*cos(n*x)+(-1)^(n+1)/n*sin(n*x),n = 1 .. 10); Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZJI1BpRyUqcHJvdGVjdGVkRyMiIiYiIiUtSSRzdW1HNiRGK0koX3N5c2xpYkdGJTYkLCYqKkkibkdGJSEiI0YqISIiLCYpRjhGNiIiIkY4RjtGOy1JJGNvc0dGMTYjKiZGNkY7OSRGO0Y7Ri0qKClGOCwmRjtGO0Y2RjtGO0Y2RjgtSSRzaW5HRjFGPkY7RjsvRjY7RjsiIzVGO0YlRiVGJQ== plot([f(x),g1(x),g2(x)],x=-Pi..Pi,linestyle=[1,3,3],color=[red,blue,black], legend=["Function","Fourier Series for N=3","Fourier Series for N=10"],thickness=[2,2,2]); 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g3:=x->5/4*Pi+sum(5/(n^2*Pi)*((-1)^n-1)*cos(n*x)+(-1)^(n+1)/n*sin(n*x),n = 1 .. 50); Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZJI1BpRyUqcHJvdGVjdGVkRyMiIiYiIiUtSSRzdW1HNiRGK0koX3N5c2xpYkdGJTYkLCYqKkkibkdGJSEiI0YqISIiLCYpRjhGNiIiIkY4RjtGOy1JJGNvc0dGMTYjKiZGNkY7OSRGO0Y7Ri0qKClGOCwmRjtGO0Y2RjtGO0Y2RjgtSSRzaW5HRjFGPkY7RjsvRjY7RjsiI11GO0YlRiVGJQ== g4:=x->5/4*Pi+sum(5/(n^2*Pi)*((-1)^n-1)*cos(n*x)+(-1)^(n+1)/n*sin(n*x),n = 1 .. 100); Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZJI1BpRyUqcHJvdGVjdGVkRyMiIiYiIiUtSSRzdW1HNiRGK0koX3N5c2xpYkdGJTYkLCYqKkkibkdGJSEiI0YqISIiLCYpRjhGNiIiIkY4RjtGOy1JJGNvc0dGMTYjKiZGNkY7OSRGO0Y7Ri0qKClGOCwmRjtGO0Y2RjtGO0Y2RjgtSSRzaW5HRjFGPkY7RjsvRjY7RjsiJCsiRjtGJUYlRiU= plot([f(x),g3(x)],x=-Pi..Pi,linestyle=[1,3],color=[red,blue], legend=["Function","Fourier Series for N=50"],thickness=[2,2]); 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<Text-field style="Heading 1" layout="Heading 1">Bibliography</Text-field> Piskunov, N., "C\303\241lculo Diferencial e Integral", Volume II, Spanish Edition, Mir Mosc\303\272 Editorial, Mosc\303\272, Russian, 1986
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