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Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Trigonometric functions and Fourier series (part II) Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients Vipul Naik The square wave The saw-tooth Issues of convergence February 13, 2007 Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Outline Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Issues of convergence Using kernels to prove pointwise convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Fourier series on R Real-time streaming Inner product for complex vector spaces Let V be a C-vector space. An inner product is a bilinear form h , i : V × V → C satisfying: I C-linearity in the first variable: h a1 , b i + h a2 , b i = h a1 + a2 , b i h λa , b i = λ h a , b i Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Inner product for complex vector spaces Let V be a C-vector space. An inner product is a bilinear form h , i : V × V → C satisfying: I C-linearity in the first variable: h a1 , b i + h a2 , b i = h a1 + a2 , b i h λa , b i = λ h a , b i Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth I Conjugate-linearity in the second variable: Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming h a , b1 i + h a , b2 i = h a , b1 + b2 i h a , λb i = λ h a , b i Inner product for complex vector spaces Let V be a C-vector space. An inner product is a bilinear form h , i : V × V → C satisfying: I C-linearity in the first variable: h a1 , b i + h a2 , b i = h a1 + a2 , b i h λa , b i = λ h a , b i Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth I Conjugate-linearity in the second variable: Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming h a , b1 i + h a , b2 i = h a , b1 + b2 i h a , λb i = λ h a , b i Inner product for complex vector spaces (contd) Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform I Hermitian symmetry: ha, b i=hb , ai Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Inner product for complex vector spaces (contd) Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform I Hermitian symmetry: ha, b i=hb , ai Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth I Positive definiteness: h a , a i ∈ R+ for a 6= 0 Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Orthonormal set for functions to complex numbers Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform For the space of functions from S 1 to C, an orthonormal set is the set of functions x 7→ e inx where n ∈ Z. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Orthonormal set for functions to complex numbers Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform For the space of functions from S 1 to C, an orthonormal set is the set of functions x 7→ e inx where n ∈ Z. Under pointwise multiplication, these functions form a group isomorphic to the group Z of integers. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Orthonormal set for functions to complex numbers Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform For the space of functions from S 1 to C, an orthonormal set is the set of functions x 7→ e inx where n ∈ Z. Under pointwise multiplication, these functions form a group isomorphic to the group Z of integers. The Fourier series for a function f is thus an expression: ∞ X Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence an e inx n=−∞ Using kernels to prove pointwise convergence Fourier series on R Real-time streaming where the infinite summation converges to f . Expression for the Fourier coefficients Trigonometric functions and Fourier series (part II) Vipul Naik e inx Because the are orthonormal, the value an is simply the signed magnitude of the projection of f along e inx , namely: 1 an = π Z 0 2π f (x)e −inx dx The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Expression for the Fourier coefficients Trigonometric functions and Fourier series (part II) Vipul Naik e inx Because the are orthonormal, the value an is simply the signed magnitude of the projection of f along e inx , namely: 1 an = π Z 2π f (x)e −inx dx 0 The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Now the map n 7→ an can be viewed as a function from Z to C which is uniquely determined by f . Thus, changing notation, we denote by fˆ(n) the value an corresponding to f . Then fˆ is a map Z → C. Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The Fourier transform Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform The transform which takes as input a function f from S 1 to C and gives as output the function fˆ, is termed the Fourier transform(defined). Thus the Fourier transform takes us from functions on S 1 to functions on Z. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Fourier transform in a little more generality Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Let G be a topological Abelian group. Then consider the set of homomorphisms from G to C. This set gets equipped with a natural group structure under pointwise addition. Call this the dual group(defined) to G . Then, given a suitable inner product on G , we can get a map analogous to a Fourier transform: which takes a continuous function f on G and outputs a function fˆ on Ĝ . Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Double dual same as original For a locally compact topological Abelian group G , the dual of the dual group is again G . This suggests a kind of duality between functions on G and functions on Ĝ . Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Double dual same as original For a locally compact topological Abelian group G , the dual of the dual group is again G . This suggests a kind of duality between functions on G and functions on Ĝ . The questions now, for the case of the Fourier transform, are thus: I What are the functions on S 1 for which the Fourier transform gives a well-defined function on Z? Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Double dual same as original For a locally compact topological Abelian group G , the dual of the dual group is again G . This suggests a kind of duality between functions on G and functions on Ĝ . The questions now, for the case of the Fourier transform, are thus: I I What are the functions on S 1 for which the Fourier transform gives a well-defined function on Z?The L2 functions What are the functions on Z that arise as Fourier transforms of well-defined functions on S 1 ? Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Double dual same as original For a locally compact topological Abelian group G , the dual of the dual group is again G . This suggests a kind of duality between functions on G and functions on Ĝ . The questions now, for the case of the Fourier transform, are thus: I What are the functions on S 1 for which the Fourier transform gives a well-defined function on Z?The L2 functions I What are the functions on Z that arise as Fourier transforms of well-defined functions on S 1 ? I For which functions on S 1 is it true that the sum of the Fourier series given by the Fourier transform actually converges? Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Double dual same as original For a locally compact topological Abelian group G , the dual of the dual group is again G . This suggests a kind of duality between functions on G and functions on Ĝ . The questions now, for the case of the Fourier transform, are thus: I What are the functions on S 1 for which the Fourier transform gives a well-defined function on Z?The L2 functions I What are the functions on Z that arise as Fourier transforms of well-defined functions on S 1 ? I For which functions on S 1 is it true that the sum of the Fourier series given by the Fourier transform actually converges? I For which functions on Z is it true that the Fourier transform of the limit of its Fourier series is the same as the original function? Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The additive group modulo p Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform At the very other end from groups like S 1 , are groups like (Fp , +), (additive group over a prime field). Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The additive group modulo p Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform At the very other end from groups like S 1 , are groups like (Fp , +), (additive group over a prime field). For Fp , the dual group, called Fˆp , is simply the group of one-dimensional characters of Fp under multiplication. This is isomorphic to Fp – however, there is no natural isomorphism. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The additive group modulo p Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform At the very other end from groups like S 1 , are groups like (Fp , +), (additive group over a prime field). For Fp , the dual group, called Fˆp , is simply the group of one-dimensional characters of Fp under multiplication. This is isomorphic to Fp – however, there is no natural isomorphism. The Fourier transform here thus takes functions on the group and returns functions on the character group. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Outline Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Issues of convergence Using kernels to prove pointwise convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Fourier series on R Real-time streaming The [−π , π] convention Trigonometric functions and Fourier series (part II) Vipul Naik In this section, we shall shift from the [0 , 2π] situation to the [−π , π] situation, viz where the function is periodically extended from its definition on the closed interval [−π , π]. Some clear advantages are: The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The [−π , π] convention Trigonometric functions and Fourier series (part II) Vipul Naik In this section, we shall shift from the [0 , 2π] situation to the [−π , π] situation, viz where the function is periodically extended from its definition on the closed interval [−π , π]. Some clear advantages are: I The integral of any odd periodic function over this interval is zero. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The [−π , π] convention Trigonometric functions and Fourier series (part II) Vipul Naik In this section, we shall shift from the [0 , 2π] situation to the [−π , π] situation, viz where the function is periodically extended from its definition on the closed interval [−π , π]. Some clear advantages are: I I The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups The integral of any odd periodic function over this interval is zero. Computing the Fourier coefficients Thus, the coefficients of the sin functions in the Fourier series expansion of an even function are 0 Issues of convergence The square wave The saw-tooth Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The [−π , π] convention Trigonometric functions and Fourier series (part II) Vipul Naik In this section, we shall shift from the [0 , 2π] situation to the [−π , π] situation, viz where the function is periodically extended from its definition on the closed interval [−π , π]. Some clear advantages are: I I I The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups The integral of any odd periodic function over this interval is zero. Computing the Fourier coefficients Thus, the coefficients of the sin functions in the Fourier series expansion of an even function are 0 Issues of convergence Similarly the coefficients of the cos function in the Fourier series expansion of an odd function are 0 Fourier series on R The square wave The saw-tooth Using kernels to prove pointwise convergence Real-time streaming The square wave The square wave is defined as: Trigonometric functions and Fourier series (part II) Vipul Naik f (x) = −1 for − π ≤ x < 0 f (x) = 0 for x = 0 f (x) = 1 for 0 < x ≤ π The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The square wave The square wave is defined as: Trigonometric functions and Fourier series (part II) Vipul Naik f (x) = −1 for − π ≤ x < 0 f (x) = 0 for x = 0 f (x) = 1 for 0 < x ≤ π Clearly f is an odd function, hence its Fourier expansion must only contain sin terms. The coefficient of sin nx is: Z 1 π f (x) sin nx dx π −π which simplifies to 0 if n is even and 4/(nπ) if n is odd. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The square wave The square wave is defined as: Trigonometric functions and Fourier series (part II) Vipul Naik f (x) = −1 for − π ≤ x < 0 f (x) = 0 for x = 0 f (x) = 1 for 0 < x ≤ π Clearly f is an odd function, hence its Fourier expansion must only contain sin terms. The coefficient of sin nx is: Z 1 π f (x) sin nx dx π −π which simplifies to 0 if n is even and 4/(nπ) if n is odd. Thus the Fourier series for the square wave looks like: 4 X sin(2n − 1)t π 2n − 1 n∈N The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Under linear changes Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Suppose f is a 2π-periodic function. Then the function x 7→ af (x) + b is also a 2π-periodic function. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Under linear changes Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Suppose f is a 2π-periodic function. Then the function x 7→ af (x) + b is also a 2π-periodic function. The Fourier coefficients of this new function are related to the old function as follows: I The constant term a0 gets transformed to aa0 + b I All the other terms get multiplied by a We can use this to directly write the Fourier series for the 0-1 square wave. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming What this tells us Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform The square wave looks almost like something flat – the only difference being that there is a jump from −1 to 1. It is interesting that so many sine functions are hidden underneath this apparently flat behaviour. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming What this tells us Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform The square wave looks almost like something flat – the only difference being that there is a jump from −1 to 1. It is interesting that so many sine functions are hidden underneath this apparently flat behaviour. For the square wave, the coefficient of sin nx is only inverse linear in n, so by the typical standards of Fourier series, it does not taper very fast. This is to be expected since the square wave is about as democratic as one can get between frequencies. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The even saw-tooth Define f (x) = |x| on the closed interval [−π , π] and extend f periodically to the whole real line. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The even saw-tooth Define f (x) = |x| on the closed interval [−π , π] and extend f periodically to the whole real line. Since f is even, all the sin coefficients are zero. This leaves us to compute the cosine coefficient and the constant term. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The even saw-tooth Define f (x) = |x| on the closed interval [−π , π] and extend f periodically to the whole real line. Since f is even, all the sin coefficients are zero. This leaves us to compute the cosine coefficient and the constant term. By simple integration, the constant term is π/2 (this is the average height) Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Trigonometric functions and Fourier series (part II) The even saw-tooth Define f (x) = |x| on the closed interval [−π , π] and extend f periodically to the whole real line. Since f is even, all the sin coefficients are zero. This leaves us to compute the cosine coefficient and the constant term. By simple integration, the constant term is π/2 (this is the average height) Also: Z 2 π an = x cos(nx) dx π 0 This simplifies to We thus get: −4 πn2 for odd n and 0 for even n. Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming π 4 X cos(2n − 1)t − 2 π (2n − 1)2 n∈N Intuitive justification Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Unlike the square wave function, which was very democratic among all the functions, the saw-tooth clearly prefers lower frequencies – this is evidenced in the relatively faster rate at which the Fourier coefficients decline. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Outline Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Issues of convergence Using kernels to prove pointwise convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Fourier series on R Real-time streaming The convergence results we want Trigonometric functions and Fourier series (part II) Vipul Naik We ideally want to say that for functions of interest the Fourier series converges at every point. This is equivalent to requiring that if we have a function which is orthogonal to all the sine and cosine functions, then it is identically zero. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The convergence results we want Trigonometric functions and Fourier series (part II) Vipul Naik We ideally want to say that for functions of interest the Fourier series converges at every point. This is equivalent to requiring that if we have a function which is orthogonal to all the sine and cosine functions, then it is identically zero. The idea now is to focus at any particular point and show that at that point the function takes the value zero. For this purpose, we try to try to obtain a sequence of functions that approaches the Dirac delta for the point. In other words we try to locate a sequence of functions fn such that h f , fn i → f (p) for that point p. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The convergence results we want Trigonometric functions and Fourier series (part II) Vipul Naik We ideally want to say that for functions of interest the Fourier series converges at every point. This is equivalent to requiring that if we have a function which is orthogonal to all the sine and cosine functions, then it is identically zero. The idea now is to focus at any particular point and show that at that point the function takes the value zero. For this purpose, we try to try to obtain a sequence of functions that approaches the Dirac delta for the point. In other words we try to locate a sequence of functions fn such that h f , fn i → f (p) for that point p. If each of these fn is a finite linear combination of the sine and cosine bunches, then h f , fn i is zero for all n and hence f (p) = 0. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Trigonometric functions and Fourier series (part II) The Dirichlet kernel Vipul Naik This is simply the average of the first n functions, and is given by: t 7→ sin((n + 1/2)t) sin(t/2) with the suitable limiting value at t = 0, namely 2n + 1 The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Trigonometric functions and Fourier series (part II) The Dirichlet kernel Vipul Naik This is simply the average of the first n functions, and is given by: t 7→ sin((n + 1/2)t) sin(t/2) with the suitable limiting value at t = 0, namely 2n + 1 This is the same as: t 7→ 1 + 2 n X k=1 The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence cos kt Fourier series on R Real-time streaming Properties of the Dirichlet kernel Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform The Dirichlet kernel has the following important properties: I The integral is uniformly bounded. In fact, for any n, the integral of the corresponding Dirichlet kernel on any sub-interval is not more than 4π. I As n → ∞, the limiting value at 0 is ∞. I The value of the Dirichlet kernel drops very rapidly just slightly away form 0. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Riemann-Lebesgue property Trigonometric functions and Fourier series (part II) Vipul Naik To quantify the last statement (the value of the Dirichlet kernel drops very rapidly just slightly after 0) the following notion was introduced. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Riemann-Lebesgue property Trigonometric functions and Fourier series (part II) Vipul Naik To quantify the last statement (the value of the Dirichlet kernel drops very rapidly just slightly after 0) the following notion was introduced. A sequence fn of 2π-periodic functions is said to have the Riemann-Lebesgue property(defined) if for any r > 0: Z π lim f (t)fn (t) dt = 0 n→∞ r for f any L1 function. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Riemann-Lebesgue property Trigonometric functions and Fourier series (part II) Vipul Naik To quantify the last statement (the value of the Dirichlet kernel drops very rapidly just slightly after 0) the following notion was introduced. A sequence fn of 2π-periodic functions is said to have the Riemann-Lebesgue property(defined) if for any r > 0: Z π lim f (t)fn (t) dt = 0 n→∞ r L1 for f any function. It turns out that the Dirichlet kernels do possess the Riemann-Lebesgue property. This, along with the fact that the value at 0 goes to infinity and the total area is bounded, suffices to show that limn→∞ f (t)Dn (t) dt is actually f (0). The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Convolution product at a point Here’s a rough sketch of the nature of result we want to establish: I Construct a sequence of functions fn , each a finite linear combination of sines and cosines, with the Riemann-Lebesgue property. The sequence that we constructed was that of Dirichlet kernels. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Convolution product at a point Here’s a rough sketch of the nature of result we want to establish: I I Construct a sequence of functions fn , each a finite linear combination of sines and cosines, with the Riemann-Lebesgue property. The sequence that we constructed was that of Dirichlet kernels. Now, using the Riemann-Lebesgue property, argue that R for any function f , limn→∞ f (t)fn (t) is f (0). Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Convolution product at a point Here’s a rough sketch of the nature of result we want to establish: I I I Construct a sequence of functions fn , each a finite linear combination of sines and cosines, with the Riemann-Lebesgue property. The sequence that we constructed was that of Dirichlet kernels. Now, using the Riemann-Lebesgue property, argue that R for any function f , limn→∞ f (t)fn (t) is f (0). R More specifically argue that limn→infty f (t + x)fn (t) is f (x). Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Convolution product at a point Here’s a rough sketch of the nature of result we want to establish: I I I I Construct a sequence of functions fn , each a finite linear combination of sines and cosines, with the Riemann-Lebesgue property. The sequence that we constructed was that of Dirichlet kernels. Now, using the Riemann-Lebesgue property, argue that R for any function f , limn→∞ f (t)fn (t) is f (0). R More specifically argue that limn→infty f (t + x)fn (t) is f (x). Thus if f is orthogonal to all the sin and cos functions, f (x) must be equal to 0 for all x. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Convolution product at a point Here’s a rough sketch of the nature of result we want to establish: I I I I I Construct a sequence of functions fn , each a finite linear combination of sines and cosines, with the Riemann-Lebesgue property. The sequence that we constructed was that of Dirichlet kernels. Now, using the Riemann-Lebesgue property, argue that R for any function f , limn→∞ f (t)fn (t) is f (0). R More specifically argue that limn→infty f (t + x)fn (t) is f (x). Thus if f is orthogonal to all the sin and cos functions, f (x) must be equal to 0 for all x. In particular, if we take an arbitrary function, consider its Fourier series and call the difference f , then we can use the above argument to show that the difference is 0. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Riemann localization principle Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform The Riemann localization principle basically states that if two functions are locally similar, then the behaviour of their Fourier series in those neighbourhoods would be similar. More precisely: Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Riemann localization principle Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform The Riemann localization principle basically states that if two functions are locally similar, then the behaviour of their Fourier series in those neighbourhoods would be similar. More precisely: Suppose f and g are 2π-periodic functions and further that f = g almost everywhere on an open interval (t − r , t + r ) with r > 0. Then the Fourier series both converge or both diverge at t. Moreover, if they converge, they take the same value on the whole interval. Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming The Fejer kernel Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups The Fejer kernel is the average of the values on the Dirichlet kernel. Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Convergence result Trigonometric functions and Fourier series (part II) Vipul Naik We can show that for a periodic function f , if the left-hand and right-hand limit exists at each point, and the limiting value of derivative with respect to this is also well-defined on each side, the Fourier series converges pointwise to the mean of the left-hand and the right-hand limit. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Convergence result Trigonometric functions and Fourier series (part II) Vipul Naik We can show that for a periodic function f , if the left-hand and right-hand limit exists at each point, and the limiting value of derivative with respect to this is also well-defined on each side, the Fourier series converges pointwise to the mean of the left-hand and the right-hand limit. Actually, we can prove a much stronger result: given any 2π-periodic function with bounded variation, the Fourier series converges to: 1 f (t − ) + f (t + ) 2 The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Convergence result Trigonometric functions and Fourier series (part II) Vipul Naik We can show that for a periodic function f , if the left-hand and right-hand limit exists at each point, and the limiting value of derivative with respect to this is also well-defined on each side, the Fourier series converges pointwise to the mean of the left-hand and the right-hand limit. Actually, we can prove a much stronger result: given any 2π-periodic function with bounded variation, the Fourier series converges to: 1 f (t − ) + f (t + ) 2 The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Finally, we can show that the Fourier series for any L2 function converges pointwise almost everywhere. Outline Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Issues of convergence Using kernels to prove pointwise convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Fourier series on R Real-time streaming Streaming music Suppose a continuous stream of sound is coming in. A stream of sound can be viewed as a function f : R → R where the domain R represents the time coordinate and the range R represents the instantaneous magnitude. Note that the study of such sound streams should be: I Invariant under time-translation viz if the whole sound stream is displaced in time by a certain amount, the analysis should not differ Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Streaming music Suppose a continuous stream of sound is coming in. A stream of sound can be viewed as a function f : R → R where the domain R represents the time coordinate and the range R represents the instantaneous magnitude. Note that the study of such sound streams should be: I Invariant under time-translation viz if the whole sound stream is displaced in time by a certain amount, the analysis should not differ I Invariant under spatial translation: That is, if a constant sound stream (white noise) is added to the sound stream, the analysis should not differ (except to take out the white noise) Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Streaming music Suppose a continuous stream of sound is coming in. A stream of sound can be viewed as a function f : R → R where the domain R represents the time coordinate and the range R represents the instantaneous magnitude. Note that the study of such sound streams should be: I Invariant under time-translation viz if the whole sound stream is displaced in time by a certain amount, the analysis should not differ I Invariant under spatial translation: That is, if a constant sound stream (white noise) is added to the sound stream, the analysis should not differ (except to take out the white noise) I Covariant under spatial/time dilation: If the instantaneous magnitude at every point is multipled by a constant factor, or if the whole sound stream is stretched or compressed along the time axis, there should simply be some proportionate changes in the constants rather than a qualitative change. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming As a combination of frequencies Musical sounds arise as superpositions (pointwise sums) of different frequencies. There are the following complications: I The frequencies may not even be related rationally. Thus, we may not be able to find any period for the superposition. For instance, the function √ sin x + sin 2x has no period Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming As a combination of frequencies Musical sounds arise as superpositions (pointwise sums) of different frequencies. There are the following complications: I The frequencies may not even be related rationally. Thus, we may not be able to find any period for the superposition. For instance, the function √ sin x + sin 2x has no period I The sounds may start and end at different times. Thus, unlike the ideal sine wave which was always there and will always be there, sounds in real life will start off and then peter off. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming As a combination of frequencies Musical sounds arise as superpositions (pointwise sums) of different frequencies. There are the following complications: I The frequencies may not even be related rationally. Thus, we may not be able to find any period for the superposition. For instance, the function √ sin x + sin 2x has no period I I The sounds may start and end at different times. Thus, unlike the ideal sine wave which was always there and will always be there, sounds in real life will start off and then peter off. There are variable levels of noise Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Can we do a Fourier transform on R? Here’s the idea. Given f : R → R, try writing f as an infinite linear combination of functions of the form x 7→ e irx for every real number r . In other words we need a map f 7→ fˆ that converts a function from R to R to another function from R to R, but where the new function represents the amplitude corresponding to a given frequency. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Can we do a Fourier transform on R? Here’s the idea. Given f : R → R, try writing f as an infinite linear combination of functions of the form x 7→ e irx for every real number r . In other words we need a map f 7→ fˆ that converts a function from R to R to another function from R to R, but where the new function represents the amplitude corresponding to a given frequency. Here’s a naive attempt at writing an expression for this: Z ∞ fˆ(r ) = f (x)e −irx dx −∞ The problem of course is that even for the nicest continuous functions, the right-hand side may blow up to infinity. So we need to restrict to continuous functions that become zero outside some finite domain – the so-called continuous functions of compact support. Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Schwarz functions and Schwarz spaces Trigonometric functions and Fourier series (part II) Vipul Naik The Schwarz space is the space of all functions f : R → R such that for any polynomial function p, the map f (x)p(x) approaches ∞. If f is also infinitely differentiable, this is equivalent to demanding that all the derivatives approach 0 as x → ∞. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Schwarz functions and Schwarz spaces Trigonometric functions and Fourier series (part II) Vipul Naik The Schwarz space is the space of all functions f : R → R such that for any polynomial function p, the map f (x)p(x) approaches ∞. If f is also infinitely differentiable, this is equivalent to demanding that all the derivatives approach 0 as x → ∞. It turns out that the Fourier transform of any continuous function with compact support is a Schwarz function. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming Schwarz functions and Schwarz spaces Trigonometric functions and Fourier series (part II) Vipul Naik The Schwarz space is the space of all functions f : R → R such that for any polynomial function p, the map f (x)p(x) approaches ∞. If f is also infinitely differentiable, this is equivalent to demanding that all the derivatives approach 0 as x → ∞. It turns out that the Fourier transform of any continuous function with compact support is a Schwarz function. In other words, if we take a sound stream that tapers off in finite time, the contribution of very high frequencies to that sound stream goes down superpolynomially fast. The Fourier transform Fourier series for functions to complex numbers The Fourier transform for topological groups The Fourier transform for discrete groups Computing the Fourier coefficients The square wave The saw-tooth Issues of convergence Using kernels to prove pointwise convergence Fourier series on R Real-time streaming