Suppose that f: T C is continuous, with Fourier coefficients n = /27 f(x) T...

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Suppose that f: T→ C is continuous, with Fourier coefficients În = √√/27 √ f(x) T₁ e- Let fx denote the mean of the first (N+1) partial sums of the Fourier series of f, meaning that KN(T) = fx(z)= (a) Show that fn = KN * f where Ky : TR is the Fejér kernel, given by KN(x) = (b) Show that KN (2) may KN (0) = N + 1 -ina dr. 3 + ) 3 ² 1 1 2π N + 1 2π Σ jme²m²). also be written as 1 1 [sin ((N+1) x/2)] 2π N + 1 sin (1/2) 2T/2)]² (N+1). N Σ (N+1-|n) einz n=-N x = 0, (c) Show that KN is an approximate identity. What can you say about the convergence of fy to f as N→ ∞? Suppose that f: T→ C is continuous, with Fourier coefficients În = √√/27 √ f(x) T₁ e- Let fx denote the mean of the first (N+1) partial sums of the Fourier series of f, meaning that KN(T) = fx(z)= (a) Show that fn = KN * f where Ky : TR is the Fejér kernel, given by KN(x) = (b) Show that KN (2) may KN (0) = N + 1 -ina dr. 3 + ) 3 ² 1 1 2π N + 1 2π Σ jme²m²). also be written as 1 1 [sin ((N+1) x/2)] 2π N + 1 sin (1/2) 2T/2)]² (N+1). N Σ (N+1-|n) einz n=-N x = 0, (c) Show that KN is an approximate identity. What can you say about the convergence of fy to f as N→ ∞?

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