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The Space of Continuous Functions
Theorems of Korovkin and StoneWeierstrass
Fourier Series of Continuous Functions
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algebra assume Banach space bounded C(II called Cauchy sequence cells cl(A closed interval compact carrier complex numbers constant continuous function converges uniformly defined denoted derivative differentiable Dirichlet kernel disjoint Dv(x equal equivalence class example Exercise exists a function exists a number f and g f vanishes Fejér kernel finite measure finite variation formula Fourier coefficients Fourier series Fourier transform function f Furthermore given hand side Hence Hölder's inequality implies inequality Lebesgue integral Lebesgue measure Lebesgue summable Lemma Let f linear operator measurable functions n-th natural number non-negative norm notation Note null set observed partial sum proof prove real continuous function Riemann integrable satisfying semiring series of f Show Similarly Stone-Weierstrass theorem subset summable function summable step function tends to zero trigonometric polynomial u-almost vector space