## Continuity, integration, and Fourier theory |

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### Contents

The Space of Continuous Functions | 1 |

Theorems of Korovkin and StoneWeierstrass | 21 |

Fourier Series of Continuous Functions | 39 |

Copyright | |

28 other sections not shown

### Common terms and phrases

27r-periodic function algebra approximate identity assume Banach space bounded C(II called Cauchy sequence cells cl(A closed interval compact carrier complex numbers constant continuous function converges uniformly Cr(A defined denoted derivative differentiable Dirichlet kernel disjoint equal equivalence class everywhere example Exercise exists a function exists a number fdfi Fejer kernel finite measure finite variation Fn(x formula Fourier coefficients Fourier series Fourier transform function f Furthermore G Lp given hand side Hence i-almost i-measurable implies Lebesgue integral Lebesgue measure Lebesgue summable Lemma Let f linear operator measurable functions measurable sets n-th natural number non-negative norm notation Note null set observe partial sum proof prove real continuous function result Riemann integrable satisfying semiring Show Similarly Stone-Weierstrass theorem subset summable function summable step function tends to zero trigonometric polynomial uniquely determined vanishes vector space