## A treatise on trigonometric series, Volume 1 |

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### Contents

THE SUMMABILITY OF FOURIER SERIES ADDITIONAL INVESTIGATION INTO PROBLEMS OF CONVERGENCE | 1 |

The application to Fourier series of methods of summation with triangular | 2 |

Numerical series summation | 5 |

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absolute constant absolute convergence absolutely continuous bn sinnx bounded function bounded variation Chapter continuous function converges absolutely converges almost everywhere converges everywhere converges to zero converges uniformly convex Corollary cosnx defined denote derivative Dirichlet kernel diverges estimate example exists fact Fejer kernel finite number following theorem Fourier coefficients Fourier formulae Fourier series converges fulfilled function f(x function of bounded given iff(x inequality interval Introductory Material lacunary Lebesgue lemma means metric modulus of continuity monotonically Moreover Natanson necessary and sufficient non-negative obtain orthogonal system partial sums period 2n point x0 possesses possible to find problem proof properties Riemann right-hand side satisfies condition series converges uniformly series o(f Sn(x sufficiently large summable function tends to zero theorem is proved trigono trigonometric polynomial trigonometric series trigonometric system uniform convergence valid values whence whilst