## Boundary effects on smoothing splines |

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### Contents

Introduction | 3 |

The Asymptotic Representation | 15 |

The Modified Estimate | 53 |

2 other sections not shown

### Common terms and phrases

2rrk achieve the optimal approximation error asymptotic asymptotically optimal behavior of f bias squared boundary behavior boundary bias effects boundary bias term boundary conditions 2.7 boundary effects bounded by 1/n bounded in absolute completes the proof Corollary Doctor of Philosophy Dominated Convergence Theorem Eh(x error bound establish esti estimate of f explicitly f satisfies follows Fourier analysis Fourier series function f gence gives Hence integrand Jensen's inequality k+sn kernel estimate l)dx Lemma mate mean squared error modified estimate non-periodic Notice optimal rate order terms Poisson summation formula problem proof of Lemma proof of Theorem Proposition 9 rate of convergence Recall remainder term Rice and Rosenblatt Riemann sum Rosenblatt 1983 San Diego satisfying boundary conditions second term Silverman smaller order smoothing spline estimate solution Speckman sum is bounded term B(x term which depends University of California variance Wahba