Which two of the functions f(x)=1, g(x)=x, h(x)=sinπx are closest to each other on the interval

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Which two of the functions f(x)=1, g(x)=x, h(x)=sinπx are...

Which two of the functions $f(x)=1, g(x)=x, h(x)=\sin \pi x$ are closest to each other on the interval $[0,1]$ under (a) the $\mathrm{L}^1$ norm?
(b) the $\mathrm{L}^2$ norm?
(c) the $\mathrm{L}^{\infty}$ norm?

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Key Concepts

L^? Norm

The L^? norm, or supremum norm, quantifies the maximum absolute difference between functions over the domain. This norm focuses on the greatest pointwise deviation, making it useful in situations where the worst-case error is a critical concern.

L^p Norms

L^p norms are a family of measures used in functional analysis to quantify the size or distance between functions. For a function defined on a certain domain, these norms are computed as the p-th root of the integral of the absolute value of the function raised to the power p. They are important in understanding convergence, stability, and approximation in various function spaces.

L^1 Norm

The L^1 norm of a function measures the total accumulated absolute difference by integrating the absolute value of the function over the domain. In the context of comparing functions, the L^1 norm emphasizes the overall area between the curves and is useful when one is interested in the average deviation across the interval.

L^2 Norm

The L^2 norm, often associated with energy or variance, is computed as the square root of the integral of the square of the difference between functions. It is particularly significant in applications such as least squares approximations, where minimizing the L^2 norm corresponds to reducing the mean squared error between functions.

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