Approximation and Convolution in the Spaces Open image in new window

Abstract

Here we use convolution to carry over the results of Chap.  7 concerning the pointwise and the uniform approximation of functions to the case of approximation “in the mean”.

Preliminarily, in Sect. 9.1 we introduce a family of \(\mathcal{L}^{p}\)-metrics and establish basic properties of the spaces \(\mathcal{L}^{p}\) for 1⩽p⩽∞. In Sect. 9.2, we study approximations with respect to the \(\mathcal{L}^{p}\)-metric by continuous and smooth functions. We also prove that summable functions are continuous in mean, and, as a consequence, obtain the classical theorem of Riemann–Lebesgue. In Sect. 9.3, we prove several inequalities connected with convolution, in particular, Young’s inequality. We prove that a function in \(\mathcal{L}^{p}\), 1⩽p<+∞, can be approximated by its convolution with an approximate identity and describe a class of approximate identities giving the almost everywhere convergence. As an example of the results obtained, we prove Lagrange’s lemma, which is important in the calculus of variations.