Approximation and Convolution in the Spaces Open image in new window

  • Boris Makarov
  • Anatolii Podkorytov
Part of the Universitext book series (UTX)


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.


Lebesgue Measure Triangle Inequality Maximal Function Uniform Approximation Summable Function 
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Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Boris Makarov
    • 1
  • Anatolii Podkorytov
    • 1
  1. 1.Mathematics and Mechanics FacultySt Petersburg State UniversitySt PetersburgRussia

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