Approximation related to quotient functionals

Simon Setzer, Gabriele Steidl, Tanja Teuber, Guido Moerkotte

Abstract: We examine the best approximation of componentwise positive vectors or positive continuous functions f<math><mi is="true">f</mi></math> by linear combinations fˆ=∑jαjφj<math><mover accent="true" is="true"><mrow is="true"><mi is="true">f</mi></mrow><mrow is="true"><mo is="true">ˆ</mo></mrow></mover><mo is="true">=</mo><msub is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub><msub is="true"><mrow is="true"><mi is="true">α</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub><msub is="true"><mrow is="true"><mi is="true">φ</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub></math> of given vectors or functions φj<math><msub is="true"><mrow is="true"><mi is="true">φ</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub></math> with respect to functionals Qp<math><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">p</mi></mrow></msub></math>, 1≤p≤∞<math><mn is="true">1</mn><mo is="true">≤</mo><mi is="true">p</mi><mo is="true">≤</mo><mi is="true">∞</mi></math>, involving quotients max{f/fˆ,fˆ/f}<math><mo is="true">max</mo><mrow is="true"><mo is="true">{</mo><mi is="true">f</mi><mo is="true">/</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">f</mi></mrow><mrow is="true"><mo is="true">ˆ</mo></mrow></mover><mo is="true">,</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">f</mi></mrow><mrow is="true"><mo is="true">ˆ</mo></mrow></mover><mo is="true">/</mo><mi is="true">f</mi><mo is="true">}</mo></mrow></math> rather than differences |f−fˆ|<math><mrow is="true"><mo is="true">|</mo><mi is="true">f</mi><mo is="true">−</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">f</mi></mrow><mrow is="true"><mo is="true">ˆ</mo></mrow></mover><mo is="true">|</mo></mrow></math>. We verify the existence of a best approximating function under mild conditions on {φj}j=1n<math><msubsup is="true"><mrow is="true"><mrow is="true"><mo is="true">{</mo><msub is="true"><mrow is="true"><mi is="true">φ</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub><mo is="true">}</mo></mrow></mrow><mrow is="true"><mi is="true">j</mi><mo is="true">=</mo><mn is="true">1</mn></mrow><mrow is="true"><mi is="true">n</mi></mrow></msubsup></math>. For discrete data, we compute a best approximating function with respect to Qp<math><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">p</mi></mrow></msub></math>, p=1,2,∞<math><mi is="true">p</mi><mo is="true">=</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">2</mn><mo is="true">,</mo><mi is="true">∞</mi></math> by second order cone programming. Special attention is paid to the Q∞<math><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">∞</mi></mrow></msub></math> functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional Q∞<math><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">∞</mi></mrow></msub></math> we give an equivalent characterization of the best approximation by using its extremal set. Then we apply this characterization to prove the uniqueness of the best Q∞<math><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">∞</mi></mrow></msub></math> approximation for Chebyshev sets {φj}j=1n<math><msubsup is="true"><mrow is="true"><mrow is="true"><mo is="true">{</mo><msub is="true"><mrow is="true"><mi is="true">φ</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub><mo is="true">}</mo></mrow></mrow><mrow is="true"><mi is="true">j</mi><mo is="true">=</mo><mn is="true">1</mn></mrow><mrow is="true"><mi is="true">n</mi></mrow></msubsup></math>.