We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are cont...We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C<span style="white-space:normal;"> [<em style="white-space:normal;">a<span style="white-space:normal;">,<em style="white-space:normal;">b<span style="white-space:normal;">] and s<sub>1</sub>* & <span style="white-space:normal;">s<sub>2</sub>* are the best uniform approximations to F<sub>1</sub> and F<sub>2</sub> from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F<sub>1</sub><span style="white-space:nowrap;">−s<sub>1</sub>* and <em style="white-space:normal;">F<sub style="white-space:normal;">2</sub>−<em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">*, <em style="white-space:normal;">s<sub style="white-space:normal;">1</sub><span style="white-space:normal;">* or <em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">* is also a best A(1) simultaneous approximation to F<sub>1</sub> and F<sub>2</sub> from S with F<sub>1</sub><span style="white-space:nowrap;">≥F<sub>2</sub> and n=2.展开更多
文摘We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F<sub>1</sub> and F<sub>2</sub> are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C<span style="white-space:normal;"> [<em style="white-space:normal;">a<span style="white-space:normal;">,<em style="white-space:normal;">b<span style="white-space:normal;">] and s<sub>1</sub>* & <span style="white-space:normal;">s<sub>2</sub>* are the best uniform approximations to F<sub>1</sub> and F<sub>2</sub> from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F<sub>1</sub><span style="white-space:nowrap;">−s<sub>1</sub>* and <em style="white-space:normal;">F<sub style="white-space:normal;">2</sub>−<em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">*, <em style="white-space:normal;">s<sub style="white-space:normal;">1</sub><span style="white-space:normal;">* or <em style="white-space:normal;">s<sub style="white-space:normal;">2</sub><span style="white-space:normal;">* is also a best A(1) simultaneous approximation to F<sub>1</sub> and F<sub>2</sub> from S with F<sub>1</sub><span style="white-space:nowrap;">≥F<sub>2</sub> and n=2.
