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