Real vs Complex Rational Chebyshev Approximation on an Interval
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چکیده
منابع مشابه
A Unified Theory for Real vs Complex Rational Chebyshev Approximation on an Interval
A unified approach is presented for determining all the constants Ym.n (m > 0, n > 0) which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that Ym,m+2 = 1/3 (m > 0), a problem which had remained open.
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I f f E C[-I, I] is real-valued, let Er( f ) and E'( f ) be the errors in best approximation to f in the supremum norm by rational functions of type ( m , n ) with real and complex coefficients, respectively. It has recently been observed that E'( f ) < Er( f ) can occur for any n > 1, but for no n 1 is it known whether y,,,, = inf, E'( f ) / E r ( f ) is zero or strictly positive. Here we show...
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When a function f(x) is holomorphic on an interval x ∈ [a, b], its roots on the interval can be computed by the following three-step procedure. First, approximate f(x) on [a, b] by a polynomial fN (x) using adaptive Chebyshev interpolation. Second, form the Chebyshev– Frobenius companion matrix whose elements are trivial functions of the Chebyshev coefficients of the interpolant fN (x). Third, ...
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