Author | Blum, E. K. ♦ Curtis, P. C. |
Source | ACM Digital Library |
Content type | Text |
Publisher | Association for Computing Machinery (ACM) |
File Format | |
Copyright Year | ©1961 |
Language | English |
Subject Domain (in DDC) | Computer science, information & general works ♦ Data processing & computer science |
Abstract | Let ƒ $(\textit{x})$ be a real-valued function defined and continuous on [- 1, +1] and let $\textit{T}(\textit{x})$ = $\textit{a}0/2$ + $∑∞\textit{k}=1$ $\textit{akTk}$ $(\textit{x})$ be the Fourier-Tschebycheff expansion for ƒ $(\textit{x}),$ that is, T k $(\textit{x})$ = cos $\textit{kl&thgr;},$ $\textit{ak}$ = 2/π ∫π0 ƒ (cos $\textit{&thgr;})$ cos k&thgr; d&thgr; where $\textit{x}$ = cos $\textit{&thgr;}.$ If $\textit{Pn}$ is the class of real polynomials in $\textit{x}$ of degree less than or equal to $\textit{n},$ we let $\textit{Mn}$ = $inf\textit{pePn}$ $sup-1≤\textit{x}≤1$ | ƒ $(\textit{x})$ - $\textit{p}$ $(\textit{x})$ |. $\textit{Mn}$ is called the best Tschebycheff approximation to ƒ $(\textit{x})$ by a polynomial of degree less than or equal to $\textit{n}.$ A rule of thumb in computing is that | $\textit{Mn}/\textit{a}\textit{n}+1$ | → 1 as $\textit{n}$ → ∞. In other words, the best Tschebycheff approximation to ƒ $(\textit{x})$ of degree $\textit{n}$ is asymptotically equal to the $(\textit{n}$ + 1)th coefficient in the Tschebycheff expansion. It is the purpose of this note to give upper and lower bounds for $\textit{Mn}$ in terms of the coefficients ${\textit{ak}},$ which will enable us to make precise statements about the validity of this asymptotic result. It has already been observed by Bernstein [1, p. 115] that under appropriate hypotheses on $ƒ(\textit{x})$ there exists a subsequence ${\textit{kj}}$ for which | $\textit{Mkj}/\textit{a}\textit{kj}+1$ | → 1. We show here that if ${\textit{akj}}$ is the subsequence of ${\textit{ak}}$ consisting of the nonzero coefficients, then a sufficient condition for $lim\textit{j}→∞$ | $\textit{M}\textit{kj}-1/\textit{akj}$ | = 1 (1) is for lim $\textit{j}→∞\textit{a}\textit{kj}+1$ | = 0. If this latter condition holds ƒ $(\textit{z})$ = 1/2 $\textit{a}0$ + $∑∞\textit{n}=1$ $\textit{anTn}$ $(\textit{z})$ is an entire function. An easy example shows that (1) is not valid, however, for all entire functions. We note first some easily derived bounds for $\textit{Mn}.$ If the series $\textit{T}$ $(\textit{x})$ converges to ƒ $(\textit{x}),$ then $\textit{Mn}$ ≤ $sup-1≤\textit{x}≤1$ | ƒ $(\textit{x})$ - 1/2 $\textit{a}0$ - $∑\textit{n}\textit{k}=1$ $\textit{akTk}(\textit{x})$ | ≤ $∑∞\textit{k}=\textit{n}+1$ | $\textit{ak}$ |.Therefore, if $∑∞\textit{k}=1$ | $\textit{ak}$ | < ∞, $∑∞\textit{k}=\textit{n}+1$ | $\textit{ak}$ | is an upper bound for $\textit{Mn}.$ Again, if the series $\textit{T}(\textit{x})$ converges, a lower bound for $\textit{Mn}$ is given by La Vallee Poussin [2, p. 107], i.e., $\textit{Mn}$ ≧ | $\textit{a}\textit{n}+1$ + $\textit{a}3(\textit{n}+1)$ + $\textit{a}5(\textit{n}+1)$ + ··· |. (2) Another lower bound for $\textit{Mn}$ in terms of the coefficients $\textit{an}$ may also be obtained. From the fact that the $\textit{n}th$ partial sum of the Fourier cosine series gives the best least squares approximation to ƒ (cos $\textit{&thgr;})$ we have 1/π ∫π0 ƒ(cos $\textit{&thgr;})$ - $\textit{a}0/2$ - $∑\textit{n}1$ $\textit{ak}$ cos $\textit{k&thgr;}]2$ $\textit{d&thgr;}$ ≤ $\textit{Mn}2.$ But by Parseval's theorem 1/π ∫π0 ƒ(cos $\textit{&thgr;})$ - $\textit{a}0/2$ - $∑\textit{n}1$ $\textit{ak}$ cos $\textit{k&thgr;}]2$ $\textit{d&thgr;}$ = 1/2 $∑∞\textit{n}=1$ $\textit{ak}2.$ Therefore, $\textit{Mn}$ ≧ { 1/2 $∑∞\textit{n}+1\textit{ak}2}1/2$ and, in particular, $\textit{Mn}$ ≧ 1/√2 | $\textit{am}$ | for $\textit{m}$ > $\textit{n}.$ (3) We now can prove the following result.THEOREM. $\textit{Let}$ ${\textit{akj}}$ be the subsequence of the ${\textit{ak}}$ consisting of all the nonzero coefficeints. If for all j ≧ $\textit{j}0$ | $\textit{a}\textit{kj}+1/\textit{akj}$ | ≤ &rgr; < 1, then for j ≧ $\textit{j}0,$ max (1/√2, 1 - 2&rgr;/1 - &rgr;) ≤ $\textit{M}\textit{kj}-1/|$ $\textit{akj}$ |) ≤ 1/1 - &rgr; (4) $\textit{and}$ 1/√2 ≤ $\textit{Mkj}/|$ $\textit{a}\textit{kj}+1$ | ≤ 1/1 - &rgr;. (5)PROOF. If for all $\textit{j}$ ≧ $\textit{j}0,$ $|\textit{a}\textit{kj}+1/\textit{akj}|$ ≤ &rgr; < 1, then $∑∞\textit{k}=1|\textit{ak}$ | < ∞. Therefore, for $\textit{j}$ ≧ $\textit{j}0,$ $\textit{M}\textit{kj}-1$ ≤ $∑∞\textit{n}=\textit{kj}$ | $\textit{an}$ | = ∑∞ $\textit{n}=\textit{j}|\textit{akn}$ | ≤ | $\textit{akj}$ | (1 + &rgr; + &rgr;2 + ··· = | $\textit{akj}$ | 1/1&rgr; Similarly $\textit{Mkj}$ ≤ | $\textit{a}\textit{kj}+1$ | (1/1 - &rgr;). An application of (3) proves that $\textit{Mkj}/|$ $\textit{a}\textit{kj}+1$ | ≧ 1/ √2, establishing (5). On the other hand by (2), $\textit{M}\textit{kj}-1$ ≧ | $\textit{akj}$ + $\textit{a}3\textit{kj}$ + ··· | ≧ | $\textit{akj}$ | - | $\textit{a}3\textit{kj}$ | ··· ≧ | $\textit{akj}$ | {1 - &rgr; - &rgr;2 - ···} = | $\textit{akj}$ | 1 - 2&rgr;/1 - &rgr;. Therefore, by (3), $\textit{M}\textit{kj}-1/|$ $\textit{akj}$ | ≧ max (1/√2, 1 - 2&rgr;/1 - &rgr;) and the theorem is proved.COROLLARY. $\textit{If}$ $lim\textit{j}→∞$ | $\textit{a}\textit{kj}+1/\textit{a}\textit{kj}$ | = 0, $\textit{then}$ $lim\textit{j}→∞$ | $\textit{M}\textit{kj}-1/\textit{akj}$ | = 1. REMARK. Suppose $\textit{an}$ ≠ 0 for all $\textit{n}.$ Then if $lim\textit{j}→∞$ | $\textit{a}\textit{n}+1/\textit{an}$ | = 0, $\textit{f}$ $(\textit{z})$ ≡ $1/2\textit{a}0$ + ∑ $∞\textit{n}=1$ $\textit{anTn}$ $(\textit{z})$ is an entire function. In light of the above corollary, one might conjecture that for any entire function such that $\textit{an}$ ≠ 0, $lim\textit{n}→∞$ | $\textit{M}\textit{n}-1/\textit{an}$ | = 1. This, however, is false as the following simple argument shows. Certainly for such entire functions $\textit{Mn}$ ≧ $\textit{M}\textit{n}+1$ > 0. Therefore $lim\textit{n}→∞$ $\textit{M}\textit{n}+1/Mn$ ≤ 1. But, $\textit{M}\textit{n}+1/\textit{Mn}$ = $\textit{M}\textit{n}+1/\textit{a}\textit{n}+2$ . $\textit{a}\textit{n}+2/\textit{a}\textit{n}+1$ . $\textit{a}\textit{n}+1/\textit{Mn}.$ Therefore, if $lim\textit{n}→∞$ | $\textit{M}\textit{n}-1/\textit{an}$ | = 1, $lim\textit{n}→∞$ | $\textit{M}\textit{n}+1/\textit{Mn}$ | = 1 $lim\textit{n}$ → | $\textit{a}\textit{n}+2/\textit{a}\textit{n}+1$ |. But one can easily construct examples of entire functions such that $lim\textit{n}→∞$ | $\textit{a}\textit{n}+2/\textit{a}\textit{n}+1$ | ≧ $\textit{k}$ > 1; e.g., let $\textit{a}2\textit{n}$ = $(2\textit{n})-2\textit{n}+1,$ $\textit{a}2\textit{n}+1$ = $\textit{k}$ $(2\textit{n})-2\textit{n}+1.$ |
ISSN | 00045411 |
Age Range | 18 to 22 years ♦ above 22 year |
Educational Use | Research |
Education Level | UG and PG |
Learning Resource Type | Article |
Publisher Date | 1961-10-01 |
Publisher Place | New York |
e-ISSN | 1557735X |
Journal | Journal of the ACM (JACM) |
Volume Number | 8 |
Issue Number | 4 |
Page Count | 3 |
Starting Page | 645 |
Ending Page | 647 |
Ministry of Human Resource Development (MHRD) under its National Mission on Education through Information and Communication Technology (NMEICT) has initiated the National Digital Library of India (NDLI) project to develop a framework of virtual repository of learning resources with a single-window search facility. Filtered and federated searching is employed to facilitate focused searching so that learners can find out the right resource with least effort and in minimum time. NDLI is designed to hold content of any language and provides interface support for leading vernacular languages, (currently Hindi, Bengali and several other languages are available). It is designed to provide support for all academic levels including researchers and life-long learners, all disciplines, all popular forms of access devices and differently-abled learners. It is being developed to help students to prepare for entrance and competitive examinations, to enable people to learn and prepare from best practices from all over the world and to facilitate researchers to perform inter-linked exploration from multiple sources. It is being developed at Indian Institute of Technology Kharagpur.
NDLI is a conglomeration of freely available or institutionally contributed or donated or publisher managed contents. Almost all these contents are hosted and accessed from respective sources. The responsibility for authenticity, relevance, completeness, accuracy, reliability and suitability of these contents rests with the respective organization and NDLI has no responsibility or liability for these. Every effort is made to keep the NDLI portal up and running smoothly unless there are some unavoidable technical issues.
Ministry of Human Resource Development (MHRD), through its National Mission on Education through Information and Communication Technology (NMEICT), has sponsored and funded the National Digital Library of India (NDLI) project.
Phone: +91-3222-282435
For any issue or feedback, please write to ndl-support@iitkgp.ac.in