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Difference between revisions of "Multiplicative system"

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An [[Orthonormal system|orthonormal system]] of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654801.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654802.png" /> satisfying the conditions:
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An [[Orthonormal system|orthonormal system]] of functions $\{\phi_n\}$ on $[a,b]$ satisfying the conditions:
  
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1) for any two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654804.png" /> the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654805.png" /> contains their product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654806.png" />;
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1) for any two functions $\phi_k$ and $\phi_l$ the system $\{\phi_n\}$ contains their product $\phi_m(x)=\phi_k(x)\phi_l(x)$;
  
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2) for each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654807.png" /> the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654808.png" /> contains the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m0654809.png" />.
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2) for each function $\phi_k$ the system $\{\phi_n\}$ contains the function $\phi_m(x)=1/\phi_k(x)$.
  
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Examples of multiplicative systems are the exponential system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m06548010.png" />, which is orthogonal on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065480/m06548011.png" />, and the [[Walsh system|Walsh system]] of functions.
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Examples of multiplicative systems are the exponential system $\{e^{i2\pi nx}\}_{n=-\infty}^\infty$, which is orthogonal on $[0,1]$, and the [[Walsh system|Walsh system]] of functions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.F. Harmuth,  "Transmission of information by orthogonal functions" , Springer  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.W. Zeek (ed.)  A.E. Showalter (ed.) , ''Applications of Walsh functions (Proc. Symp. Washington, April 1971)'' , Univ. Maryland  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.F. Harmuth,  "Transmission of information by orthogonal functions" , Springer  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.W. Zeek (ed.)  A.E. Showalter (ed.) , ''Applications of Walsh functions (Proc. Symp. Washington, April 1971)'' , Univ. Maryland  (1971)</TD></TR></table>

Latest revision as of 14:57, 19 August 2014

An orthonormal system of functions $\{\phi_n\}$ on $[a,b]$ satisfying the conditions:

1) for any two functions $\phi_k$ and $\phi_l$ the system $\{\phi_n\}$ contains their product $\phi_m(x)=\phi_k(x)\phi_l(x)$;

2) for each function $\phi_k$ the system $\{\phi_n\}$ contains the function $\phi_m(x)=1/\phi_k(x)$.

Examples of multiplicative systems are the exponential system $\{e^{i2\pi nx}\}_{n=-\infty}^\infty$, which is orthogonal on $[0,1]$, and the Walsh system of functions.

References

[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[2] H.F. Harmuth, "Transmission of information by orthogonal functions" , Springer (1972)
[3] R.W. Zeek (ed.) A.E. Showalter (ed.) , Applications of Walsh functions (Proc. Symp. Washington, April 1971) , Univ. Maryland (1971)
How to Cite This Entry:
Multiplicative system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicative_system&oldid=33019
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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