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One of the classical orthonormal systems of continuous functions. The Franklin system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413301.png" /> (see [[#References|[1]]] or [[#References|[2]]]) is obtained by applying the Schmidt orthogonalization process (cf. [[Orthogonalization method|Orthogonalization method]]) on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413302.png" /> to the [[Faber–Schauder system|Faber–Schauder system]], which is constructed using the set of all dyadic rational points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413303.png" />; in this case the Faber–Schauder system is, up to constant multiples, the same as the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413305.png" /> is the [[Haar system|Haar system]]. The Franklin system was historically the first example of a [[Basis|basis]] in the space of continuous functions that had the property of orthogonality. This system is also a basis in all the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413306.png" />, $1\le p<\infty$ (see [[#References|[3]]]). If a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413308.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413309.png" /> has modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133011.png" /> is the partial sum of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133012.png" /> of the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133013.png" /> with respect to the Franklin system, then
+
One of the classical orthonormal systems of continuous functions. The Franklin system $ \{f_n (t)\}_{n=1}^\infty $ (see [[#References|[1]]] or [[#References|[2]]]) is obtained by applying the Schmidt orthogonalization process (cf. [[Orthogonalization method|Orthogonalization method]]) on the interval $[0,1]$ to the [[Faber–Schauder system|Faber–Schauder system]], which is constructed using the set of all dyadic rational points in $[0,1]$ in this case the Faber–Schauder system is, up to constant multiples, the same as the system $ \{1, \int_0^t \chi_n(x)\,dx \} $, where $ \{\chi_n(x)\}_{n=1}^\infty $ is the [[Haar system|Haar system]]. The Franklin system was historically the first example of a [[Basis|basis]] in the space of continuous functions that had the property of orthogonality. This system is also a basis in all the spaces $ L_p[0,1] $, $1\le p<\infty$ (see [[#References|[3]]]). If a continuous function $ f $ on $ [0,1] $ has modulus of continuity $ \omega(\delta, f) $, and $ S_n(t, f) $ is the partial sum of order $ n $ of the Fourier series of $ f $ with respect to the Franklin system, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133014.png" /></td> </tr></table>
+
\[
 +
\max_{0\le t \le 1} |f(t) - S_n(t, f)| \le 8 \omega \left( \frac{1}{n}, f \right), \quad n=1, 2, \dots
 +
\]
  
Here the Fourier–Franklin coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133016.png" /> satisfy the inequalities
+
Here the Fourier–Franklin coefficients $ a_n(f) $ of $ f $ satisfy the inequalities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133017.png" /></td> </tr></table>
+
\[
 
+
|a_n(f)| \le \frac{12\sqrt{3}}{\sqrt{2^m}}\omega \left( \frac{1}{2^m}, f \right), \quad n=2^m+k,\quad k=1, \dots, 2^m,\quad m=0, 1, \dots, 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133018.png" /></td> </tr></table>
+
\]
  
 
and the conditions
 
and the conditions
 +
: a) $ \max_{0\le t\le 1} |f(t) - S_n(t, f)| = O(n^{-\alpha}), n \to \infty $
 +
: b) $ a_n(f) = O(n^{-\alpha-1/2}), n \to \infty $
 +
: c) $ \omega(\delta, f) = O(\delta^\alpha), \delta\to +0 $
 +
are equivalent for $ 0 < \alpha < 1 $.
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133020.png" />;
+
If the continuous function $ f $ is such that
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133022.png" />;
+
\[
 
+
\sum_{n=1}^\infty \frac{1}{n}\omega\left(\frac{1}{n}, f \right) < \infty,
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133024.png" />; are equivalent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133025.png" />.
+
\]
 
 
If the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133026.png" /> is such that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133027.png" /></td> </tr></table>
 
  
 
then the series
 
then the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133028.png" /></td> </tr></table>
+
\[
 +
\sum_{n=1}^{\infty} |a_n(f) f_n(t)|
 +
\]
  
converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133029.png" />, and if
+
converges uniformly on $ [0,1] $, and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133030.png" /></td> </tr></table>
+
\[
 +
\sum_{n=1}^{\infty}n^{-1/2}\omega\left(\frac{1}{n}, f \right) < \infty,
 +
\]
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133031.png" /></td> </tr></table>
+
\[
 +
\sum_{n=1}^{\infty} |a_n(f)| < \infty.
 +
\]
  
 
All these properties of the Franklin system are proved by using the inequalities
 
All these properties of the Franklin system are proved by using the inequalities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133032.png" /></td> </tr></table>
+
\[
 
+
\max_{0\le t\le 1}\sum_{k=1}^{2^n} |f_{2^n+k}(t)| \le C \sqrt{2^n}, \qquad n=0, 1, \dots, \quad C = 2^5\sqrt{3}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133033.png" /></td> </tr></table>
+
\]
  
The Franklin system is an unconditional basis in all the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133034.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133035.png" /> and, moreover, in all reflexive Orlicz spaces (see [[#References|[5]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133036.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133038.png" />, then one has the inequality
+
The Franklin system is an unconditional basis in all the spaces $ L_p[0,1] \quad (1 < p < \infty) $ and, moreover, in all reflexive Orlicz spaces (see [[#References|[5]]]). If $ f $ belongs to $ L_p[0,1], 1 < p < \infty $, then one has the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133039.png" /></td> </tr></table>
+
\[
 +
A_p \|f\|_p \le \left\| \left( \sum_{k=1}^{\infty} a_k^2(f) f_k^2(t) \right)^{1/2} \right\|_p \le B_p \|f\|_p,
 +
\]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133040.png" /> denotes the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133041.png" />, and the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133042.png" /> depend only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133043.png" />.
+
where $ \|.\|_p $ denotes the norm in $ L_p[0,1] $, and the constants $ A_p, B_p > 0 $ depend only on $ p $.
  
The Franklin system has had important applications in various questions in analysis. In particular, bases in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133044.png" /> (see [[#References|[4]]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133045.png" /> (see [[#References|[5]]]) have been constructed using this system. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133046.png" /> is the space of all continuously-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133047.png" /> on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133048.png" /> with the norm
+
The Franklin system has had important applications in various questions in analysis. In particular, bases in the spaces $ C^1(I^2) $ (see [[#References|[4]]]) and $ A(D) $ (see [[#References|[5]]]) have been constructed using this system. Here $ C^1(I^2) $ is the space of all continuously-differentiable functions $ f(x, y) $ on the square $ I^2 = [0, 1] \times [0, 1] $ with the norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133049.png" /></td> </tr></table>
+
\[
 +
\|f\| = \max|f(x, y)| + \max \left|\frac{\partial f}{\partial x} \right| + \max \left| \frac{\partial f}{\partial y}\right|,
 +
\]
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133050.png" />, the disc space, is the space of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133051.png" /> that are analytic in the open disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133052.png" /> in the complex plane and continuous in the closed disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133053.png" /> with the norm
+
and $ A(D) $, the disc space, is the space of all functions $ f(z) $ that are analytic in the open disc $ D = \{z: |z| < 1 \} $ in the complex plane and continuous in the closed disc $ \overline D = \{ z: |z|\le 1 \} $ with the norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133054.png" /></td> </tr></table>
+
\[
 +
\|f\| = \max_{|z|\le 1} |f(z)|.
 +
\]
  
The questions of whether there are bases in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133056.png" /> were posed by S. Banach [[#References|[6]]].
+
The questions of whether there are bases in $ C^1(I^2) $ and $ A(D) $ were posed by S. Banach [[#References|[6]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Franklin,  "A set of continuous orthogonal functions"  ''Math. Ann.'' , '''100'''  (1928)  pp. 522–529</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Z. Ciesielski,  "Properties of the orthogonal Franklin system"  ''Studia Math.'' , '''23''' :  2  (1963)  pp. 141–157</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Z. Ciesielski,  "A construction of a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133057.png" />"  ''Studia Math.'' , '''33''' :  2  (1969)  pp. 243–247</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.V. Bochkarev,  "Existence of a basis in the space of functions analytic in the disk, and some properties of Franklin's system"  ''Math. USSR-Sb.'' , '''24''' :  1  (1974)  pp. 1–16  ''Mat. Sb.'' , '''95''' :  1  (1974)  pp. 3–18</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.S. Banach,  "Théorie des opérations linéaires" , Chelsea, reprint  (1955)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Franklin,  "A set of continuous orthogonal functions"  ''Math. Ann.'' , '''100'''  (1928)  pp. 522–529</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Z. Ciesielski,  "Properties of the orthogonal Franklin system"  ''Studia Math.'' , '''23''' :  2  (1963)  pp. 141–157</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Z. Ciesielski,  "A construction of a basis in $C^{(1)}(I^2)$"  ''Studia Math.'' , '''33''' :  2  (1969)  pp. 243–247</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.V. Bochkarev,  "Existence of a basis in the space of functions analytic in the disk, and some properties of Franklin's system"  ''Math. USSR-Sb.'' , '''24''' :  1  (1974)  pp. 1–16  ''Mat. Sb.'' , '''95''' :  1  (1974)  pp. 3–18</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.S. Banach,  "Théorie des opérations linéaires" , Chelsea, reprint  (1955)</TD></TR></table>

Latest revision as of 22:07, 1 June 2013


One of the classical orthonormal systems of continuous functions. The Franklin system $ \{f_n (t)\}_{n=1}^\infty $ (see [1] or [2]) is obtained by applying the Schmidt orthogonalization process (cf. Orthogonalization method) on the interval $[0,1]$ to the Faber–Schauder system, which is constructed using the set of all dyadic rational points in $[0,1]$ in this case the Faber–Schauder system is, up to constant multiples, the same as the system $ \{1, \int_0^t \chi_n(x)\,dx \} $, where $ \{\chi_n(x)\}_{n=1}^\infty $ is the Haar system. The Franklin system was historically the first example of a basis in the space of continuous functions that had the property of orthogonality. This system is also a basis in all the spaces $ L_p[0,1] $, $1\le p<\infty$ (see [3]). If a continuous function $ f $ on $ [0,1] $ has modulus of continuity $ \omega(\delta, f) $, and $ S_n(t, f) $ is the partial sum of order $ n $ of the Fourier series of $ f $ with respect to the Franklin system, then

\[ \max_{0\le t \le 1} |f(t) - S_n(t, f)| \le 8 \omega \left( \frac{1}{n}, f \right), \quad n=1, 2, \dots \]

Here the Fourier–Franklin coefficients $ a_n(f) $ of $ f $ satisfy the inequalities

\[ |a_n(f)| \le \frac{12\sqrt{3}}{\sqrt{2^m}}\omega \left( \frac{1}{2^m}, f \right), \quad n=2^m+k,\quad k=1, \dots, 2^m,\quad m=0, 1, \dots, \]

and the conditions

a) $ \max_{0\le t\le 1} |f(t) - S_n(t, f)| = O(n^{-\alpha}), n \to \infty $
b) $ a_n(f) = O(n^{-\alpha-1/2}), n \to \infty $
c) $ \omega(\delta, f) = O(\delta^\alpha), \delta\to +0 $

are equivalent for $ 0 < \alpha < 1 $.

If the continuous function $ f $ is such that

\[ \sum_{n=1}^\infty \frac{1}{n}\omega\left(\frac{1}{n}, f \right) < \infty, \]

then the series

\[ \sum_{n=1}^{\infty} |a_n(f) f_n(t)| \]

converges uniformly on $ [0,1] $, and if

\[ \sum_{n=1}^{\infty}n^{-1/2}\omega\left(\frac{1}{n}, f \right) < \infty, \]

then

\[ \sum_{n=1}^{\infty} |a_n(f)| < \infty. \]

All these properties of the Franklin system are proved by using the inequalities

\[ \max_{0\le t\le 1}\sum_{k=1}^{2^n} |f_{2^n+k}(t)| \le C \sqrt{2^n}, \qquad n=0, 1, \dots, \quad C = 2^5\sqrt{3} \]

The Franklin system is an unconditional basis in all the spaces $ L_p[0,1] \quad (1 < p < \infty) $ and, moreover, in all reflexive Orlicz spaces (see [5]). If $ f $ belongs to $ L_p[0,1], 1 < p < \infty $, then one has the inequality

\[ A_p \|f\|_p \le \left\| \left( \sum_{k=1}^{\infty} a_k^2(f) f_k^2(t) \right)^{1/2} \right\|_p \le B_p \|f\|_p, \]

where $ \|.\|_p $ denotes the norm in $ L_p[0,1] $, and the constants $ A_p, B_p > 0 $ depend only on $ p $.

The Franklin system has had important applications in various questions in analysis. In particular, bases in the spaces $ C^1(I^2) $ (see [4]) and $ A(D) $ (see [5]) have been constructed using this system. Here $ C^1(I^2) $ is the space of all continuously-differentiable functions $ f(x, y) $ on the square $ I^2 = [0, 1] \times [0, 1] $ with the norm

\[ \|f\| = \max|f(x, y)| + \max \left|\frac{\partial f}{\partial x} \right| + \max \left| \frac{\partial f}{\partial y}\right|, \]

and $ A(D) $, the disc space, is the space of all functions $ f(z) $ that are analytic in the open disc $ D = \{z: |z| < 1 \} $ in the complex plane and continuous in the closed disc $ \overline D = \{ z: |z|\le 1 \} $ with the norm

\[ \|f\| = \max_{|z|\le 1} |f(z)|. \]

The questions of whether there are bases in $ C^1(I^2) $ and $ A(D) $ were posed by S. Banach [6].

References

[1] P. Franklin, "A set of continuous orthogonal functions" Math. Ann. , 100 (1928) pp. 522–529
[2] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[3] Z. Ciesielski, "Properties of the orthogonal Franklin system" Studia Math. , 23 : 2 (1963) pp. 141–157
[4] Z. Ciesielski, "A construction of a basis in $C^{(1)}(I^2)$" Studia Math. , 33 : 2 (1969) pp. 243–247
[5] S.V. Bochkarev, "Existence of a basis in the space of functions analytic in the disk, and some properties of Franklin's system" Math. USSR-Sb. , 24 : 1 (1974) pp. 1–16 Mat. Sb. , 95 : 1 (1974) pp. 3–18
[6] S.S. Banach, "Théorie des opérations linéaires" , Chelsea, reprint (1955)
How to Cite This Entry:
Franklin system. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Franklin_system&oldid=29773
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article