Franklin system

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^1) $ (see [4]) and $ A(D) $ (see [5]) have been constructed using this system. Here $ C^1(I^1) $ 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)