# Franklin system

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One of the classical orthonormal systems of continuous functions. The Franklin system $\{f_n (t)\}_{n=1}^\infty$ (see  or ) 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 ). 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 ). 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 ) and $A(D)$ (see ) 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 .