View source for Franklin system

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" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f0413307.png" /> (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

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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

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and the conditions

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" />;

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" />;

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

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then the series

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converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041330/f04133029.png" />, and if

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then

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

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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

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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" />.

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

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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

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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]]].

====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>