Criterion of Equivalence of the Haar and Franklin Systems in R.I. Spaces

Abstract

The Franklin system EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaada % GacaqaaiaadAgacaWGTbaacaGL9baaaiaawUhaamaaDaaaleaacaWG %TbaabaGaeyOhIukaaOGaeyypa0ZaaSbaaSqaaiaaicdaaeqaaaaa!3E8F!]]</EquationSource><EquationSource Format="TEX"><![CDATA[ $$\left\{ {\left. {fm} \right\}} \right._m^\infty { = _0} $$ is an orthonormal system obtained by the Gram-Schmidt orthonormalization from the Faber-Schauder system EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaada % GacaqaaiaadIgacaWGTbaacaGL9baaaiaawUhaamaaDaaaleaacaaI % WaaabaGaeyOhIukaaaaa!3C63!]]</EquationSource><EquationSource Format="TEX"><![CDATA[ $$\left\{ {\left. {hm} \right\}} \right._0^\infty $$ Recall that

EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaaBa % aaleaacaaIWaaabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadUga % daWgaaWcbaWaamWaaeaacaaIWaGaaiilaiaaigdaaiaawUfacaGLDb % aaaeqaaOGaaiikaiaadshacaGGPaaaaa!42B9!]]</EquationSource><EquationSource Format="TEX"><![CDATA[ $${h_0}(t) = {k_{\left[ {0,1} \right]}}(t) $$

EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaaBa % aaleaacaWGTbaabeaakiaacIcacaWG0bGaaiykaiabg2da9maapeaa % baWaa0baaSqaaiaaicdaaeaacaWG0baaaOGaamiEamaaBaaaleaaca % WGTbaabeaakiaacIcacaWGZbGaaiykaiaadsgacaWGZbGaaiilamaa % BaaaleaaaeqaaaqabeqaniabgUIiYdGccaWG0bGaeyicI48aamWaae % aacaaIWaGaaiilaiaaigdaaiaawUfacaGLDbaacaGGSaWaaSbaaSqa % aaqabaGccaWGTbWaaSbaaSqaaaqabaGccqGHiiIZdaWgaaWcbaaabe % aakiaad6eacaGGUaaaaa!525A!]]</EquationSource><EquationSource Format="TEX"><![CDATA[ $${h_m}(t) = \int {_0^t{x_m}(s)ds{,_{}}} t \in \left[ {0,1} \right]{,_{}}{m_{}}{ \in _{}}N. $$