Volume 8 | Issue 3 |

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Muradabad Janpad ke nagrik

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Muradabad Janpad ke nagrik

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ABSTRACT Objective: To determine the Statin (Fluvastatin) in pure form simple and cost effective spectrophotometric method was developed. Methods: The UV spectrum of Fluvastatin in methanol showed absorption maximum at 304 nm and obeys beer�s law in the concentration range 5-30 �g/ml. The absorbance was found to be increases concentration with increasing linearity which is calculated by correlation coefficient value of 0.998. This method was validated for the accuracy, linearity, precision, ruggedness and robustness. Results: The method has demonstrated excellent linearity over the range of 5-30 �g/ml with regression equation y = 0.022x + 0.053 and regression coefficient r2 = 0.998. Furthermore, the method was found to be highly sensitive with LOD (1.501�g/ml) and LOQ (4.550�g/ml) Conclusion: On the basis of the results, this method was successfully applied for the assay of Fluvastatin in different pharmaceutical dosage forms.

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Keywords: Fluvastatin, Spectrophotometry, Dimethyl formamide, validation.

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Autism spectrum disorder is characterized by persistent deficits in the ability to initiate and to sustain reciprocal social interaction and social communication, and by a range of restricted, repetitive, and inflexible patterns of behaviour and interests. Sensory symptoms are prevalent in autism spectrum disorder, and prevalence rates of tactile dysfunction in children with Autism have significantly to the extent of 77%. A descriptive and qualitative study on "Poor Tactile Dysfunction in Children with Autism" was carried out to study the presence of poor tactile perception and discrimination with regard to various tactile components. The Sensory Processing Disorder Checklist by Carol Stock Kranowitz (1995) has been used for this study. Data was collected from three special schools by using direct observation, parental interview, interaction with special educators and caregivers. Data were descriptively and qualitatively analyzed. Children with poor tactile dysfunction experience difficulty in motor planning and body awareness.

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Autism, Sensory Symptoms, Poor Tactile Dysfunction, Motor Planning, Body Awareness

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ABSTRACT Cold-formed steel sections are widely used for light-gauge structural beams and roof purlins due to their high strength-to-weight ratio and ease of installation on site. The commonly used profiles have a wide variety of cross sectional shapes, e.g. C, Z, �top hat� and sigma sections. Amongst these popular sections, the sigma section possesses several structural advantages, such as high cross-sectional resistance and large torsional rigidity compared with standard Z or C sections. These also have shear Center close to the web. Commercially in market, �?� sigma sections are available which are predominantly used as lightly loaded and medium span elements in roofing systems. This study aims to make the flexural member with two point loading, equally strong under compression and tension by opting suitable geometry as Cold Formed Steel has the versatility of being cast into three different sigma sections of equal cross section. The coupon test was conducted to obtain material property. Finite element analysis has been carried out using the software ABAQUS 6.14-1.The investigation has thus enabled to study the effect of dimension and position of web stiffener on the flexural behaviour and strength prediction of cold formed steel sigma section.

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Beam, Cold-formed steel, Sigma sections, Finite element and Tension coupon test

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Disaster prediction is a challenging research area, where a future occurrence of the devastating catastrophe is predicted. In the project a simpler way of detecting the occurrence of landslide and flood has been introduced. It is based on collecting WSN data and using the API�s where weather information API is used to fetch live weather details. The collected live weather of a particular place is used to predict the disaster. If there is a chance of occurrence of the disaster (landslide, flood) then an alert message is sent to the concerned authority to create awareness among people.

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WSN , API, Landslide, Flood

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In today�s era, there is a large amount of increase in the crime rate due to the research gap between technologies and the optimal usage of Investigation. Identifying and analysing the patterns for crime prevention is one of the big challenge. Also, due to some technological limitations, having large amount of data it is difficult to analyse crimes. The goal of this paper is to propose employment of algorithms that works efficiently on large amount of data. This paper is concentrated on crime prevention by concerning various incidents occurred in various states.

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Crime Analysis, Location Based Systems, Data Mining, Database, K Mean.

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Supply chain management in pharmaceutical industry is a complex process which comprises of managing tasks like manufacturing, storage, and sales of medical drugs. Though supply chain management is a challenge on every industry, in health care, compromised supply chain adds risk to consumers� safety. Increased adoption of technology, globalization and industry populated with multiple stakeholders in various jurisdictions has given rise to a complicated health supply chain. In this operation process, due to imbalance and asymmetry of information among the stakeholders arises a fraud problem such as compromising the consumer drug information. In such cases, in order to improve the security and reliability of drug information we suggest blockchain based data storage technology. In this paper we discuss about hoe fraudulent drug information problem can be resolved using blockchain and by the way presenting reliable information to the concerning consumers. We discuss about smart contract based on Consensus algorithm. Blockchain in pharmaceutical supply chain not only reduces the risk of counterfeiting and theft, but also allows efficient inventory management

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Supply chain, blockchain, pharmaceutical industry, proof of work, medical drug information, drug counterfeiting.

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ANPAD RAMPUR ME JAL SANSADHANO KA BHAUGOLIC VISHLESHAN

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ANPAD RAMPUR ME JAL SANSADHANO KA BHAUGOLIC VISHLESHAN

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Emotion recognition is that a part of speech recognition which is gaining more popularity and wish for it increases enormously. Although there are methods to know emotion using machine learning techniques, this project attempts to use deep learning and image classification method to acknowledge emotion and classify the emotion consistent with the speech signals. Our proposed model outperforms previous state-of-the-art methods in assigning data to at least one of 4 emotion categories (i.e., angry, happy, sad and neutral) when the model is applied to the RAVDESS dataset, as re?ected by accuracies starting from 68.8% to 71.8%. Various datasets are investigated and explored for training emotion recognition model are explained during this paper.

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Emotion recognition

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Creative Commons Attribution 4.0 and The Open Definition

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documentclass[11pt]{article} usepackage{fullpage} usepackage{amsfonts} usepackage{amsmath} usepackage{amssymb} usepackage{amsthm} ewtheorem{theorem}{Theorem}[section] egin{document} itle{Hardy Hilbert spaces and its applications} author{Praveen Sharma Department of Mathematics, University of Delhi} date{} maketitle section*{Abstract} In this paper , we discuss the Hardy Hilbert Space on the open disk with center origin and radius unity. We have proved that $H^2$ Space is isomorphic to proper subspace of $L^2$ Space which has various applications in Quantumm Mechanics. section{Preliminaries} subsubsection{Definition(Inner Product Space)} An extit{inner product space} is an vector space $W$ (over field $K=mathbb{R}ormathbb{C}$) with an inner product defined on it. Here, an inner product is an function $<,> : W imes W o K$ which satisfies the following properties:-- egin{enumerate} item $<alpha u+v,w>hspace{1cm} =hspace{1cm} alpha<u,w> + <v,w>$ item $ overline{<u,v>} hspace{2cm} =hspace{1cm} <v,u> $ item $<u,u> hspace{2cm} geq 0$ item $ <u,u> = 0 hspace{1cm} Leftrightarrow u = 0$ hspace{1cm} ( extbf{ for all scalers $alphain K$ and for all vectors $u,v,win W$}) paragraph{Note1} extbf{ Every inner product space is an normed spaces with the norm induced by the inner product is given by} egin{center} $||u|| = sqrt{<u,u>}$ end{center} paragraph{Note2} extbf{An normed space $(W,||.||)$ is said to be complete if each cauchy sequence coverges in $W$.} end{enumerate} subsection{Hilbert Space} An Hilbert Space is defined as the complete inner product space.smallskip extbf{Example:-} smallskip egin{center} egin{large} extbf{$l^2 = {(x_0,x_1,ldots ) : x_n in mathbb{C}, sum_{n=0}^infty |x_n|^2 <infty }$}end{large} end{center} i.e. all the elements of $l^2$ are the sequence of all the complex numbers that are square-summable. Inner product on $l^2$ is given by :- egin{center} extbf{$ <(x_n)_{n=0}^infty , (y_n)_{n=0}^infty> = sum_{n=0}^infty{x_noverline{y_n}}$} hspace{1cm} ( extsc{it is an hilbert sequence space)} end{center} subsection{Definition(Orthonormal sets and sequences)} An subset $X$ of an inner product space is said to be orthonormal if for all $u, v in X $ we have , $$ <u,v> = left{egin{array}{ll} 0 & mathrm{if} u eq v ||u||^2 & mathrm{if} u=v end{array} ight. .$$ paragraph{Note} extbf{If norm of each element of an orthogonal set $X$ is 1 then the set is said to be orthogonal. i.e for all $u,v in X$ we have, } $$ <u,v> = left{ egin{array}{ll} 0 & mathrm{if} u eq v 1 & mathrm{if} u =v end{array} ight. $$ subsection{Definition(Orthonormal basis)} An orthonormal subset $X$ of Hilbert space $W$ is said to be an orthonormal basis if span of $X$ is dense in $W$.i.e. $$overline{spanX} = W $$ paragraph{Note} Every Hilbert space $W eq{0}$ has an orthonormal basis. subsection{Definition(Separable Hilbert Space)} An Hilbert-Space $W$ is said to be extit{separable} if there exist an countable set which is dense in $W$.smallskip extbf{Example:} $l^2$ is an separable Hilbert space paragraph{Note} Each orthonormal basis of an separable Hilbert space are countable.Therefore orthonormal basis of $l^2$ are countable subsection*{Recall} egin{enumerate} item An orthonormal sequence $(e_n)_{n=0}^infty$ is an orthonormal basis of an Hilbert - Space $W$ egin{LARGE}$$Leftrightarrow$$end{LARGE} for all $uin W$ we havehspace{2cm}$ sum_{n=0}^infty|<u,e_n>|^2 = ||u||^2$ cite{kreyszig1978introductory} hspace{0.5cm} extbf{Parseval identity} item Let $(e_n)$ be an orthonormal sequence in an Hilbert-space then $$ sum_{n=0}^inftyalpha_ne_n$$ converges in $W$ $$iff$$ the series $$sum_{n=0}^infty|alpha_n|^2$$ converges in $mathbb{R}$ end{enumerate} section{THE HARDY-HILBERT SPACE} subsection{DEFINITION} { It is defined as the space of all the analytic functions which have a power series representation about origin with square-summable complex coefficients. It is denoted by extbf{ $H^2$}. $$ H^2 = { f : f(z) = sum_{n=0}^infty alpha_nz^n : sum_{n=0}^infty|alpha_n|^2 <infty}$$ Inner Product on $H^2$ is given by $$<f,g> =sum_{n=0}^infty a_noverline{b_n}$$ for $ f(z) = sum_{n=0}^infty a_nz^n hspace{0.5cm} and hspace{0.5cm} g(z) = sum_{n=0}^infty b_nz^nhspace{0.3cm} in hspace{0.2cm} H^2$ egin{theorem} extbf{The Hardy-Hilbert space is an separable Hilbert Space.} end{theorem} egin{proof} Define an function;-- $$ phi:l^2 o H^2$$ given by $$ (a_n)_{n=0}^infty o sum_{n=0}^infty a_nz^n$$ egin{itemize} item extbf{underline{ $phi$ is well defined}} since hspace{1cm} $ (a_n)_{n=0}^infty in l^2 Rightarrow sum_{n=0}^infty |a_n|^2 <infty Rightarrow sum_{n=0}^infty a_nz^n $ hspace{1cm} which being an power series is an analytic function whose coefficients are square summable hence is in $H^2$ $ herefore phi$ is well defined item extbf{underline{Clearly $phi$ is linear}} item extbf{underline{ $phi$ is isometric}} Fix $(a_n)_{n=0}^infty in l^2 $ then we have $$phi((a_n)_{n=0}^infty) = ||sum_{n=0}^infty a_nz^n||_{H^2} = sqrt{sum_{n=0}^infty |a_n|^2} =||(a_n)_{n=0}^infty||_{l^2} $$ $ herefore phi$ is an isometric $ herefore phi$ preserves the norm so that the inner product item extbf{since isometry property implies one one property$ herefore phi$ is one one } cite{bhatia2009notes} item extbf{underline{$phi$ is onto}}smallskip Let $f in H^2$ then $f(z) = sum_{n=0}^infty a_nz^n$ where $sum_{n=0}^infty|a_n|^2<infty$medskip extbf{ define} $x=(a_0,a_1,ldots)$ extbf{Since} $$||x||^2 =sum_{n=0}^infty |a_n|^2 <infty$$ extbf{$$ herefore xin l^2$$} extbf{and}$$phi(x) = f $$ extbf{$$ hereforephi hspace{0.5cm} is hspace{0.5cm} onto$$} end{itemize} extbf{Therefore $phi$ is an vector space isomorphism which also preserves the inner product. Since $l^2$ is an separable Hilbert space hence $H^2$ is also an separable Hilbert Space } end{proof} paragraph{Notations} $mathbb{D}= {z:|z|<1}$ denotes the open unit disk about origin in $mathbb{C}$ $mathbb{S^1} ={z:|z|=1}$ denotes the unit circle about origin in $mathbb{C}$ egin{theorem} extbf{Radius of convergence of each function in $H^2$ is atleast $1$} (i.e. each function in $H^2$ is analytic in the open unit disk $mathbb{D} $) end{theorem} egin{proof} Let $z_0in mathbb{D}$ is fixed $Rightarrowhspace{0.5cm} |z_0|<1$ $ herefore$ the geometric series $sum_{n=0}^infty |z_0|^n$ converges. Let $fin H^2$ is arbitrary.Then $$f(z) = sum_{n=0}^infty a_nz^n hspace{2cm} where hspace{1cm} sum_{n=0}^infty|a_n|^2 <infty$$ Since the series $sum_{n=0}^infty|a_n|^2$ converges $Rightarrow$ $|a_n|^2 longrightarrow 0 Rightarrow|a_n|longrightarrow 0$ $ herefore (|a_n|)_{n=0}^infty$ is an convergent sequence hence bounded. $ herefore exists M>0 $ such that $$ |a_n|leq M hspace{2cm} forall hspace{1cm} ngeq0$$ Now $$sum_{n=0}^infty|a_nz_0^n| leq Msum_{n=0}^infty|z_0|^n $$ where being an geometric series right hand side converges. $ herefore$ By Comparison test the series $ sum_{n=0}^infty a_nz_0^n$ converges absolutely . Since in Hilbert space absolute convergence implies convergence. $ herefore$ the series $sum_{n=0}^infty a_nz_0^n$ converges in $H^2$ since $z_0in H^2$ is arbitrary $ herefore$ each function in $H^2$ is analytic in the unit disk $mathbb{D}$ end{proof} subsection{Definition ($L^2(mathbb{S^1})$ space)} It is defined as the space of all the equivalence classes of functions cite{royden2010real} that are Lebesgue measurable on $S^mathbb{1}$ and square integrable on $S^mathbb{1}$ with respect to Lebesgue measure normalized such that measure of $S^mathbb{1}$ is $1$. $$ L^2(S^mathbb{1}) = {f: f hspace{0.2cm} ishspace{0.2cm} Lesbesgue hspace{0.2cm} measurable hspace{0.2cm} onhspace{0.2cm} mathbb{S^1} hspace{0.2cm} andhspace{0.3cm} frac{1}{2pi}int_0^{2pi}|f(e^{iota heta)}|^2d heta<infty } $$ Inner product on $L^2(mathbb{S^1})$ is given by - $$<f,g> = frac{1}{2pi}int_0^{2pi} f(e^{iota heta})overline{g(e^{iota heta})}d heta$$ paragraph{Note} $L^2(mathbb{S^1})$ is an Hilbert-space with the orthonormal basis given by ${e_n: nin mathbb{Z}}$ where $e_n(e^{iota heta})= e^{iota n heta}.$ extbf{Therefore} $$L^2(mathbb{S^1}) =left {f:f=sum_{n=-infty}^{n=infty}<f,e_n>e_n ight}.$$.cite{martinez2007introduction} subsubsection{Definition ($widehat{H^2}$ space)} $$widehat{H^2} = {f in L^2(mathbb{S^1}): <f,e_n> =0hspace{0.2cm} forhspace{0.2cm} negativehspace{0.2cm} valuehspace{0.2cm} ofhspace{0.2cm} n }$$ $$widehat{H^2} = left{f in L^2(mathbb{S^1}) : f= sum_{n=0}^infty <f,e_n>e_n ight }.$$ extbf{$widehat{H^2}$ is an subspace of $L^2(mathbb{S^1})$ whose negative Fourier coefficients are 0 $ herefore $ ${e_n : n=0,1,ldots }$ are orthonormal basis of $widehat{H^2}$} egin{theorem} egin{LARGE} extbf{$widehat{H^2}$ is an Hilbert-space}end{LARGE} end{theorem} egin{proof} Let $f in overline{widehat{H^2}}$ then there exist an sequence $(f_n)_{n=0}^infty$medskip such that hspace{2cm} $f_n longrightarrow f$ as $nlongrightarrowinfty$medskip Since hspace{2cm} $f_nin widehat{H^2}hspace{1cm} forall hspace{1cm} ngeq0$medskip $ herefore hspace{3cm} <f_n,e_k>=0 hspace{1cm} forall ngeq0 hspace{1cm} and hspace{1cm} forall k<0$medskip extbf{Now for each $k<0$ we have}medskip $|<f_n,e_k> - <f,e_k>| leq |<(f_n-f,e_k>| leq||f_n - f||longrightarrow 0 hspace{0.3cm} ashspace{0.1cm} nlongrightarrowinfty$(Schwarz Inequality cite{kreyszig1978introductory})medskip since $ hspace{3cm} <f_n,e_k>=0 hspace{1cm} forall ngeq0 hspace{1cm} Rightarrow hspace{1cm} <f,e_k>=0 $medskip Since $k<0$ is arbitrary $ herefore hspace{1cm}<f,e_k>=0 hspace{1cm}forallhspace{1cm} k<0$medskip $$ hereforehspace{1cm} fin widehat{H^2}$$ extbf{Therefore $widehat{H^2}$ is an closed subspace of $L^2(mathbb{S^1})$ Hence an Hilbert-Space} end{proof} egin{theorem} egin{large} extbf{The Hardy-Hilbert space can be identified as a subspace of $L^2(mathbb{S^1})$} end{large} end{theorem} egin{proof} Define an function extbf{$$psi:H^2 owidehat{H^2}$$} $$f o ilde{f}$$ where $f(z)=sum_{n=0}^infty a_nz^n$hspace{1cm} and hspace{1cm} $ ilde{f}=sum_{n=0}^infty a_ne_n$ egin{itemize} item extbf{underline{$psi$ is well defined}}medskip Let $f in H^2$ hspace{0.5cm} Then hspace{0.5cm} $f(z)=sum_{n=0}^infty a_nz^n$ hspace{0.5cm} where hspace{0.5cm} $sum_{n=0}^infty |a_n|^2 <infty$medskip Then by extbf{(recall 2)} the series $ ilde{f} =sum_{n=0}^infty a_ne_n $ converges in $widehat{H^2}$medskip $ hereforepsi$ hspace{0.5cm} is hspace{0.5cm} wellhspace{0.5cm} defined item extbf{underline{Clearly $psi$ is linear}}medskip item extbf{underline{$psi$ is an isometry}}medskip For any arbitrary $fin H^2$ where $f(z)=sum_{n=0}^infty a_nz^n$ we have:- $$ || psi(f)|| = || ilde{f}|| = frac{1}{2pi}int_0^{2pi}| ilde{ f}(e^{iota heta}|^2d heta $$medskip extbf{Now} $$frac{1}{2pi}int_0^{2pi} | ilde{f}(e^{iota heta})|^2 d heta = frac{1}{2pi}int_0^{2pi}(sum_{n=0}^infty a_ne^{iota n heta})(overline{sum_{m=0}^infty a_me^{iota m heta}}) $$ hspace{7cm} = $frac{1}{2pi}int_0^{2pi}sum_{n=0}^infty sum_{m=0}^infty a_noverline{a_m}e^{iota(n-m) heta} d heta$ hspace{7cm} = $sum_{n=0}^infty |a_n|^2$ hspace{1cm} extbf{(since $frac{1}{2pi} int_0^{2pi} e^{iota(n-m) heta} = delta_{nm}$)} hspace{7cm} = $||f||^2$ Since $fin H^2$ is arbitrary extbf{$$ herefore ||psi(f)||hspace{1cm} = hspace{1cm}||f||hspace{1cm} forall hspace{0.5cm} fin H^2$$} egin{large} extbf{Therefore $psi$ is an isometry. Hence it preserves the inner product Isometry $Rightarrow$ one one property. $ herefore psi$ is one one.} end{large} item extbf{underline{$psi$ is Onto}} Let $ ilde{f}in widehat{H^2}$. Then $ ilde{f}=sum_{n=0}^infty<f,e_n>e_n$ where $<f,e_1>,<f,e_2>,ldots$ are Fourier coefficients of f with respect to the orthonormal basis ${e_n: nin mathbb{N}}.$ extbf{Then by Parseval relation we have} $$sum_{n=0}^infty |<f,e_n>|^2 = ||f||^2 < infty $$ extbf{ Define} $$ f =sum_{n=0}^infty a_nz^n hspace{1cm} where hspace{1cm} a_n = <f,e_n>hspace{1cm} forallhspace{1cm} ngeq0$$ extbf{ Since} $$ sum_{n=0}^infty |a_n|^2 = sum_{n=0}^infty |<f,e_n>|^2 = ||f||^2 < infty $$ extbf{Therefore} $$fin H^2$$ extbf{That is for each $ ilde{f}in widehat{H^2}$ there exist $fin H^2$ such that $psi(f) = ilde{f}$ Therefore $psi$ is onto}medskip extbf{ That is $psi$ is an vector space isomorphism which also preserves the norm. Therefore $H^2$ can br identified as a subspace of the $L^2(mathbb{S^1})$ space} end{itemize} end{proof} section{ extbf{ Applications} } egin{enumerate} item In the mathematical rigrous formulation of Quantum Mechanics, developed by extbf{Joh Von Neumann}' the position and momentum states for a single non relavistic spin 0 Particle is the space of all the square integrable functions($L^2$). But $L^2$ have some undesirable properties and $H^2$ is much well behaved space so we work with $H^2$ instead of $L^2$. end{enumerate} ibliographystyle{plain} ibliography{my} end{document} documentclass[11pt]{article} usepackage{fullpage} usepackage{amsfonts} usepackage{amsmath} usepackage{amssymb} usepackage{amsthm} ewtheorem{theorem}{Theorem}[section] egin{document} itle{Hardy Hilbert spaces and its applications} author{Praveen Sharma Department of Mathematics, University of Delhi} date{} maketitle section*{Abstract} In this paper , we discuss the Hardy Hilbert Space on the open disk with center origin and radius unity. We have proved that $H^2$ Space is isomorphic to proper subspace of $L^2$ Space which has various applications in Quantumm Mechanics. section{Preliminaries} subsubsection{Definition(Inner Product Space)} An extit{inner product space} is an vector space $W$ (over field $K=mathbb{R}ormathbb{C}$) with an inner product defined on it. Here, an inner product is an function $<,> : W imes W o K$ which satisfies the following properties:-- egin{enumerate} item $<alpha u+v,w>hspace{1cm} =hspace{1cm} alpha<u,w> + <v,w>$ item $ overline{<u,v>} hspace{2cm} =hspace{1cm} <v,u> $ item $<u,u> hspace{2cm} geq 0$ item $ <u,u> = 0 hspace{1cm} Leftrightarrow u = 0$ hspace{1cm} ( extbf{ for all scalers $alphain K$ and for all vectors $u,v,win W$}) paragraph{Note1} extbf{ Every inner product space is an normed spaces with the norm induced by the inner product is given by} egin{center} $||u|| = sqrt{<u,u>}$ end{center} paragraph{Note2} extbf{An normed space $(W,||.||)$ is said to be complete if each cauchy sequence coverges in $W$.} end{enumerate} subsection{Hilbert Space} An Hilbert Space is defined as the complete inner product space.smallskip extbf{Example:-} smallskip egin{center} egin{large} extbf{$l^2 = {(x_0,x_1,ldots ) : x_n in mathbb{C}, sum_{n=0}^infty |x_n|^2 <infty }$}end{large} end{center} i.e. all the elements of $l^2$ are the sequence of all the complex numbers that are square-summable. Inner product on $l^2$ is given by :- egin{center} extbf{$ <(x_n)_{n=0}^infty , (y_n)_{n=0}^infty> = sum_{n=0}^infty{x_noverline{y_n}}$} hspace{1cm} ( extsc{it is an hilbert sequence space)} end{center} subsection{Definition(Orthonormal sets and sequences)} An subset $X$ of an inner product space is said to be orthonormal if for all $u, v in X $ we have , $$ <u,v> = left{egin{array}{ll} 0 & mathrm{if} u eq v ||u||^2 & mathrm{if} u=v end{array} ight. .$$ paragraph{Note} extbf{If norm of each element of an orthogonal set $X$ is 1 then the set is said to be orthogonal. i.e for all $u,v in X$ we have, } $$ <u,v> = left{ egin{array}{ll} 0 & mathrm{if} u eq v 1 & mathrm{if} u =v end{array} ight. $$ subsection{Definition(Orthonormal basis)} An orthonormal subset $X$ of Hilbert space $W$ is said to be an orthonormal basis if span of $X$ is dense in $W$.i.e. $$overline{spanX} = W $$ paragraph{Note} Every Hilbert space $W eq{0}$ has an orthonormal basis. subsection{Definition(Separable Hilbert Space)} An Hilbert-Space $W$ is said to be extit{separable} if there exist an countable set which is dense in $W$.smallskip extbf{Example:} $l^2$ is an separable Hilbert space paragraph{Note} Each orthonormal basis of an separable Hilbert space are countable.Therefore orthonormal basis of $l^2$ are countable subsection*{Recall} egin{enumerate} item An orthonormal sequence $(e_n)_{n=0}^infty$ is an orthonormal basis of an Hilbert - Space $W$ egin{LARGE}$$Leftrightarrow$$end{LARGE} for all $uin W$ we havehspace{2cm}$ sum_{n=0}^infty|<u,e_n>|^2 = ||u||^2$ cite{kreyszig1978introductory} hspace{0.5cm} extbf{Parseval identity} item Let $(e_n)$ be an orthonormal sequence in an Hilbert-space then $$ sum_{n=0}^inftyalpha_ne_n$$ converges in $W$ $$iff$$ the series $$sum_{n=0}^infty|alpha_n|^2$$ converges in $mathbb{R}$ end{enumerate} section{THE HARDY-HILBERT SPACE} subsection{DEFINITION} { It is defined as the space of all the analytic functions which have a power series representation about origin with square-summable complex coefficients. It is denoted by extbf{ $H^2$}. $$ H^2 = { f : f(z) = sum_{n=0}^infty alpha_nz^n : sum_{n=0}^infty|alpha_n|^2 <infty}$$ Inner Product on $H^2$ is given by $$<f,g> =sum_{n=0}^infty a_noverline{b_n}$$ for $ f(z) = sum_{n=0}^infty a_nz^n hspace{0.5cm} and hspace{0.5cm} g(z) = sum_{n=0}^infty b_nz^nhspace{0.3cm} in hspace{0.2cm} H^2$ egin{theorem} extbf{The Hardy-Hilbert space is an separable Hilbert Space.} end{theorem} egin{proof} Define an function;-- $$ phi:l^2 o H^2$$ given by $$ (a_n)_{n=0}^infty o sum_{n=0}^infty a_nz^n$$ egin{itemize} item extbf{underline{ $phi$ is well defined}} since hspace{1cm} $ (a_n)_{n=0}^infty in l^2 Rightarrow sum_{n=0}^infty |a_n|^2 <infty Rightarrow sum_{n=0}^infty a_nz^n $ hspace{1cm} which being an power series is an analytic function whose coefficients are square summable hence is in $H^2$ $ herefore phi$ is well defined item extbf{underline{Clearly $phi$ is linear}} item extbf{underline{ $phi$ is isometric}} Fix $(a_n)_{n=0}^infty in l^2 $ then we have $$phi((a_n)_{n=0}^infty) = ||sum_{n=0}^infty a_nz^n||_{H^2} = sqrt{sum_{n=0}^infty |a_n|^2} =||(a_n)_{n=0}^infty||_{l^2} $$ $ herefore phi$ is an isometric $ herefore phi$ preserves the norm so that the inner product item extbf{since isometry property implies one one property$ herefore phi$ is one one } cite{bhatia2009notes} item extbf{underline{$phi$ is onto}}smallskip Let $f in H^2$ then $f(z) = sum_{n=0}^infty a_nz^n$ where $sum_{n=0}^infty|a_n|^2<infty$medskip extbf{ define} $x=(a_0,a_1,ldots)$ extbf{Since} $$||x||^2 =sum_{n=0}^infty |a_n|^2 <infty$$ extbf{$$ herefore xin l^2$$} extbf{and}$$phi(x) = f $$ extbf{$$ hereforephi hspace{0.5cm} is hspace{0.5cm} onto$$} end{itemize} extbf{Therefore $phi$ is an vector space isomorphism which also preserves the inner product. Since $l^2$ is an separable Hilbert space hence $H^2$ is also an separable Hilbert Space } end{proof} paragraph{Notations} $mathbb{D}= {z:|z|<1}$ denotes the open unit disk about origin in $mathbb{C}$ $mathbb{S^1} ={z:|z|=1}$ denotes the unit circle about origin in $mathbb{C}$ egin{theorem} extbf{Radius of convergence of each function in $H^2$ is atleast $1$} (i.e. each function in $H^2$ is analytic in the open unit disk $mathbb{D} $) end{theorem} egin{proof} Let $z_0in mathbb{D}$ is fixed $Rightarrowhspace{0.5cm} |z_0|<1$ $ herefore$ the geometric series $sum_{n=0}^infty |z_0|^n$ converges. Let $fin H^2$ is arbitrary.Then $$f(z) = sum_{n=0}^infty a_nz^n hspace{2cm} where hspace{1cm} sum_{n=0}^infty|a_n|^2 <infty$$ Since the series $sum_{n=0}^infty|a_n|^2$ converges $Rightarrow$ $|a_n|^2 longrightarrow 0 Rightarrow|a_n|longrightarrow 0$ $ herefore (|a_n|)_{n=0}^infty$ is an convergent sequence hence bounded. $ herefore exists M>0 $ such that $$ |a_n|leq M hspace{2cm} forall hspace{1cm} ngeq0$$ Now $$sum_{n=0}^infty|a_nz_0^n| leq Msum_{n=0}^infty|z_0|^n $$ where being an geometric series right hand side converges. $ herefore$ By Comparison test the series $ sum_{n=0}^infty a_nz_0^n$ converges absolutely . Since in Hilbert space absolute convergence implies convergence. $ herefore$ the series $sum_{n=0}^infty a_nz_0^n$ converges in $H^2$ since $z_0in H^2$ is arbitrary $ herefore$ each function in $H^2$ is analytic in the unit disk $mathbb{D}$ end{proof} subsection{Definition ($L^2(mathbb{S^1})$ space)} It is defined as the space of all the equivalence classes of functions cite{royden2010real} that are Lebesgue measurable on $S^mathbb{1}$ and square integrable on $S^mathbb{1}$ with respect to Lebesgue measure normalized such that measure of $S^mathbb{1}$ is $1$. $$ L^2(S^mathbb{1}) = {f: f hspace{0.2cm} ishspace{0.2cm} Lesbesgue hspace{0.2cm} measurable hspace{0.2cm} onhspace{0.2cm} mathbb{S^1} hspace{0.2cm} andhspace{0.3cm} frac{1}{2pi}int_0^{2pi}|f(e^{iota heta)}|^2d heta<infty } $$ Inner product on $L^2(mathbb{S^1})$ is given by - $$<f,g> = frac{1}{2pi}int_0^{2pi} f(e^{iota heta})overline{g(e^{iota heta})}d heta$$ paragraph{Note} $L^2(mathbb{S^1})$ is an Hilbert-space with the orthonormal basis given by ${e_n: nin mathbb{Z}}$ where $e_n(e^{iota heta})= e^{iota n heta}.$ extbf{Therefore} $$L^2(mathbb{S^1}) =left {f:f=sum_{n=-infty}^{n=infty}<f,e_n>e_n ight}.$$.cite{martinez2007introduction} subsubsection{Definition ($widehat{H^2}$ space)} $$widehat{H^2} = {f in L^2(mathbb{S^1}): <f,e_n> =0hspace{0.2cm} forhspace{0.2cm} negativehspace{0.2cm} valuehspace{0.2cm} ofhspace{0.2cm} n }$$ $$widehat{H^2} = left{f in L^2(mathbb{S^1}) : f= sum_{n=0}^infty <f,e_n>e_n ight }.$$ extbf{$widehat{H^2}$ is an subspace of $L^2(mathbb{S^1})$ whose negative Fourier coefficients are 0 $ herefore $ ${e_n : n=0,1,ldots }$ are orthonormal basis of $widehat{H^2}$} egin{theorem} egin{LARGE} extbf{$widehat{H^2}$ is an Hilbert-space}end{LARGE} end{theorem} egin{proof} Let $f in overline{widehat{H^2}}$ then there exist an sequence $(f_n)_{n=0}^infty$medskip such that hspace{2cm} $f_n longrightarrow f$ as $nlongrightarrowinfty$medskip Since hspace{2cm} $f_nin widehat{H^2}hspace{1cm} forall hspace{1cm} ngeq0$medskip $ herefore hspace{3cm} <f_n,e_k>=0 hspace{1cm} forall ngeq0 hspace{1cm} and hspace{1cm} forall k<0$medskip extbf{Now for each $k<0$ we have}medskip $|<f_n,e_k> - <f,e_k>| leq |<(f_n-f,e_k>| leq||f_n - f||longrightarrow 0 hspace{0.3cm} ashspace{0.1cm} nlongrightarrowinfty$(Schwarz Inequality cite{kreyszig1978introductory})medskip since $ hspace{3cm} <f_n,e_k>=0 hspace{1cm} forall ngeq0 hspace{1cm} Rightarrow hspace{1cm} <f,e_k>=0 $medskip Since $k<0$ is arbitrary $ herefore hspace{1cm}<f,e_k>=0 hspace{1cm}forallhspace{1cm} k<0$medskip $$ hereforehspace{1cm} fin widehat{H^2}$$ extbf{Therefore $widehat{H^2}$ is an closed subspace of $L^2(mathbb{S^1})$ Hence an Hilbert-Space} end{proof} egin{theorem} egin{large} extbf{The Hardy-Hilbert space can be identified as a subspace of $L^2(mathbb{S^1})$} end{large} end{theorem} egin{proof} Define an function extbf{$$psi:H^2 owidehat{H^2}$$} $$f o ilde{f}$$ where $f(z)=sum_{n=0}^infty a_nz^n$hspace{1cm} and hspace{1cm} $ ilde{f}=sum_{n=0}^infty a_ne_n$ egin{itemize} item extbf{underline{$psi$ is well defined}}medskip Let $f in H^2$ hspace{0.5cm} Then hspace{0.5cm} $f(z)=sum_{n=0}^infty a_nz^n$ hspace{0.5cm} where hspace{0.5cm} $sum_{n=0}^infty |a_n|^2 <infty$medskip Then by extbf{(recall 2)} the series $ ilde{f} =sum_{n=0}^infty a_ne_n $ converges in $widehat{H^2}$medskip $ hereforepsi$ hspace{0.5cm} is hspace{0.5cm} wellhspace{0.5cm} defined item extbf{underline{Clearly $psi$ is linear}}medskip item extbf{underline{$psi$ is an isometry}}medskip For any arbitrary $fin H^2$ where $f(z)=sum_{n=0}^infty a_nz^n$ we have:- $$ || psi(f)|| = || ilde{f}|| = frac{1}{2pi}int_0^{2pi}| ilde{ f}(e^{iota heta}|^2d heta $$medskip extbf{Now} $$frac{1}{2pi}int_0^{2pi} | ilde{f}(e^{iota heta})|^2 d heta = frac{1}{2pi}int_0^{2pi}(sum_{n=0}^infty a_ne^{iota n heta})(overline{sum_{m=0}^infty a_me^{iota m heta}}) $$ hspace{7cm} = $frac{1}{2pi}int_0^{2pi}sum_{n=0}^infty sum_{m=0}^infty a_noverline{a_m}e^{iota(n-m) heta} d heta$ hspace{7cm} = $sum_{n=0}^infty |a_n|^2$ hspace{1cm} extbf{(since $frac{1}{2pi} int_0^{2pi} e^{iota(n-m) heta} = delta_{nm}$)} hspace{7cm} = $||f||^2$ Since $fin H^2$ is arbitrary extbf{$$ herefore ||psi(f)||hspace{1cm} = hspace{1cm}||f||hspace{1cm} forall hspace{0.5cm} fin H^2$$} egin{large} extbf{Therefore $psi$ is an isometry. Hence it preserves the inner product Isometry $Rightarrow$ one one property. $ herefore psi$ is one one.} end{large} item extbf{underline{$psi$ is Onto}} Let $ ilde{f}in widehat{H^2}$. Then $ ilde{f}=sum_{n=0}^infty<f,e_n>e_n$ where $<f,e_1>,<f,e_2>,ldots$ are Fourier coefficients of f with respect to the orthonormal basis ${e_n: nin mathbb{N}}.$ extbf{Then by Parseval relation we have} $$sum_{n=0}^infty |<f,e_n>|^2 = ||f||^2 < infty $$ extbf{ Define} $$ f =sum_{n=0}^infty a_nz^n hspace{1cm} where hspace{1cm} a_n = <f,e_n>hspace{1cm} forallhspace{1cm} ngeq0$$ extbf{ Since} $$ sum_{n=0}^infty |a_n|^2 = sum_{n=0}^infty |<f,e_n>|^2 = ||f||^2 < infty $$ extbf{Therefore} $$fin H^2$$ extbf{That is for each $ ilde{f}in widehat{H^2}$ there exist $fin H^2$ such that $psi(f) = ilde{f}$ Therefore $psi$ is onto}medskip extbf{ That is $psi$ is an vector space isomorphism which also preserves the norm. Therefore $H^2$ can br identified as a subspace of the $L^2(mathbb{S^1})$ space} end{itemize} end{proof} section{ extbf{ Applications} } egin{enumerate} item In the mathematical rigrous formulation of Quantum Mechanics, developed by extbf{Joh Von Neumann}' the position and momentum states for a single non relavistic spin 0 Particle is the space of all the square integrable functions($L^2$). But $L^2$ have some undesirable properties and $H^2$ is much well behaved space so we work with $H^2$ instead of $L^2$. end{enumerate} ibliographystyle{plain} ibliography{my} end{document} In this paper , we discuss the Hardy Hilbert Space on the open disk with center origin and radius unity. We have proved that H2 Space is isomorphic to proper subspace of L2 Space which has various applications in Quantumm Mechanics.

Licence: creative commons attribution 4.0

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Creative Commons Attribution 4.0 and The Open Definition

 Keywords

Lebesgue , Parseval Identity , inner product space , separable

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Creative Commons Attribution 4.0 and The Open Definition