What is the view of Islam within the Catholic Church? Ever since Islam arose what has been the view of Islam within the Catholic Church?

The desired answer will include papal and Church magisterium and pronouncements, the view of saints, and while not required, perhaps views of a few prestigious Catholic writers.

The detailed answer will also have a mention on Muhammad the founder, and the Qur'an.



Please note the answer is not the one that will have ALL the pronouncements ever made, but the one that is a succinct summary. For example if Pope A &amp; B say the same thing, summarize and state what the say in common and make a footnote saying all who said the same thing.
Muslims have an incorrect view of God, as they do not recognize the Trinity, nor the divinity of Christ. Muhammad was not a true prophet like Moses, Isaiah, etc. 

However, Muslims do worship the "same" God, and Catholics should treat Muslims with love and appreciate the good things they believe.



The most authoritative and well-recognized doctrine comes from the Catecism of the Catholic Church, 841 (Part 1, Section 2, Chapter 3, Article 9, Paragraph 3, III):


  The Church's relationship with the Muslims. "The plan of salvation also includes those who acknowledge the Creator, in the first place amongst whom are the Muslims; these profess to hold the faith of Abraham, and together with us they adore the one, merciful God, mankind's judge on the last day."


The  United States Conference of Catholic Bishops’
Committee for Ecumenical and Interreligious Affairs published in 2013 a detailed explanation Understanding Islam: A Guide for Catholic Educators. A few excerpts:


  Muslims and Christians share many
  common beliefs in their worship of a single Creator God who loves creation and who commands
  that His most cherished creations, human beings, love Him, one another, and His creation.
  
  Christians do not accord
  Muhammad the same status as the biblical prophets, but may regard him as a prophetic figure on
  such issues as charity and the protection of the poor, widows and orphans.


Pope Paul VI issued the Declaration on 
the Relation of the Church to Non-Christian Religions in 1965, which included references to Jews, Hindus, and Muslims.


  The Church regards with esteem also the Moslems. They adore the one God, living and subsisting in Himself; merciful and all- powerful, the Creator of heaven and earth,(5) who has spoken to men; they take pains to submit wholeheartedly to even His inscrutable decrees, just as Abraham, with whom the faith of Islam takes pleasure in linking itself, submitted to God. Though they do not acknowledge Jesus as God, they revere Him as a prophet. They also honor Mary, His virgin Mother; at times they even call on her with devotion. In addition, they await the day of judgment when God will render their deserts to all those who have been raised up from the dead. Finally, they value the moral life and worship God especially through prayer, almsgiving and fasting.
  
  It is...the burden of the Church's preaching to proclaim the cross of Christ as the sign of God's all-embracing love and as the fountain from which every grace flows.
  
  The Church reproves, as foreign to the mind of Christ, any discrimination against men or harassment of them because of their race, color, condition of life, or religion.


More recently, Pope Francis said in Evangelii Gaudium 252-3. 


  The sacred writings of Islam have retained some Christian teachings; Jesus and Mary receive profound veneration and it is admirable to see how Muslims both young and old, men and women, make time for daily prayer and faithfully take part in religious services. Many of them also have a deep conviction that their life, in its entirety, is from God and for God. They also acknowledge the need to respond to God with an ethical commitment and with mercy towards those most in need.




Your question is rather broad, but I think I have covered the most important points, with authoritative sources.