WASHINGTON — Images of angry mobs in Arab cities burning American flags and attacking U.S. diplomatic posts suggest the Muslim world is no less enraged at the United States than when President George W. Bush had to duck shoes hurled at him in Baghdad.

But more than three years after President Barack Obama declared in Cairo that he would seek “a new beginning” in U.S.-Muslim relations, a closer look reveals strides as well as setbacks.

One U.S.-led war is over and another is receding, although there are questions about whether America has made lasting gains in Afghanistan. The Arab Spring revolution, a spontaneous combustion that happened independent of Western influence, has given people new power and hope as well as democratic elections the U.S. supports.

But peace between Israel and the Palestinians is nowhere in sight, Iran is seen as a menace and broad mistrust with America is still deep and explosive across much of the Muslim world.

As nations across North Africa and the Middle East move chaotically toward democracy, they and Washington have settled into a wary, redefined relationship. Obama is not ready to call Mohammad Morsi, the popularly elected Egyptian president, an ally, and the democratically elected Iraqi president, Nouri al-Maliki, has dismissed U.S. demands that he stop Iran from using Iraqi airspace to fly weapons to Syria for use against anti-government rebels.

Such is the complicated progress report that Obama carries toward the United Nations General Assembly next week, his final moment on a world stage before the U.S. election on Nov. 6. For that election, Pew Research Center polling shows Obama has a clear edge over Republican Mitt Romney in handling foreign policy in general and problems in the Middle East specifically.

Across the world his standing remains markedly lower in predominantly Muslim nations. However, Leila Hilal, a Mideast expert at the New America Foundation, said Obama may have made more progress toward improving relations than critics say.

“Obama inherited a very damaged U.S. credibility in the region,” she said, and so it would be unrealistic to think that his “new beginning” would take hold fast.

“There’s only so much one president can do, given the history” of perceived insults by the U.S., she said. Those range all the way from the American invasion of Iraq to, more recently, the privately made anti-Islam video that ridicules the prophet Muhammad and triggered major protests across the Muslim world.

The question of the Obama administration’s relationship with that Muslim world came under new election year scrutiny when four Americans, including Ambassador Chris Stevens, were killed in a Sept. 11 attack on the U.S. Consulate in Benghazi, Libya.

Obama found himself eulogizing the dead, pledging that the work of U.S. diplomacy would go on undaunted — and prodding his Muslim partners to accept responsibilities.

“As they emerge into new forms of government, part of what they’re going to have to do is to recognize that democracy is not just casting a ballot,” Obama said this week. “It’s respecting freedom of speech and tolerating people with different points of view.”

Obama’s critics say he misunderstands the nature of the threat to moderation in the Mideast. Sen. John McCain, R-Ariz., said the White House is demonstrating this by overstating the role of the anti-Islam video in igniting the violence that killed Stevens in Benghazi.

“It has nothing to do with videos. It has everything to do with Islamists trying to hijack these revolutions in places like Libya,” McCain, Obama’s 2008 challenger, said Wednesday. “And it shows the abysmal ignorance of this administration of what’s really going on in the Middle East.”

Abdeslam Maghraoui, the director of undergraduate studies in Duke University’s political science department, says the protests that have erupted in Egypt, Libya, Tunisia, Yemen and in other Arab countries had more to do with local conditions than with U.S. policies. “The current anti-American backlash in the region is the byproduct of genuine misunderstanding, real ignorance and political jockeying among Islamic groups,” he said.

Obama warned from the start that it would be a long slog.

In his Cairo speech on June 4, 2009, Obama noted that it was a “time of tension” between the U.S. and Muslims around the world — “tension rooted in historical forces that go beyond any current policy debate.”

At the time, Egyptians had not yet ousted their authoritarian leader, Hosni Mubarak, a decades-long U.S. ally, and popular rebellions had not yet sprung up across the region.

“I have come here to seek a new beginning between the United States and Muslims around the world, one based upon mutual interest and mutual respect,” Obama declared.

Assessing such an enormous promise is hard to quantify.

“It’s vital to keep in mind that how Obama is perceived by the average person in Egypt or Iraq or Pakistan is not going to be the same as the way he’s perceived by the diplomats or the opposition party,” said Kecia Ali, an Islamic studies expert at Boston University. “To assume that there is a Muslim world view or a Middle Eastern view or even an Egyptian view of Obama makes no sense at all. There’s not even an American view of Obama.”

Then how about actions and results?

He has been unable to gather an agreed international response to Syria, where an Arab Spring revolt has devolved into a civil war that has killed 23,000 people, and the U.S. is unwilling to go it alone there. Without lethal aid from the West, the Syrian rebels have begun to accept arms and other assistance from more extreme factions, possibly including terror groups. That leaves open the possibility that if the rebels succeed in ousting President Bashar Assad, the country could be run by factions sympathetic to extremists.

On other big issues that help define U.S.-Muslim relations — Iran, the stalled Israeli-Palestinian peace talks and the Arab Spring — the president has seen a combination of setback, stalemate and frustration.

Iran stands out as perhaps the most clear-cut failure. Early in his presidency Obama offered an open hand to Iran’s leaders, hoping to negotiate limits on their nuclear program. He said in June 2009 that the nuclear standoff had reached a “decisive point,” and that what was at stake was preventing a nuclear arms race in the Mideast.

But the Iranians gave him the cold shoulder, and after a series of inconclusive attempts at negotiations, they are thought to be progressing toward a nuclear weapons capability. As he nears the end of his term, Obama has little to show for his Iranian outreach beyond a strengthening of international sanctions and a chilled relationship with Israeli Prime Minister Benjamin Netanyahu. The Israeli has complained publicly about U.S. inaction and has given Romney a warm welcome in his country.

Obama did, as promised, reduce the U.S. military’s presence in Muslim countries by removing all troops from Iraq and beginning to wind down the war in Afghanistan. But relations with Pakistan are arguably worse.

Obama priorities have been not just to mend relations with the broader Muslim world but also to sharpen the focus of U.S. policy toward defeating al-Qaida through the use of less blunt instruments of military power. And in joining NATO allies and the Arab League to get rid of Libyan strongman Moammar Gadhafi, Obama succeeded without committing U.S. ground troops. But there are limits to the power of a U.S. president to shape relations with Muslim nations, even longstanding allies.

Steven A. Cook, a Mideast expert at the Council on Foreign Relations, says Washington has long tended to make political demands of Egypt and other Arab countries that they cannot reasonably be expected to meet.

“Americans consistently fail to recognize,” he recently wrote, “that Arabs have their own politics and have the ability to calculate their own interests independently of what Washington demands.”