I know that vowels are nasalized before a nasal in the same syllable in English. I am wondering if this would include the schwa [ə] as well? For example, would the schwa in "restriction" [rɪstrɪkʃən] be nasalized?
The short answer is yes--schwa before a nasal in the same syllable tends to be nasalized.
The more nuanced answer is that nasalization in English is not really as straightforward as is sometimes taught in Intro to Linguistics classes. There are a couple of issues to bear in mind:
Perhaps precisely because nasalization is not contrastive in English, there is a lot of variation when it comes to the degree to which (and consistency with which) pre-(and post-)nasal vowels are nasalized in that language. If I take a rendition of 'restriction' in which the schwa is nasalized and splice in a non-nasalized schwa instead, native listeners will still hear it as 'restriction'; in fact, most listeners won't even notice a difference if all other acoustic properties of the schwa are appropriate.
Traditionally, phonologists characterized nasalization as a phonological (and therefore categorical) phenomenon, but phonetically the facts are more "messy". Articulatorily speaking, nasalization is not an all-or-nothing, binary thing (in any language). If one tracks nasal flow during speech, it becomes clear that it takes time for the velum to lower and raise, and especially in continuous speech it is common for it to be only partially lowered during the realization of segments abutting nasals. Abby Cohn showed this in her dissertation and later papers.
The above issues apply to all segments in the vicinity of nasals, but since schwa is by definition unstressed in English and therefore short in duration and not very loud (relative to all the segments around it) it is even more susceptible to variation and "non-committal" articulation. From a perceptual perspective it is reasonable to suppose that this is because listeners don't rely as much on the information they are given in unstressed syllables to parse speech.
The question has is a presupposition which may be false, depending on dialect, namely that there actually is a schwa before a tautosyllabic nasal. An alternative narrow transcription of "restriction" is [ɹʷəˈstɹʷɪkʃn̩], with a syllabic n. There is one trend of transcribing <ər, əl, ən, əm>, and a competing trend of transcribing those as <ɹ̩, l̩, m̩, n̩>. Although those syllable peaks may derive from the reduction of full vowel plus sonorant (tyrant vs. tyrannical), there is phonological evidence that schwa is eventually deleted. That evidence is a rule reducing t to glottal stop before syllabic and non-syllabic n, as in [ˈwɪʔnəs] "witness", [fæʔn̩] "fatten". Positing that there is a schwa between /t/ and /n/ needlessly obscures the debuccalization rule.
There may, of course, easily be dialects where there is unquestionably a schwa before those sonorants (i.e. there may be dialects where "prism" is [ˈpɹʷɪzəm]), and there just aren't any syllabic sonorants in that dialect. One may feel (and thus assert) that there clearly is a vowel in the first syllable of "contemporary", but what you may be observing is the release of the preceding consonant, Unless you consider the alternative that the first syllable is [kn̩] rather than [kən] and given a reason to reject [kn̩], you are stuck with an indeterminacy of analysis.
In other words, surface phonetic transcriptions are still abstractions, which need to be justified.
I agree that your example [fæʔn̩] "fatten" shows it is possible to have no vowel before [n] in the antecedent form [fæt'n̩], I don't see how this or the other examples show that surface phonetic transcriptions are abstractions. In the first syllable of "contemporary", either there is some time between when the [k] ends and the [n] begins during which the tongue does not contact the top of the mouth, or there isn't. It's a question of fact.– Greg LeeApr 25, 2015 at 23:51
@GregLee: even if there is a transitional period, is that transition necessarily a vowel? Is it the same phonetically as other transitions that can be transcribed with [ə], like in "butter" [bɐɾəɹ] or "barrel" [beɹəl]? If it's different, then the fact that it is transcribed the same is due to an abstraction, right? Apr 26, 2015 at 0:07
What sumelic said. The abstraction lies in how you categorize the acoustic output between the silence of /k/ and the 'steady state' of the sonorant -- as "release" (a property of the onset consonant), or as a vowel segment.– user6726Apr 26, 2015 at 0:19
I don't see the abstraction there. If it's a tiny little vowel period, you can characterize it as a very very short vowel, if you like. It's a matter of convenience. There is no reason to categorize it as vowel or non-vowel, other than convenience, is there? (This is like arguing against Bloomfield, who taught the structuralists never to make phonetic transcriptions, because they are inherently abstract, therefore not to be trusted.)– Greg LeeApr 26, 2015 at 1:16
@GregLee: apparently you have a very different understanding of the concept "abstract".– user6726Apr 26, 2015 at 1:39