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Periodic functions: relationship Relationship between f(t,x) as t and f(t/ϵ,x/ϵ2) as ϵ0 (periodic functions)

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Limits for periodic Periodic functions: relationship between f(t,x) as t and f(t/ϵ,x/ϵ2) as ϵ0

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Let f:(0,)×RR be 1-periodic in the second variable and in L((0,)×R). If it is necessary, we can also assume f to be continuous.

  1. Suppose that f(t,x)aR in L on compact sets as t. Do we have that f(t/ϵ,x/ϵ2)a in L on compact sets as ϵ0?

  2. Suppose that f(t/ϵ,x/ϵ2)aR as ϵ0 in L on compact sets. Do we have that f(t,x)a in L on compact sets as t?

  3. If 1. and 2. are not true, is there a reasonable set of assumptions that make the statements true?

Let f:(0,)×RR be 1-periodic in the second variable and in L((0,)×R).

  1. Suppose that f(t,x)aR in L on compact sets as t. Do we have that f(t/ϵ,x/ϵ2)a in L on compact sets as ϵ0?

  2. Suppose that f(t/ϵ,x/ϵ2)aR as ϵ0 in L on compact sets. Do we have that f(t,x)a in L on compact sets as t?

  3. If 1. and 2. are not true, is there a reasonable set of assumptions that make the statements true?

Let f:(0,)×RR be 1-periodic in the second variable and in L((0,)×R). If it is necessary, we can also assume f to be continuous.

  1. Suppose that f(t,x)aR in L on compact sets as t. Do we have that f(t/ϵ,x/ϵ2)a in L on compact sets as ϵ0?

  2. Suppose that f(t/ϵ,x/ϵ2)aR as ϵ0 in L on compact sets. Do we have that f(t,x)a in L on compact sets as t?

  3. If 1. and 2. are not true, is there a reasonable set of assumptions that make the statements true?

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