HW8_2012

# HW8_2012 - tion of a vibrating string T x y x λρ x y x =...

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Math 602, 2012 Homework 8 Please note the following very useful inequalities: | f ( t ) | ≤ k f k for all t S if k f k = sup x S | f ( x ) | and ± ± Z b a f ( t ) dt ± ± Z b a ± ± f ( t ) ± ± dt 1. Consider the sequence of functions f n ( x ) = x n . a) Show that the sequence f n is point-wise convergent for all x [0 , 1], that is, show that for all x [0 , 1] the limit lim n →∞ f n ( x ) = f ( x ) exists; what is the limit function f ( x )? b) The functions f n belong to the space C [0 , 1] of continuous functions on [0 , 1]. Do they converge in the sup norm in this space? c) The functions f n belong to the space L 2 [0 , 1]. Do they converge in the L 2 norm in this space? 2. Consider the integral operator I : C [0 , 2] C [0 , 2] given by I ( f ) = Z x 0 f ( t ) dt a) Show that I is a linear transformation. b) Show that I is a bounded operator, that is, there exists a number B so that k I ( f ) k B k f k for all f C [0 , 2] and ﬁnd the (smallest) number B . 3. Consider the eigenvalue problem (separation of variables in the equa-
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Unformatted text preview: tion of a vibrating string) ( T ( x ) y ( x )) + λρ ( x ) y ( x ) = 0 , x ∈ [0 ,L ] , where T ( x ) ,ρ ( x ) > with boundary conditions y (0) = 0 , my ( L ) + k y ( L ) = 0 (the endpoint x = 0 is kept ﬁxed, and the other end, x = L is accelerated up and down with no transversal force). Reformulate this problem using a self-adjoint diﬀerential operator on an appropriate Hilbert space: give the operator, its domain, and show that it is self-adjoint. Is this operator positive deﬁnite? Without explicitly solving the problems, what can you say about the eigenvalues and eigenfunctions of this problem? 1...
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