TRB PG Assistant - Riesz–Fischer theorem, Bessel's inequality, Parseval's theorem Online Test (MCQ Quiz)

1. What is the primary statement of the Riesz–Fischer theorem in the context of Fourier series?

A) Every convergent Fourier series represents a continuous function. B) If a function is square-integrable, its Fourier series converges to the function almost everywhere. C) A square-summable sequence corresponds to the Fourier coefficients of some L^2 function. D) Bessel's inequality holds for all periodic functions.

2. The Riesz–Fischer theorem establishes a connection between which two mathematical objects?

A) Polynomials and their roots B) Sequences of real numbers and continuous functions C) Square-summable sequences and L^2 functions D) Complex numbers and their magnitudes

3. In the context of Fourier series, what condition must a sequence of coefficients (c_n) satisfy to be the Fourier coefficients of an L^2 function?

A) The sequence must be absolutely convergent. B) The sum of the squares of the coefficients must be finite (square-summable). C) The sequence must be periodic. D) The sequence must consist of only integers.

4. Bessel's inequality, for a function f(x) with Fourier coefficients a_n and b_n, states that:

A) The sum of the squares of the coefficients is always equal to the integral of f(x)^2. B) The sum of the absolute values of the coefficients is finite. C) The sum of the squares of the Fourier coefficients is less than or equal to the mean square value of the function. D) The Fourier coefficients must decrease faster than 1/n.

5. For a function f(x) integrable over [-π, π], Bessel's inequality is mathematically expressed as:

A) a_0^2 / 2 + sum(a_n^2 + b_n^2) = (1/π) integral(f(x)^2 dx) B) a_0^2 / 2 + sum(a_n^2 + b_n^2) <= (1/π) integral(f(x)^2 dx) C) sum(|a_n| + |b_n|) <= (1/π) integral(|f(x)| dx) D) a_0^2 / 2 + sum(|a_n| + |b_n|) <= (1/π) integral(f(x)^2 dx)

6. What does Bessel's inequality imply about the Fourier coefficients of a square-integrable function?

A) They must all be zero. B) Their sum must converge to zero. C) They must tend to zero as n approaches infinity. D) They must be bounded by a constant.

7. Parseval's theorem is a generalization of which mathematical concept?

A) The Pythagorean theorem B) The Cauchy-Schwarz inequality C) The Fundamental Theorem of Calculus D) The Mean Value Theorem

8. Parseval's theorem relates the integral of the square of a function to the sum of the squares of its Fourier coefficients. What is the statement of Parseval's theorem for a function f(x) defined on [-π, π]?

A) a_0^2 / 2 + sum(a_n^2 + b_n^2) < (1/π) integral(f(x)^2 dx) B) a_0^2 / 2 + sum(a_n^2 + b_n^2) = (1/π) integral(f(x)^2 dx) C) sum(a_n^2 + b_n^2) = (1/π) integral(f(x)^2 dx) D) a_0^2 + sum(a_n^2 + b_n^2) = (1/π) integral(f(x)^2 dx)

9. What is the condition on the function f(x) for Parseval's theorem to hold in its standard form for Fourier series?

A) f(x) must be continuous. B) f(x) must be periodic with period 2π. C) f(x) must be square-integrable over the interval. D) f(x) must be differentiable.

10. If a function f(x) is square-integrable and its Fourier series converges to f(x) in the L^2 sense, then Parseval's theorem implies:

A) The integral of f(x)^2 is infinite. B) The sum of the squares of its Fourier coefficients diverges. C) The integral of f(x)^2 over the interval is equal to π times the sum of the squares of its coefficients. D) The function f(x) must be identically zero.

11. Consider a function f(x) = x on the interval [-π, π]. Its Fourier series coefficients are a_n = 0 for n >= 0 and b_n = 2(-1)^n / n for n >= 1. What does Parseval's theorem tell us about the integral of x^2?

A) integral(-π to π) x^2 dx = π * sum( (2(-1)^n / n)^2 ) B) integral(-π to π) x^2 dx = π * sum( (2(-1)^n / n) ) C) integral(-π to π) x^2 dx = sum( (2(-1)^n / n)^2 ) D) integral(-π to π) x^2 dx = π * (0^2 / 2 + sum( (2(-1)^n / n)^2 ))

12. The Riesz–Fischer theorem is crucial for proving the existence of the Fourier series for which class of functions?

A) Continuous functions only B) Differentiable functions only C) Square-integrable functions (L^2 functions) D) Analytic functions

13. Bessel's inequality provides an upper bound for the sum of squares of Fourier coefficients. What happens if this upper bound is finite and the equality holds in Bessel's inequality?

A) The function is not square-integrable. B) The Fourier series converges pointwise to the function. C) The Fourier series converges to the function in the L^2 sense (completeness). D) The function must be identically zero.

14. Which theorem is often referred to as the 'completeness' theorem for the Fourier series of L^2 functions?

A) Riesz–Fischer theorem B) Bessel's inequality C) Parseval's theorem D) Dirichlet's theorem

15. The Riesz–Fischer theorem guarantees that if we have a sequence of Fourier coefficients (a_n, b_n) such that sum(a_n^2 + b_n^2) is finite, then there exists a function f(x) in L^2 whose Fourier coefficients are precisely (a_n, b_n). This function f(x) is constructed using:

A) Pointwise summation of the Fourier series B) The integral of the function's square C) A specific method involving partial sums and limits in the L^2 norm D) Direct integration of the sequence

16. If f(x) is a periodic function with period 2π and f(x) belongs to L^2([-π, π]), Bessel's inequality states:

A) Integral(|f(x)|) < infinity B) Integral(f(x)^2) is finite C) a_0^2 / 2 + sum(a_n^2 + b_n^2) <= (1/π) Integral(f(x)^2 dx) D) a_0^2 / 2 + sum(a_n^2 + b_n^2) >= (1/π) Integral(f(x)^2 dx)

17. Parseval's theorem is fundamentally an energy conservation principle. In signal processing terms, it equates the total energy of a signal to:

A) The sum of the squares of its time-domain samples. B) The sum of the squares of its frequency-domain components (Fourier coefficients). C) The integral of its absolute value. D) The maximum amplitude of the signal.

18. For a function f(x) = 1 on [-π, π], the Fourier coefficients are a_0 = 2, and a_n = 0, b_n = 0 for n >= 1. Applying Parseval's theorem:

A) (2^2 / 2) + 0 = (1/π) integral(-π to π) 1^2 dx B) 0^2 / 2 + 0 = (1/π) integral(-π to π) 1^2 dx C) (2^2) + 0 = (1/π) integral(-π to π) 1^2 dx D) (2 / 2) + 0 = (1/π) integral(-π to π) 1^2 dx

19. The Riesz–Fischer theorem is essential for establishing the existence of a function corresponding to a given set of Fourier coefficients. This ensures that the space of square-integrable functions is 'complete' with respect to the L^2 norm. What is the definition of the L^2 norm squared for a function f(x) on [-π, π]?

A) Integral(|f(x)|) dx B) Integral(f(x)^2) dx C) (1/π) Integral(f(x)^2) dx D) Integral(|f(x)|^2) dx

20. Bessel's inequality is a necessary condition for a sequence to be Fourier coefficients, but not sufficient. What additional condition is needed for sufficiency?

A) The sequence must be periodic. B) The sequence must converge to zero. C) The sum of the squares of the sequence terms must be finite (satisfying the condition for Riesz-Fischer theorem). D) The sequence must consist of integers.

21. If a function f(x) is such that its Fourier series converges uniformly to f(x), then f(x) must be continuous. Does Parseval's theorem hold under uniform convergence?

A) No, Parseval's theorem only applies to L^2 convergence. B) Yes, uniform convergence implies L^2 convergence, so Parseval's theorem holds. C) No, Parseval's theorem requires the function to be discontinuous. D) Yes, but only if the function is constant.

22. The Riesz–Fischer theorem can be stated using complex Fourier coefficients c_n = (a_n - i*b_n)/2 for n != 0 and c_0 = a_0/2. The theorem states that a sequence {c_n} is the sequence of Fourier coefficients of some f in L^2 if and only if:

A) sum(|c_n|) is finite B) sum(c_n) is finite C) sum(|c_n|^2) is finite D) sum(c_n^2) is finite

23. Consider the Fourier series of a function f(x). Bessel's inequality states that the sum of the squares of the Fourier coefficients is bounded by the mean square value of the function. If the function is identically zero, what does Bessel's inequality imply?

A) The sum of squares of coefficients is positive. B) The sum of squares of coefficients is finite and less than or equal to zero. C) The sum of squares of coefficients is infinite. D) The sum of squares of coefficients is equal to the integral of zero, which is undefined.

24. Parseval's theorem is particularly useful for calculating sums of infinite series. For instance, if f(x) = x on [-π, π], Parseval's theorem leads to a value for:

A) sum(1/n^2) B) sum(1/n^3) C) sum(1/n^4) D) sum(1/n)

25. The Riesz–Fischer theorem connects the abstract Hilbert space L^2 with the sequence space l^2. What is the significance of this connection?

A) It shows that all functions are polynomials. B) It implies that Fourier series can always be computed explicitly. C) It confirms the completeness of the trigonometric system in L^2. D) It proves that all L^2 functions are continuous.

26. If a function f(x) is square-integrable, Bessel's inequality guarantees that the Fourier coefficients a_n and b_n satisfy:

A) a_n -> 0 and b_n -> 0 as n -> infinity B) sum(a_n) -> 0 and sum(b_n) -> 0 as n -> infinity C) a_n^2 + b_n^2 -> 0 as n -> infinity D) a_n^2 + b_n^2 is bounded

27. Parseval's theorem for Fourier series states that the average power of a periodic signal is equal to the sum of the average powers of its harmonic components. Mathematically, for f(x) in L^2([-π, π]):

A) (1/π) integral(f(x)^2 dx) = a_0^2 / 2 + sum(a_n^2 + b_n^2) B) (1/2π) integral(-π to π) f(x)^2 dx = a_0^2 / 2 + sum(a_n^2 + b_n^2) C) (1/π) integral(f(x)^2 dx) = sum(a_n^2 + b_n^2) D) (1/π) integral(f(x)^2 dx) = a_0^2 + sum(a_n^2 + b_n^2)

28. The Riesz–Fischer theorem is fundamental in Fourier analysis because it validates the process of associating a function with its Fourier coefficients. Which space of functions does it primarily deal with?

A) Space of continuous functions C[a, b] B) Space of differentiable functions C^1[a, b] C) Space of square-integrable functions L^2 D) Space of bounded functions

29. Bessel's inequality is often written as sum( |c_n|^2 ) <= ||f||_L^2^2, where c_n are complex Fourier coefficients. What is the relationship between the real coefficients (a_n, b_n) and complex coefficients (c_n)?

A) c_n = a_n + i*b_n B) c_n = (a_n + i*b_n)/2 for n>0, c_0 = a_0 C) c_n = (a_n - i*b_n)/2 for n!=0, c_0 = a_0/2 D) c_n = (a_n + i*b_n)/2 for all n

30. Parseval's theorem has a direct analogue in finite-dimensional vector spaces. If v is a vector and {u_i} is an orthonormal basis, Parseval's theorem corresponds to:

A) ||v||^2 = sum( |<v, u_i>|^2 ) B) ||v||^2 = sum( |<v, u_i>| ) C) ||v||^2 = sum( <v, u_i>^2 ) D) ||v||^2 = sum( |<v, u_i>|^3 )

31. The Riesz–Fischer theorem ensures that the mapping from an L^2 function to its sequence of Fourier coefficients is surjective onto the space of square-summable sequences. What does surjective mean in this context?

A) Every square-summable sequence corresponds to at least one L^2 function. B) Every square-summable sequence corresponds to exactly one L^2 function. C) Every square-summable sequence corresponds to some L^2 function. D) Every L^2 function maps to a square-summable sequence.

32. If Bessel's inequality becomes an equality for a function f(x), i.e., a_0^2 / 2 + sum(a_n^2 + b_n^2) = (1/π) integral(f(x)^2 dx), this implies:

A) The function f(x) is not in L^2. B) The Fourier series converges to f(x) in the L^2 sense. C) The function f(x) must be zero. D) The Fourier series converges pointwise everywhere.

33. Parseval's theorem relates the 'energy' of a function in the time (or spatial) domain to its energy in the frequency domain. For a periodic function f(x) with period 2π, the total energy is given by:

A) integral(-π to π) f(x) dx B) integral(-π to π) f(x)^2 dx C) (1/π) integral(-π to π) f(x)^2 dx D) π * integral(-π to π) f(x)^2 dx

34. The Riesz–Fischer theorem is a cornerstone for the theory of Fourier series because it guarantees the existence of the series representation for a wider class of functions than previously considered. Which theorem does it generalize?

A) Dirichlet's theorem on pointwise convergence B) Cauchy's theorem on convergence C) Parseval's theorem on L^2 convergence D) Weierstrass's approximation theorem

35. Bessel's inequality for a function f(x) in L^2([-π, π]) is: ||f||_L^2^2 >= (π/2)a_0^2 + π * sum_{n=1 to inf} (a_n^2 + b_n^2). What does ||f||_L^2^2 represent?

A) The integral of |f(x)| B) The integral of f(x)^2 C) The mean square value (1/π) integral(f(x)^2 dx) D) The maximum value of f(x)

36. If a function f(x) is such that its Fourier series converges to f(x) in the L^2 sense, then Parseval's theorem implies that the sum of the squares of the Fourier coefficients is finite. Conversely, if the sum of the squares of the Fourier coefficients is finite, does it imply L^2 convergence?

A) Yes, by definition of Parseval's theorem. B) No, it only implies Bessel's inequality. C) Yes, by the Riesz–Fischer theorem and Parseval's theorem. D) No, it implies pointwise convergence.

37. The Riesz–Fischer theorem is particularly important for proving the convergence properties of Fourier series. It establishes a bijection between L^2 functions and:

A) Convergent sequences B) Absolutely convergent series C) Square-summable sequences D) Periodic sequences

38. Consider a function f(x) that is square-integrable. Bessel's inequality states that the sum of the squares of its Fourier coefficients is bounded. What is the implication if this sum is finite?

A) The function must be discontinuous. B) The function belongs to L^2. C) The function is not periodic. D) The function is identically zero.

39. Parseval's theorem for Fourier series is a statement about the conservation of energy. If f(x) is a periodic function with period 2π, the theorem equates the average power of the signal to:

A) The integral of |f(x)| over one period. B) The sum of the squares of its complex Fourier coefficients, multiplied by 2π. C) The sum of the squares of its real Fourier coefficients (scaled appropriately). D) The integral of f(x)^2 over one period.

40. The Riesz–Fischer theorem can be viewed as asserting the completeness of the trigonometric system {1, cos(nx), sin(nx)} in which function space?

A) L^1([-π, π]) B) L^infinity([-π, π]) C) L^2([-π, π]) D) C([-π, π])

41. If f(x) is a square-integrable function, Bessel's inequality ensures that the sum of the squares of its Fourier coefficients converges. This convergence is essential for:

A) Proving pointwise convergence of the Fourier series B) Establishing the existence of the function from its coefficients (Riesz–Fischer) C) Showing that the function is continuous D) Calculating the integral of the function

42. Parseval's theorem provides a powerful tool for evaluating certain infinite sums. For a function f(x) with Fourier coefficients c_n, the theorem states ||f||^2 = 2π * sum(|c_n|^2). What is ||f||^2 in this context?

A) Integral(|f(x)|) dx B) Integral(f(x)^2) dx C) (1/2π) Integral(|f(x)|^2) dx D) Integral(|f(x)|^2) dx

43. The Riesz–Fischer theorem states that the set of Fourier coefficients of functions in L^2([-π, π]) is precisely the set of square-summable sequences. This implies that the Fourier transform is a(n):

A) Injective map from L^2 to l^2 B) Surjective map from L^2 to l^2 C) Isomorphic map from L^2 to l^2 D) Continuous map from L^2 to l^2

44. Bessel's inequality is a direct consequence of the orthogonality of the trigonometric system. If f(x) is a square-integrable function, the inequality is:

A) sum(a_n^2 + b_n^2) <= Integral(f(x)^2 dx) B) a_0^2 / 2 + sum(a_n^2 + b_n^2) <= (1/π) Integral(f(x)^2 dx) C) sum(|a_n| + |b_n|) <= Integral(|f(x)| dx) D) a_0^2 + sum(a_n^2 + b_n^2) <= Integral(f(x)^2 dx)

45. Parseval's theorem is sometimes called the 'completeness relation' for the Fourier system. What does 'completeness' mean in this context?

A) Every function can be represented by its Fourier series. B) The Fourier series converges uniformly for all functions. C) The trigonometric system is a basis for L^2 functions, meaning any L^2 function can be represented as a sum of basis functions. D) The Fourier coefficients are always zero.

46. The Riesz–Fischer theorem provides the theoretical foundation for understanding Fourier series as representations of functions in L^2. It guarantees that the space of L^2 functions is separable, meaning:

A) It contains only separable functions. B) It has a countable dense subset. C) It is not a Hilbert space. D) It is finite-dimensional.

47. If f(x) is a square-integrable function and sum(a_n^2 + b_n^2) is finite, Bessel's inequality is satisfied. The Riesz–Fischer theorem then guarantees the existence of an L^2 function whose Fourier coefficients are these a_n, b_n. What is the relationship between the integral of f(x)^2 and the sum of squares of coefficients?

A) Integral(f(x)^2 dx) = π * sum(a_n^2 + b_n^2) (assuming equality in Bessel's) B) Integral(f(x)^2 dx) = sum(a_n^2 + b_n^2) C) Integral(f(x)^2 dx) = (1/π) * sum(a_n^2 + b_n^2) D) Integral(f(x)^2 dx) = π * (a_0^2 / 2 + sum(a_n^2 + b_n^2))

48. Parseval's theorem is a statement about the norm of a function in L^2. Specifically, it relates the L^2 norm squared of f(x) to the sum of the squares of its Fourier coefficients. The theorem states that:

A) ||f||_L^2^2 = sum(|c_n|^2) B) ||f||_L^2^2 = (1/2π) sum(|c_n|^2) C) ||f||_L^2^2 = 2π * sum(|c_n|^2) D) ||f||_L^2^2 = π * sum(|c_n|^2)

Riesz–Fischer theorem, Bessel's inequality, Parseval's theorem - Online Test

Test Results