INEQUALITIES FOR ORTHORNORMAL FAMILIES OF VECTORS IN INNER PRODUCT SPACES RELATED TO BUZANO’S, RICHARD’S AND KUREPA’S RESULTS

S.S. DRAGOMIR

Abstract. Some inequalities for families of orthornormal vectors in inner product spaces that are related with Buzano’s, Richard’s and Kurepa’s results are given.

1. Introduction

In [3], M.L. Buzano obtained the following extension of the celebrated Schwarz’s inequality in a real or complex inner product space (H; ⟨·, ·⟩) :

(1.1)  |⟨a, x⟩ ⟨x, b⟩| ≤ \frac{1}{2} [∥a∥ ∥b∥ + |⟨a, b⟩|] ∥x∥^{2} ,

for any a, b, x ∈ H.

It is clear that the above inequality becomes, for a = b, the Schwarz’s inequality

(1.2)  |⟨a, x⟩|^{2} ≤ ∥a∥^{2} ∥x∥^{2} , a, x ∈ H;

in which the equality holds if and only if there exists a scalar λ ∈ 𝕂 (ℝ, ℂ) so that x = λa.

As noted by M. Fujii and I. Kubo in [5], where they provided a simple proof of (1.1) on using orthogonal projection arguments, the case of equality holds in (1.1) if

(1.3)  x = \{ \begin{matrix}α (\frac{a}{∥a∥} + \frac{⟨a,b⟩}{|⟨a,b⟩|} · \frac{b}{∥b∥}) , & when ⟨a, b⟩ ≠ 0 \\ α (\frac{a}{∥a∥} + β · \frac{b}{∥b∥}) , & when ⟨a, b⟩ = 0, \end{matrix}

where α, β ∈ 𝕂.

As noted by T. Precupanu in [8], independently of Buzano, U. Richard [9] obtained the following similar inequality holding in real inner product spaces:

(1.4)  \frac{1}{2} ∥a∥^{2} [⟨a, b⟩ − ∥a∥ ∥b∥] ≤ ⟨a, x⟩ ⟨x, b⟩ ≤ \frac{1}{2} ∥x∥^{2} [⟨a, b⟩ − ∥a∥ ∥b∥] .

In [7], J. Pečarić gave a simple direct proof of (1.4) without mentioning the work of either Buzano of Richard, but tracked down the result, in a particular form, to an earlier paper of C. Blatter [1].

Date: November 10, 2004.

1991 Mathematics Subject Classification. 46C05, 46C09, 26D15.

Key words and phrases. Schwarz’s inequality, Buzano’s inequality, Richard’s inequality, Kurepa’s inequality, Inner products, Integral inequalities.

In [4], the author has improved Buzano’s inequality by showing that, in fact, one has:

(1.5)  |⟨a, x⟩ ⟨x, b⟩| ≤ |⟨a, x⟩ ⟨x, b⟩ − \frac{1}{2} ⟨a, b⟩ ∥x∥^{2}| + \frac{1}{2} |⟨a, b⟩| ∥x∥^{2}

≤ \frac{1}{2} [∥a∥ ∥b∥ + |⟨a, b⟩|] ∥x∥^{2} .

In the same paper, the author has also improved the celebrated Kurepa inequality for the complexification of a real inner product space, namely, the inequality [6]

(1.6)  |⟨z, x⟩_{ℂ}|^{2} ≤ \frac{1}{2} ∥x∥^{2} [∥z∥_{ℂ}^{2} + |⟨z, z̄⟩_{ℂ}|] ≤ ∥x∥^{2} ∥z∥_{ℂ}^{2} ,

where x ∈ H, (H; ⟨·, ·⟩) is a real space, z ∈ H_{ℂ}, (H_{ℂ}, ⟨·, ·⟩_{ℂ}) is the complexification of (H; ⟨·, ·⟩) , and z̄ is the conjugate vector of z.

The refinement of Kurepa’s result (1.6) obtained in [4] is as follows:

(1.7)  |⟨z, x⟩_{ℂ}|^{2} ≤ |⟨z, x⟩_{ℂ}^{2} − \frac{1}{2} ⟨z, z̄⟩_{ℂ} ∥x∥^{2}| + \frac{1}{2} |⟨z, z̄⟩_{ℂ}| ∥x∥^{2}

≤ \frac{1}{2} ∥x∥^{2} [∥z∥_{ℂ}^{2} + |⟨z, z̄⟩_{ℂ}|]

≤ ∥x∥^{2} ∥z∥_{ℂ}^{2} ,

with the same assumptions as above.

The main aim of the present paper is to obtain similar results for families of orthonormal vectors in (H; ⟨·, ·⟩) , real or complex space, that are naturally connected with the celebrated Bessel inequality.

2. A Generalisation for Orthonormal Families

We say that the finite family {e_{i}}_{i∈I} (I is finite) of vectors is orthonormal if ⟨e_{i}, e_{j}⟩ = 0 if i, j ∈ I with i ≠ j and ∥e_{i}∥ = 1 for each i ∈ I. The following result may be stated:

Theorem 1. Let (H; ⟨·, ·⟩) be an inner product space over the real or complex number field 𝕂 and {e_{i}}_{i∈I} a finite orthonormal family in H. Then for any a, b ∈ H, one has the inequality:

(2.1)  |∑_{i∈I} ⟨a, e_{i}⟩ ⟨e_{i}, b⟩ − \frac{1}{2} ⟨a, b⟩| ≤ \frac{1}{2} ∥a∥ ∥b∥ .

The case of equality holds in (2.1) if and only if

(2.2)  ∑_{i∈I} ⟨a, e_{i}⟩ e_{i} = \frac{1}{2}a + (∑_{i∈I} ⟨a, e_{i}⟩ ⟨e_{i}, b⟩ − \frac{1}{2} ⟨a, b⟩) · \frac{b}{∥b∥^{2}} .

Proof. It is well known that, for e ≠ 0 and f ∈ H, the following identity holds:

(2.3)  \frac{∥f∥^{2} ∥e∥^{2} − |⟨f, e⟩|^{2}}{∥e∥^{2}} = ∥f − \frac{⟨f, e⟩ e}{∥e∥^{2}} ∥^{2} .

Therefore, in Schwarz’s inequality

(2.4)  |⟨f, e⟩|^{2} ≤ ∥f∥^{2} ∥e∥^{2} , f, e ∈ H;

the case of equality, for e ≠ 0, holds if and only if

f = \frac{⟨f, e⟩ e}{∥e∥^{2}}.

Let f := 2 ∑_{i∈I} ⟨a, e_{i}⟩ e_{i} − a and e := b. Then, by Schwarz’s inequality (2.4), we may state that

(2.5)  |⟨2 ∑_{i∈I} ⟨a, e_{i}⟩ e_{i} − a, b⟩|^{2} ≤ ∥2 ∑_{i∈I} ⟨a, e_{i}⟩ e_{i} − a∥^{2} ∥b∥^{2}

with equality, for b ≠ 0, if and only if

(2.6)  2 ∑_{i∈I} ⟨a, e_{i}⟩ e_{i} − a = ⟨2 ∑_{i∈I} ⟨a, e_{i}⟩ e_{i} − a, b⟩ \frac{b}{∥b∥^{2}} .

Since

⟨2 ∑_{i∈I} ⟨a, e_{i}⟩ e_{i} − a, b⟩ = 2 ∑_{i∈I} ⟨a, e_{i}⟩ ⟨e_{i}, b⟩ − ⟨a, b⟩

and

∥2 ∑_{i∈I} ⟨a, e_{i}⟩ e_{i} − a∥^{2} = 4 ∥∑_{i∈I} ⟨a, e_{i}⟩ e_{i}∥^{2} − 4 Re ⟨∑_{i∈I} ⟨a, e_{i}⟩ e_{i}, a⟩ + ∥a∥^{2}

= 4 ∑_{i∈I} |⟨a, e_{i}⟩|^{2} − 4 ∑_{i∈I} |⟨a, e_{i}⟩|^{2} + ∥a∥^{2}

= ∥a∥^{2} ,

hence by (2.5) we deduce the desired inequality (2.1).

Finally, as (2.2) is equivalent to

∑_{i∈I} ⟨a, e_{i}⟩ e_{i} − \frac{a}{2} = (∑_{i∈I} ⟨a, e_{i}⟩ ⟨e_{i}, b⟩ − \frac{1}{2} ⟨a, b⟩) \frac{b}{∥b∥^{2}} ,

hence the equality holds in (2.1) if and only if (2.2) is valid.

The following result is well known in the literature as Bessel’s inequality

(2.7)  ∑_{i∈I} |⟨x, e_{i}⟩|^{2} ≤ ∥x∥^{2} , x ∈ H,

where, as above, {e_{i}}_{i∈I} is a finite orthonormal family in the inner product space (H; ⟨·, ·⟩) .

If one chooses a = b = x in (2.1), then one gets the inequality

|∑_{i∈I} |⟨x, e_{i}⟩|^{2} − \frac{1}{2} ∥x∥^{2}| ≤ \frac{1}{2} ∥x∥^{2} ,

which is obviously equivalent to Bessel’s inequality (2.7). Therefore, the inequality (2.1) may be regarded as a generalisation of Bessel’s inequality as well.

Utilising the Bessel and Cauchy-Bunyakovsky-Schwarz inequalities, one may state that

(2.8)  |∑_{i∈I} ⟨a, e_{i}⟩ ⟨e_{i}, b⟩| ≤ [∑_{i∈I} |⟨a, e_{i}⟩|^{2} ∑_{i∈I} |⟨b, e_{i}⟩|^{2}]^{\frac{1}{2}} ≤ ∥a∥ ∥b∥

A different refinement of the inequality between the first and the last term in (2.8) is incorporated in the following:

Corollary 1. With the assumption of Theorem 1, we have

(2.9)  |∑_{i∈I} ⟨a, e_{i}⟩ ⟨e_{i}, b⟩| ≤ |∑_{i∈I} ⟨a, e_{i}⟩ ⟨e_{i}, b⟩ − \frac{1}{2} ⟨a, b⟩| + \frac{1}{2} |⟨a, b⟩|

≤ \frac{1}{2} [∥a∥ ∥b∥ + |⟨a, b⟩|]

≤ ∥a∥ ∥b∥ .

Remark 1. If the space (H; ⟨·, ·⟩) is real, then, obviously, (2.1) is equivalent to:

(2.10)  \frac{1}{2} (⟨a, b⟩ − ∥a∥ ∥b∥) ≤ ∑_{i∈I} ⟨a, e_{i}⟩ ⟨e_{i}, b⟩ ≤ \frac{1}{2} [∥a∥ ∥b∥ + ⟨a, b⟩] .

Remark 2. It is obvious that if the family comprises of only a single element e = \frac{x}{∥x∥}, x ∈ H, x ≠ 0, then from (2.9) we recapture the refinement of Buzano’s inequality incorporated in (1.5) while from (2.10) we deduce Richard’s result from (1.4).

The following corollary of Theorem 1 is of interest as well:

Corollary 2. Let {e_{i}}_{i∈I} be a finite orthonormal family in (H; ⟨·, ·⟩) . If x, y ∈ H\ {0} are such that there exists the constants m_{i,}n_{i}, M_{i}, N_{i} ∈ ℝ, i ∈ I such that:

(2.11)  −1 ≤ m_{i} ≤ \frac{Re ⟨x, e_{i}⟩}{∥x∥} · \frac{Re ⟨y, e_{i}⟩}{∥y∥} ≤ M_{i} ≤ 1, i ∈ I

and

(2.12)  −1 ≤ n_{i} ≤ \frac{Im ⟨x, e_{i}⟩}{∥x∥} · \frac{Im ⟨y, e_{i}⟩}{∥y∥} ≤ N_{i} ≤ 1, i ∈ I

then

(2.13)  2 ∑_{i∈I} (m_{i} + n_{i}) − 1 ≤ \begin{matrix}Re & ⟨x, y⟩ \\ ∥x∥ & ∥y∥ \end{matrix} ≤ 1 + 2 ∑_{i∈I} (M_{i} + N_{i}) .

Proof. Using Theorem 1 and the fact that for any complex number z, |z| ≥ |Re z| , we have

(2.14)  |∑_{i∈I} Re [⟨x, e_{i}⟩ ⟨e_{i}, y⟩] − \frac{1}{2} Re ⟨x, y⟩| ≤ |∑_{i∈I} ⟨x, e_{i}⟩ ⟨e_{i}, y⟩ − \frac{1}{2} ⟨x, y⟩|

≤ \frac{1}{2} ∥x∥ ∥y∥ .

Since

Re [⟨x, e_{i}⟩ ⟨e_{i}, y⟩] = Re ⟨x, e_{i}⟩ Re ⟨y, e_{i}⟩ + Im ⟨x, e_{i}⟩ Im ⟨y, e_{i}⟩ ,

hence by (2.14) we have:

− \frac{1}{2} ∥x∥ ∥y∥ + \frac{1}{2} Re ⟨x, y⟩ (2.15)

≤ ∑_{i∈I} Re ⟨x, e_{i}⟩ Re ⟨y, e_{i}⟩ + ∑_{i∈I} Im ⟨x, e_{i}⟩ Im ⟨y, e_{i}⟩

≤ \frac{1}{2} ∥x∥ ∥y∥ + \frac{1}{2} Re ⟨x, y⟩ .

Utilising the assumptions (2.11) and (2.12), we have

(2.16)  ∑_{i∈I} m_{i} ≤ ∑_{i∈I} \begin{aligned}Re & ⟨x, e_{i}⟩ & Re & ⟨y, e_{i}⟩ \\ & ∥x∥ & ∥y∥ \end{aligned} ≤ ∑_{i∈I} M_{i}

and

(2.17)  ∑_{i∈I} n_{i} ≤ ∑_{i∈I} \begin{aligned}Im & ⟨x, e_{i}⟩ & Im & ⟨y, e_{i}⟩ \\ & ∥x∥ & ∥y∥ \end{aligned} ≤ ∑_{i∈I} N_{i}.

Finally, on making use of (2.15) – (2.17), we deduce the desired result (2.13).

Remark 3. By Schwarz’s inequality, is it obvious that, in general,

−1 ≤ \begin{matrix}Re & ⟨x, y⟩ \\ ∥x∥ & ∥y∥ \end{matrix} ≤ 1.

Consequently, the left inequality in (2.13) is of interest if ∑_{i∈I} (m_{i} + n_{i}) > 0, while the right inequality in (2.13) is of interest if ∑_{i∈I} (M_{i} + N_{i}) < 0.

3. Refinements of Kurepa’s Inequality

Let (H; ⟨·, ·⟩) be a real inner product space generating the norm ∥·∥ . The complexification H_{ℂ} of H is defined as a complex linear space H × H of all ordered pairs (x, y) , x, y ∈ H endowed with the operations:

(x, y) + (x′, y′) := (x + x′, y + y′) ,  x, x′, y, y′ ∈ H;
(σ + iτ) · (x, y) := (σx − τy, τx + σy) ,  x, y ∈ H and σ, τ ∈ ℝ.

On H_{ℂ} := H × H, endowed with the above operations, one can now canonically define the scalar product ⟨·, ·⟩_{ℂ} by:

(3.1)  ⟨z, z′⟩_{ℂ} := ⟨x, x′⟩ + ⟨y, y′⟩ + i [⟨x′, y⟩ − ⟨x, y′⟩]

where z = (x, y) , z′ = (x′, y′) ∈ H_{ℂ}. Obviously,

∥z∥_{ℂ}^{2} = ∥x∥^{2} + ∥y∥^{2} , z = (x, y) ∈ H_{ℂ}.

One can also define the conjugate of a vector z = (x, y) by z̄:= (x, −y) . It is easy to see that, the elements of H_{ℂ}, under defined operations, behave as formal “complex” combinations x+iy with x, y ∈ H. Because of this, we may write z = x+iy instead of z = (x, y) . Thus, z̄ = x − iy. Under this setting, S. Kurepa [6] proved the following refinement of Schwarz’s inequality:

(3.2)  |⟨a, z⟩_{ℂ}|^{2} ≤ \frac{1}{2} ∥a∥^{2} [∥z∥_{ℂ}^{2} + |⟨z, z̄⟩_{ℂ}|] ≤ ∥a∥^{2} ∥z∥_{ℂ}^{2} ,

for any a ∈ H and z ∈ H_{ℂ}.

This was motivated by generalising the de Bruijn result for sequences of real and complex numbers obtained in [2].

The following result holds.

Theorem 2. Let {e_{j}}_{j∈I} be a finite orthonormal family in the real inner product space (H; ⟨·, ·⟩) . Then for any w ∈ H_{ℂ}, where (H_{ℂ}; ⟨·, ·⟩_{ℂ}) is the complexification of (H; ⟨·, ·⟩) , one has the following Bessel’s type inequality:

(3.3)  |∑_{j∈I} ⟨w, e_{j}⟩^{2}| ≤ |∑_{j∈I} ⟨w, e_{j}⟩_{ℂ}^{2} − \frac{1}{2} ⟨w, w̄⟩_{ℂ}| + \frac{1}{2} |⟨w, w̄⟩_{ℂ}|

≤ \frac{1}{2} [∥w∥_{ℂ}^{2} + |⟨w, w̄⟩_{ℂ}|] ≤ ∥w∥_{ℂ}^{2} .

Proof. Define f_{j} ∈ H_{ℂ}, f_{j} := (e_{j}, 0) , j ∈ I. For any k, j ∈ I we have

⟨f_{i}, f_{j}⟩_{ℂ} = ⟨(e_{k}, 0) , (e_{j}, 0)⟩_{ℂ} = ⟨e_{k}, e_{j}⟩ = δ_{kj},

therefore {f_{j}}_{j∈I} is an orthonormal family in (H_{ℂ}; ⟨·, ·⟩_{ℂ}) .

If we apply Theorem 1 for (H_{ℂ}; ⟨·, ·⟩_{ℂ}) , a = w, b = w̄, we may write:

(3.4)  |∑_{j∈I} ⟨w, e_{j}⟩_{ℂ} ⟨e_{j}, w̄⟩_{ℂ} − \frac{1}{2} ⟨w, w̄⟩_{ℂ}| ≤ \frac{1}{2} ∥w∥_{ℂ} \left∥{ \atop w̄}\right∥_{ℂ} .

However, for w := (x, y) ∈ H_{ℂ}, we have w̄ = (x, −y) and

⟨e_{j}, w̄⟩_{ℂ} = ⟨(e_{j}, 0) , (x, −y)⟩_{ℂ} = ⟨e_{j}, x⟩ − i ⟨e_{j}, −y⟩ = ⟨e_{j}, x⟩ + i ⟨e_{j}, y⟩

and

⟨w, e_{j}⟩_{ℂ} = ⟨(x, y) , (e_{j}, 0)⟩_{ℂ} = ⟨e_{j}, x⟩ − i ⟨e_{j}, −y⟩ = ⟨x, e_{j}⟩ + i ⟨e_{j}, y⟩

for any j ∈ I. Thus ⟨e_{j}, w̄⟩ = ⟨w, e_{j}⟩ for each j ∈ I and since

∥w∥_{ℂ} = \left∥{ \atop w̄}\right∥_{ℂ} = (∥x∥^{2} + ∥y∥^{2})^{\frac{1}{2}} ,

we get from (3.4) that

(3.5)  |∑_{j∈I} ⟨w, e_{j}⟩_{ℂ}^{2} − \frac{1}{2} ⟨w, w̄⟩_{ℂ}| ≤ \frac{1}{2} ∥w∥_{ℂ}^{2} .

Now, observe that the first inequality in (3.3) follows by the triangle inequality, the second is an obvious consequence of (3.5) and the last one is derived from Schwarz’s result.

Remark 4. If the family \{e_{j}\}_{j∈I} contains only a single element e = \frac{x}{∥x∥}, x ∈ H, x ≠ 0, then from (3.3) we deduce (1.7), which, in its turn, provides a refinement of Kurepa’s inequality (3.2).

4. An Application for L_{2} [−π, π]

It is well known that in the Hilbert space L_{2} [−π, π] of all functions f : [−π, π] → ℂ with the property that f is Lebesgue measurable on [−π, π] and ∫_{−π}^{π} |f (t)|^{2} dt < ∞, the set of functions

\{\frac{1}{\sqrt{2π}} , \frac{1}{\sqrt{π}} cos t, \frac{1}{\sqrt{π}} sin t, …, \frac{1}{\sqrt{π}} cos nt, \frac{1}{\sqrt{π}} sin nt, … \}

is orthonormal.

If by trig t, we denote either sin t or cos t, t ∈ [−π, π] , then on using the results from Sections 2 and 3, we may state the following inequality:

(4.1) |\frac{1}{π} ∑_{k=1}^{n} ∫_{−π}^{π} f (t) trig (kt) dt · ∫_{−π}^{π} \overline{g (t)} trig (kt) dt − \frac{1}{2} ∫_{−π}^{π} f (t) \overline{g (t)}dt|^{2}

≤ \frac{1}{4} ∫_{−π}^{π} |f (t)|^{2} dt ∫_{−π}^{π} |g (t)|^{2} dt,

where all trig (kt) is either sin kt or cos kt, k ∈ {1, …, n} and f ∈ L_{2} [−π, π] . This follows by Theorem 1. If one uses Corollary 1, then one can state the following chain of inequalities

(4.2) |\frac{1}{π} ∑_{k=1}^{n} ∫_{−π}^{π} f (t) trig (kt) dt · ∫_{−π}^{π} \overline{g (t)} trig (kt) dt|

≤ |\frac{1}{π} ∑_{k=1}^{n} ∫_{−π}^{π} f (t) trig (kt) dt · ∫_{−π}^{π} \overline{g (t)} trig (kt) dt − \frac{1}{2} ∫_{−π}^{π} f (t) \overline{g (t)}dt|

+ \frac{1}{2} |∫_{−π}^{π} f (t) \overline{g (t)}dt|

≤ \frac{1}{2} [(∫_{−π}^{π} |f (t)|^{2} dt ∫_{−π}^{π} |g (t)|^{2} dt)^{\frac{1}{2}} + |∫_{−π}^{π} f (t) \overline{g (t)}dt|]

≤ (∫_{−π}^{π} |f (t)|^{2} dt ∫_{−π}^{π} |g (t)|^{2} dt)^{\frac{1}{2}} ,

where f ∈ L_{2} [−π, π] .

Finally, by employing Theorem 2, we may state:

\frac{1}{π} |∑_{k=1}^{n} [∫_{−π}^{π} f (t) trig (kt) dt]^{2}|

≤ |\frac{1}{π} ∑_{k=1}^{n} [∫_{−π}^{π} f (t) trig (kt) dt]^{2} − \frac{1}{2} ∫_{−π}^{π} f^{2} (t) dt| + \frac{1}{2} |∫_{−π}^{π} f^{2} (t) dt|

≤ \frac{1}{2} [∫_{−π}^{π} |f (t)|^{2} dt + |∫_{−π}^{π} f^{2} (t) dt|] ≤ ∫_{−π}^{π} |f (t)|^{2} dt,

where f ∈ L_{2} [−π, π] .

References

[1] C. BLATTER, Zur Riemannschen Geometrie im Grossen auf dem Möbiusband. (German) Compositio Math. 15 (1961) 88–107.

[2] N.G. de BRUIJN, Problem 12, Wisk. Opgaven, 21 (1960), 12-14.

[3] M.L. BUZANO, Generalizzazione della disiguaglianza di Cauchy-Schwarz (Italian), Rend. Sem. Mat. Univ. e Politech. Torino, 31 (1971/73), 405-409 (1974).

[4] S.S. DRAGOMIR, Refinements of Buzano’s and Kurepa’s inequalities in inner product spaces, Preprint, RGMIA Res. Rep. Coll., 7 (2004).

[5] M. FUJII and F. KUBO, Buzano’s inequality and bounds for roots of algebraic equations, Proc. Amer. Math. Soc., 117(2) (1993), 359-361.

[6] S. KUREPA, On the Buniakowsky-Cauchy-Schwarz inequality, Glasnick Mathematički, 1(21)(2) (1966), 147-158.

[7] J.E. PEČARIĆ, On some classical inequalities in unitary spaces, Mat. Bilten, (Macedonia) 16 (1992), 63-72.

[8] T. PRECUPANU, On a generalisation of Cauchy-Buniakowski-Schwarz inequality, Anal. St. Univ. “Al. I. Cuza” Iaşi, 22(2) (1976), 173-175.

[9] U. RICHARD, Sur des inégalités du type Wirtinger et leurs application aux équationes différentielles ordinaires, Colloquium of Analysis held in Rio de Janeiro, August, 1972, pp. 233-244.

School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, MCMC 8001, VIC, Australia.

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